Classical Capacity of the Free-Space
Quantum-Optical Channel
by
Saikat Guha
Bachelor of Technology in Electrical Engineering,
Indian Institute of Technology Kanpur (2002)
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
January 2004
@ Massachusetts
Institute of Technology 2004. All rights reserved.
.................
A u thor .................
Department of Electrical Engineering and Computer Science
( January 30, 2004
Certified by.....
jH. Shapiro
'effr
Julius A. / trgtonjiofessor of Electrical Engineering
Thesis Supervisor
Accepted by ......
Arthur C. Smith
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
APR 15 2004
LIBRARIES
BARKER
Classical Capacity of the Free-Space Quantum-Optical
Channel
by
Saikat Guha
Submitted to the Department of Electrical Engineering and Computer Science
on January 30, 2004, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
Abstract
Exploring the limits to reliable communication rates over quantum channels has been
the primary focus of many researchers over the past few decades. In the present
work, the classical information carrying capacity of the free-space quantum optical
channel has been studied thoroughly in both the far-field and near-field propagation
regimes. Results have been obtained for the optimal capacity, in which information
rate is maximized over both transmitter encodings and detection schemes at the receiver, for the entanglement-assisted capacity, and also for sub-optimal systems that
employ specific transmitter and receiver structures. For the above cases, several new
broadband results have been obtained for capacity in the presence of both diffraction limited loss and additive fluctuations emanating from a background blackbody
radiation source at thermal equilibrium.
Thesis Supervisor: Jeffrey H. Shapiro
Title: Julius A. Stratton Professor of Electrical Engineering
2
Acknowledgments
First and foremost, I would like to extend my sincerest thanks to my research advisor
Prof. Jeffrey H. Shapiro. This work has been made possible only with the continual
guidance and encouragement I received from him over the past one and a half years.
I am really impressed at the way he efficiently organizes his work, and devotes time
to meet each of his graduate students individually on a regular basis, along with
managing his numerous administrative responsibilities. I particularly adore his style
of 'color-coding' his presentations on both black-board and paper.
I thank my graduate advisor Prof. Dimitri P. Bertsekas, for his suggestions and
discussions on my progress in graduate school, at the beginning of every term.
I am grateful to my office mate Brent Yen for the numerous interesting discussion
sessions we have had on a wide variety of topics ranging from quantum information
to classical physics to number theory. The discussions we had during the 'ups and
downs' of the thermal noise capacity proof were indeed fun.
I would also like to thank Dr. Vittorio Giovannetti and Dr. Lorenzo Maccone,
post-doctoral associates in RLE, for quite a few useful discussions I have had with
them.
My family and friends have always been very supportive towards all my endeavors
so far. I thank my father Prof. Shambhu N. Guha, mother Mrs. Shikha Guha and
my sister Somrita, for their continual encouragement and support over all these years.
I am really grateful to my best friends Arindam and Pooja for having been there for
me whenever I needed them, and for the nice times I relished spending with them
over the last couple of years.
Finally, I thank the Army Research Office for supporting this research through
DoD Multidisciplinary University Research Initiative Grant No. DAAD-19-00-1-0177.
3
Contents
1 Introduction
14
2 Background
18
2.1
Free-Space Propagation Geometry:
The Channel Model . . . . . . . .
3
. . . . .
18
2.1.1
Far Field . . . . . . . . . . . . . . . . . . . .
20
2.1.2
Near Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.2
Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.3
Quantum Detectors for Single Mode Bosonic States . . . . . . . . . .
22
2.3.1
Direct Detection
. . . . . . . . . . . . . . . . . . . . . . . . .
23
2.3.2
Homodyne Detection . . . . . . . . . . . . . . . . . . . . . . .
23
2.3.3
Heterodyne Detection
. . . . . . . . . . . . . . . . . . . . . .
24
2.4
Classical Capacity of a Quantum Channel . . . . . . . . . . . . . . .
24
2.5
Entanglement-Assisted Capacity . . . . . . . . . . . . . . . . . . . . .
26
2.6
Wideband Channel Model . . . . . . . . . . . . . . . . . . . . . . . .
27
Capacity using Number State Inputs and Direct Detection Receivers 30
3.1
Number State Capacity: Lossless Channel . . . . . . . . . . . . . . .
30
3.2
Number State Capacity: Lossy Channel . . . . . . . . . . . . . . . . .
31
3.2.1
SM Capacity
. . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.2.2
Wideband Capacity . . . . . . . . . . . . . . . . . . . . . . . .
34
Constant High Loss and Low Input Power . . . . . . . . . . .
35
Constant Loss with Arbitrary Input Power . . . . . . . . . . .
37
4
4
4.1
4.2
Coherent State Capacity: Lossless Channel . . . . . . . . . . . . . . .
40
4.1.1
SM Capacity: Heterodyne Detection
. . . . . . . . . . . . . .
40
4.1.2
SM Capacity: Homodyne Detection . . . . . . . . . . . . . . .
42
4.1.3
Wideband Capacity: Heterodyne and Homodyne Detection . .
43
4.1.4
Wideband Capacity: Restricted Bandwidth
. . . . . . . . . .
43
Coherent State Capacity: Lossy Channel . . . . . . . . . . . . . . . .
47
4.2.1
SM Capacity: Homodyne and Heterodyne Detection . . . . . .
49
4.2.2
Wideband Capacity . . . . . . . . . . . . . . . . . . . . . . . .
49
Frequency Independent Loss . . . . . . . . . . . . . . . . . . .
50
. . . . . . . . . .
53
Free-Space Channel: Near-Field Propagation . . . . . . . . . .
57
.
61
General Loss Model: A Wideband Analysis . . . . . . . . . . .
64
Free-Space Channel: Far-Field Propagation
Free-Space Channel: Single Spatial Mode - Ultra Wideband
5
39
Capacity using Coherent State Inputs and Structured Receivers
Capacity using Squeezed State Inputs and Homodyne Detection
5.1
5.2
68
Squeezed State Capacity: Lossless Channel . . . . . . . . . . . . . . .
69
5.1.1
SM Capacity
. . . . . . . . . . . . . . . . . . . . . . . . . . .
70
5.1.2
Wideband Capacity . . . . . . . . . . . . . . . . . . . . . . . .
71
Squeezed State Capacity: Lossy Channel . . . . . . . . . . . . . . . .
72
5.2.1
SM Capacity
. . . . . . . . . . . . . . . . . . . . . . . . . . .
74
5.2.2
Wideband Capacity . . . . . . . . . . . . . . . . . . . . . . . .
74
6 The Ultimate Classical Capacity of the Lossy Channel
76
6.1
Ultimate Capacity: SM and Wideband Lossy Channels
. . . . . . . .
77
6.2
Free-Space Channel: Far-Field Propagation . . . . . . . . . . . . . . .
78
6.3
Asymptotic Optimality of Heterodyne Detection . . . . . . . . . . . .
81
6.3.1
Heterodyne Detection Optimality: Free-Space Far-Field Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2
82
A Sufficient Condition for Asymptotic Optimality of Coherent
State Encoding and Heterodyne Detection . . . . . . . . . . .
5
83
7
7.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
7.2
Ultimate Classical Capacity: Recent Advances . . . . . . . . . . . . .
91
. . . . . . . . . . . . . .
94
. . . . . . . .
98
7.2.1
7.3
7.4
8
88
Capacity of a Lossy Channel with Thermal Noise
Number States: Majorization Result
Capacity using known Transmitter-Receiver Structures
7.3.1
Number States and Direct Detection
. . . . . . . . . . . . . .
98
7.3.2
Coherent States and Homodyne or Heterodyne Detection . . .
103
7.3.3
Squeezed States and Homodyne Detection
. . . . . . . . . . .
104
Wideband Results and Discussion . . . . . . . . . . . . . . . . . . . .
105
Conclusions and Scope for Future Work
109
111
A Blahut-Arimoto Algorithm
6
List of Figures
2-1
Free space propagation geometry. AT and AR are the transmitter and
receiver apertures respectively. . . . . . . . . . . . . . . . . . . . . . .
2-2
19
A single-mode lossy bosonic channel can be modelled as a beam-splitter
with transmissivity rj. , is the input mode,
I
is the environment, which
remains unexcited (in vacuum state) in the absence of any extraneous
noise other than pure loss.
2-3
a is the
output of the channel. . . . . . . .
20
Variation of the fractional power transfers (transmissivities) of the first
five successive spatial modes as a function of normalized frequency of
transmission
f'
= 2f/f, [23, 24]. The modes have been arranged in
decreasing order, i.e., rm > r/2 > 173 > ... , and
fc
= cL/ /ATAR is a
frequency normalization. . . . . . . . . . . . . . . . . . . . . . . . . .
3-1
21
Capacity (in bits per use) of the SM lossy number state channel computed for very low average input photon numbers, (a) Numerically
using the 'Blahut-Arimoto Algorithm' for q7 = 0.001 [blue solid line],
and (b) Using approximation (3.5) [black dashed line].
3-2
. . . . . . . .
33
Capacity achieving probability distribution for r/ = 0.2 and Ft = 2.62
photons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
33
3-3
Capacity achieving probability distributions of photon numbers at the
input of the SM lossy bosonic channel for three different values of
transmissivity -
= 0.1, 0.2, and 0.3 (the black, blue and green lines
respectively). All three distributions correspond to the same average
number constraint A = 2.62. As the transmissivity decreases, the peaks
of the distribution spread out. . . . . . . . . . . . . . . . . . . . . . .
3-4
34
Capacity (in bits per use) CNS-DD(h), of the SM lossy number state
channel computed numerically using the 'Blahut-Arimoto Algorithm',
and plotted as a function of h, for r7= 0.2, 0.14, 0.1, and 0.06. . . . .
4-1
35
Optimal allocation of intensity across the transmission bandwidth to
achieve maximum capacity on a lossless channel using coherent-state
inputs and heterodyne detection. (a)
8
f m ,.
> f,, (b)
f m,
< f.
....
46
4-2
This figure illustrates the effect of forced lower and upper cut-off frequencies (fmin and fmax), on the coherent state homodyne and heterothe critical power levels
dyne capacities. Given values of fmin and fm,,
(the input power at which the critical frequency
f, hits
the upper-cut
off frequency fm., and beyond which power 'fills up' only between fmin
and
f m ,)
have been denoted by Phlr
and Pht respectively for the two
cases. In this figure, we plot the normalized capacities C' - C/fma
vs.
a normalized (dimensionless) input power P' = P/hf2.. We
normalize all frequencies by fma. and denote them by
f'
-
f/fm..
The 'dash-dotted' (black) line represents the capacity achievable using
either homodyne or heterodyne with unrestricted use of bandwidth
(Eqn. (4.20)), i.e. C = (1/ In 2) V2P/h bits per second. The 'solid'
(green) line, and the 'dashed' (blue) line represent homodyne and heterodyne capacities respectively for any set of cut-off frequencies satisfying fi
=
0.2. For this value of f'in P'd
=
0.16, and P'h
=
0.32.
Different capacity expressions hold true for power levels below and
above the respective critical powers (Shown by the thick and the thin
lines in the figure respectively).
4-3
. . . . . . . . . . . . . . . . . . . . .
48
Optimal allocation of intensity across the transmission bandwidth to
achieve maximum capacity on a channel with frequency independent
transmissivity TI, using coherent-state inputs and heterodyne detection.
(a) f m ax
4-4
>
f , (b) fmax < fc. . . . . . . . .. ... . . . . . . .
. . . .
51
Optimal allocation of intensity across the transmission bandwidth to
achieve maximum capacity on a channel with frequency dependent
transmissivity
q
OC
f2, using coherent-state inputs and heterodyne de-
tection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
54
4-5
Dependence of C' on P' for coherent-state inputs and heterodyne detection, for different values of
the red (dotted) line is for
for
fj
fi.
fAni
The blue (solid) line is for
fi
0,
=
= 0.1 and the green (dashed) line is
= 0.2. Observe that all the three curves coincide in the low
power regime, because when P' < PC'rit C' does not depend on
fnin.
When P' > Pcrit, performance of the channel deteriorates because of
restricting ourselves to a non-zero
4-6
f'..
. . . . . . . . . . . . . . . . .
Dependence of C' on P'for coherent-state inputs with heterodyne and
homodyne detection, for f'
= 0. The blue (solid) line is for hetero-
dyne detection, and the red (dashed) line is for homodyne detection. .
4-7
56
57
This figure illustrates the coherent state homodyne and heterodyne
capacities for the free-space optical channel in the near-field regime. All
the frequencies in the transmission range
f
E [fmin, fma] satisfy Df >
1. In this curve, a normalized capacity C' = C/Af,
has been plotted
against a dimensionless normalized input power P' = P/(Ahfm),
where A = ATAR/C 2 L 2 is a geometrical parameter of the channel (Fig.
the critical power levels (the input
2-1). Given values of fmin and fm,,
power at which the critical frequency
fmax,
f, hits the upper-cut
off frequency
and beyond which power 'fills up' only between fmin and
have been denoted by Phr
and Ph,
f m a)
respectively for the two cases.
The 'solid' (black and blue) line, and the 'dashed' (red and green) line
represent homodyne and heterodyne capacities respectively for any set
of cut-off frequencies satisfying
fj,1,
= 0.3. For this value of
fmi,
the
values of the normalized critical input power are: p'crit = 0.019, and
p'it = 0.076 respectively. Different capacity expressions hold true for
power levels below and above the respective critical powers (Shown by
the thick and the thin lines in the figure respectively). . . . . . . . . .
4-8
62
An example of ultra wideband capacities for coherent state encoding with homodyne or heterodyne detection, for the free-space optical
channel under the assumption of a single spatial mode transmission. .
10
65
4-9
The optimal power allocation for the wideband coherent state channel with heterodyne detection depicted as 'water-filling', for frequency
dependent transmissivity shown for ql in Fig. 2-3. . . . . . . . . . . .
5-1
67
This figure shows comparative performance of squeezed state encoding
and homodyne detection with respect to that of coherent state encoding with homodyne and heterodyne detection. The 'solid' lines (red,
green and grey) represent coherent state heterodyne, coherent state
homodyne, and squeezed state homodyne capacities respectively for
an arbitrary set of cut-off frequencies satisfying f' i=
0.2. Different
capacity expressions hold true for power levels below and above the
respective critical powers (Shown by the thick and the thin lines in the
figure respectively). For the above value of
= 0.32, p'
malized cut-off power are: P'
f' i,
the values of the nor-
= 0.08, and p'crit = 0.16.
The thick dashed (blue) line is the best achievable performance of the
coherent state channel (with either heterodyne or homodyne detection), and is given by C' = (1/ In2)
2P'. The thick dotted (red) line
is the best achievable performance of the squeezed state channel, and
is given by C' = (1/In 2)
5-2
1. . . . . . . . . . . . . . . . . . . . . . .
73
An example of ultra wideband capacities for squeezed state channel and
coherent state encoding with homodyne or heterodyne detection, for
the free-space optical channel under the assumption of a single spatial
mode transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-1
Single Mode Capacities (in bits per use of the channel) for various
cases. The transmissivity of the SM channel q = 0.16. . . . . . . . . .
6-2
75
78
Dependence of C' on P' for the ultimate classical capacity compared to
C' for coherent-state inputs with heterodyne and homodyne detection.
All curves assume f'is = 0.
The blue (solid) line is the ultimate
classical capacity, the red (dotted) line is for heterodyne detection,
and the green (dashed) line is for homodyne detection.
11
. . . . . . . .
81
7-1
The thermal noise channel can be decomposed into a lossy bosonic
channel without thermal noise (2.1), followed by a classical Gaussian
noise channel
7-2
A plot of E'
T
4
(1 ,)F (7.4).
o Pn,k
. . . . . . . . . . . . . . . . . . . . . . .
for a SM thermal noise channel with transmissivity
= 0.8, and N = 5 photons, as a function of m; for n = 0, 6, 12, 18,.
The dark (black) lines correspond to the values n = 0, 30, 60, 90.
7-3
90
. . .
96
A plot of output entropies of the SM thermal noise channel as a function
of 71 for number states inputs In)(n1, corresponding to n = 0, 1, 2, 3 and
4. The value of N = 5. We see that output entropy is minimum for
the vacuum state 10) (0 , which is shown in the figure by the dark black
line. The results for n > 0 were obtained by numerical evaluation of
the pa,
from (7.18); the n = 0 entropy result so obtained matches the
analytic result g((1 - ,)N ). . . . . . . . . . . . . . . . . . . . . . . .
7-4
97
A plot of optimum apriori probability distributions (in dB) for a pure
SM lossy channel (Tj = 0.8, h = 3.7), and a SM thermal noise channel
(T
= 0.8, h = 3.7, N = 5). In the pure lossy case, there are no peaks
at this value of transmissivity and the distribution is close to linear,
whereas for the thermal noise lossy channel, the distribution still has
multiple peaks. For the optimum distribution of the thermal noise
channel to converge sufficiently close to the Bose-Einstein distribution,
y has to be even closer to unity. . . . . . . . . . . . . . . . . . . . . .
7-5
102
A sample calculation of the number-state direct-detection capacity (using the Blahut-Arimoto algorithm) of a SM lossy thermal noise channel
with transmissivity 7 = 0.8, and mean thermal photon number N = 5. 102
7-6
A sample simulation result of all the different capacities of a SM lossy
thermal noise channel we considered so far in the thesis. The channel
SN
has transmissivity y = 0.8, and mean thermal photon number N = 5.106
12
7-7
Ultra-wideband capacity of a lossy thermal noise channel
-
optical communication, single spatial mode propagation
-
cm
2
and AR
=
Free-space
AT = 400
1 cm 2 , separated by L = 400 km of free-space; with
frequencies of transmission ranging from that associated with a wavelength of 30 microns (which is deep into the far-field regime), to that
associated with a wavelength of 300 nanometers (which is well within
the near field regime). Mean thermal photon number N = 5 is assumed
to remain constant across the entire range of frequencies. . . . . . . .
7-8
Capacities of a wideband lossy thermal noise channel
108
Free-space
-
all param-
optical communication, single spatial mode propagation
eters same as in Fig. 7-7, except that the mean thermal photon number
N is now assumed to vary with frequency of transmission
according
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
108
Blahut-Arimoto Algorithm . . . . . . . . . . . . . . . . . . . . . . . .
113
to E qn. (7.2)
A-1
f
13
Chapter 1
Introduction
The objective of any communication system is to transfer information from one point
to another in the most efficient manner, given the constraints of the available physical resources. In most communication systems, the transfer of information is done by
superimposing the information onto an electromagnetic (EM) wave. The EM wave is
known as the carrierand the process of superimposing information onto the carrier
wave is known as modulation. The modulated carrier is then transmitted to the destination through a noisy medium, called the communication channel. At the receiver,
the noisy wave is received and demodulated to retrieve the information as accurately
as possible. Such systems are often characterized by the location of the frequency of
the carrier wave in the electromagnetic spectrum. In radio systems for example, the
carrier wave is selected from the radio frequency (RF) portion of the spectrum.
In an optical communication system, the carrier wave is selected from the optical range of frequencies, which includes the infrared, visible light, and ultraviolet
frequencies. The main advantage of communicating with optical frequencies is the
potential increase in information that can be transmitted because of the possibility of harnessing an immense amount of bandwidth.
The amount of information
transmitted in any communication system depends directly on the bandwidth of the
modulated carrier, which is usually a fraction of the center frequency of the carrier
wave itself. Thus increasing the carrier frequency increases the available transmission
bandwidth. For example, the frequencies in the optical range would typically have
14
a usable transmission bandwidth about three to four orders of magnitude greater
than that of a carrier wave in the RF region. Another important advantage of optical communications relative to RF systems comes from their narrower transmitted
beams
[tRad beam divergences are possible with optical systems. These narrower
beamwidths deliver power more efficiently to the receiver aperture. Narrow beams
also enhance communication security by making it hard for an eavesdropper to intercept an appreciable amount of the transmitted power. Communicating with optical
frequencies has some challenges associated with it as well.
As optical frequencies
are accompanied by extremely small wavelengths, the design of optical components
require completely different design techniques than conventional microwave or RF
communication systems. Also, the advantage of optical communication derives from
its comparatively narrow beam, which introduces the difficulty of high-precision beam
pointing. RF beams require much less pointing precision. Progress in the theoretical study of optical communication, the advent of the laser - a high-power optical
carrier source
the developments in the field of optical fiber-based communication,
the development of novel wideband optical modulators and efficient detectors, have
made optical communication emerge as a field of immense technological interest [1].
The subject of information theory addresses ultimate limits on communication. It
tells us how to compute the maximum rate at which reliable data communication can
be achieved over a particular communication channel by appropriately encoding and
decoding the data. This rate is known as the channel capacity [20, 2, 31. It also tells
us how to compute the maximum extent a given set of data can be compressed so that
the original data can be recovered within a specific amount of distortion. Though
information theory doesn't give us the exact algorithm (or the optimal code) that
would achieve capacity on a given channel, and it doesn't tell us how to optimally
compress a given set of data, nevertheless it sets ultimate limits on communication
which are essential to meaningfully determine how well a real system is actually
performing. The field of information theory was launched by Shannon's revolutionary
1948 paper [20], in which he laid down its complete foundation.
Because light is quantum mechanical in nature, quantum information theory is
15
needed to assess the ultimate limits on optical communication. Much work has already been done on quantum information theory [13, 4], which sets ultimate limits
on the rates of reliable communication of classical information and quantum information over quantum mechanical communication channels. As in classical information
theory, quantum information theory does not tell us the transmitter and receiver
structures that would achieve the best communication rate for a given kind of quantum mechanical noise. Nevertheless the limits set by quantum information theory
are extremely useful in determining how well available technology can perform with
respect to the theoretical limit.
In this thesis, we will primarily focus on the lossy free-space quantum optical channel. Some examples of such a channel are satellite-to-satellite communication, wireless
optical communication through air with a clear line-of-sight between transmitter and
receiver units, and deep space communication. Long distance communication through
a lossy fiber or a waveguide can also be treated along similar lines. Quantum information theory work to date has not treated the lossy channel case sufficiently well. This
thesis will establish ultimate limits achievable on reliable communication of classical
information over such a channel. In addition, it will study the best possible performance of standard transmitter and receiver structures in relation to the ultimate
achievable capacity. It will also consider the effects of background thermal noise on
the ultimate capacity and other achievable capacities of the channel.
The organization of this thesis is as follows. In chapter 2, we elaborate the channel model in detail, establish our notation for the rest of the work, present a short
review on the notion of thermal noise in single mode bosonic communication, discuss
some standard quantum detection schemes, and review the notions of unassisted and
entanglement-assisted classical capacity of a quantum channel. The capacity of the
far-field lossy channel using photon number state inputs and unit-efficiency direct
detection receivers is dealt with in chapter 3. Chapter 4 evaluates the capacity of the
lossy channel using coherent state inputs with homodyne and heterodyne receivers.
The calculations in this chapter have been done for a simple frequency-independent
loss model and for a more realistic free-space optical channel model in which the
16
transmissivity of the channel depends on the frequency of transmission, in both the
near-field and the far-field regimes. We look into the achievable rates using optimized
quadrature-squeezed state inputs in Chapter 5. We establish the ultimate classical
capacity of a lossy far-field channel in chapter 6, and work out the single mode and
wideband capacities in the presence of thermal noise, in Chapter 7. The thesis concludes in chapter 8 with a note on ongoing and future directions of research in the
field.
17
Chapter 2
Background
This chapter is intended to provide all the necessary theoretical background and
framework, with ample references to previous work in the area, for the material to
be covered in the thesis. We discuss the model of the free-space quantum optical
channel both in the far-field and the near-field propagation regimes. We describe the
well known structured receivers for single mode (SM) bosonic states, and lay down
their definitions mathematically. We also present a brief discussion on the notion of
capacity for transmitting classical information over a quantum channel, followed by a
discussion on entanglement assisted classical capacity. The chapter ends with a short
note on the formulation of the power constrained wideband channel model that we
use for all our subsequent capacity calculations.
2.1
Free-Space Propagation Geometry:
The Channel Model
Free-space optical communication has attracted considerable attention recently for
a variety of applications [25, 26]. In the present work, we shall primarily focus on
maximizing the rate at which classical information can be sent reliably over a freespace optical communication link
under an average input power constraint
by
modulating various parameters characterizing the quantum states of the optical field
18
AT
AR
z-0
Free Space
z=L
Figure 2-1: Free space propagation geometry. AT and AR are the transmitter and
receiver apertures respectively.
modes at the transmitter, and by using different detection schemes at the receiver.
Consider a line-of-sight free space optical communication channel with circular
shaped transmitter and receiver apertures denoted AT and AR respectively, as shown
in Fig. 2-1. A quantized source of radiation at the transmitter produces a field E(r, t)
which is space limited to the transmitter exit aperture AT
{ (x, y)
:x 2
+y
2
d0 2 /4
and time limited to a finite signalling interval To = {t : to - T < t < to}. After
propagation through L m of free space, the field is collected by a receiver whose
entrance aperture is AR
{(x', y') :
+ y' 2 < dL2 /4}. The receiver is assumed to
time limit the field in its entrance aperture to an interval TL
t < to + L/c}.
{t : to - T + L/c <
The free space Fresnel number for transmission at frequency f is
defined as Df = (ATAR/c 2 L 2 )f 2 , where with some abuse of notation, we are now
using AT = irdo2 /4 and AR
2
= 7rdL /4
to denote the aperture areas at the transmitter
and the receiver respectively. For narrowband transmission at a center frequency
f, depending
upon the magnitude of D1 , the propagation is broadly classified to be
either in the 'far-field' (Df < 1), or the 'near-field' (Df > 1) regimes.
19
Environment
c=J0a+ 1-q~b
&
Output
Input
Figure 2-2: A single-mode lossy bosonic channel can be modelled as a beam-splitter
with transmissivity 7. e is the input mode, b is the environment, which remains
unexcited (in vacuum state) in the absence of any extraneous noise other than pure
loss. is the output of the channel.
2.1.1
Far Field
In the far-field regime, only one spatial mode of the input bosonic field couples appreciable amount of power to the receiver aperture [5], and all other spatial modes
are thus insignificant. Further, it has been shown that [5], in the Heisenberg picture,
quantum description of the propagation of a single spatial mode of the field through
this channel in the far-field regime is obtained by coupling the input field mode & into
a beam splitter of transmissivity 7 = Df, with an 'environment' mode b (Fig. 2-2),
which is in the vacuum state (2.1); i.e.
=
d&+
/1 -
7 b,
(2.1)
where
(ATAR
f2
c2 L2 )
and d,
22
, and b are the modal annihilation operators of the input, output and the noise
modes respectively. Several authors like to call the above model a single-mode lossy
bosonic channel, where 1 - q is identified as the 'loss'. Physically, q is the fraction
of the input power that gets coupled to the receiver aperture.
quadratic dependence on the carrier frequency
20
f
in the far-field.
Note that, q has a
.
1.000
0.00
0
0
-
0.600-
IU
0.400
0.200
-
eta_1
eta_2
eta_3
eta_4
eta_5
-
-
0.000
0.000
2.000
4.000
6.000
8.000
10.000
12.000
normalized frequency (f-prime)
Figure 2-3: Variation of the fractional power transfers (transmissivities) of the first
five successive spatial modes as a function of normalized frequency of transmission
f = 2f/fc [23, 24]. The modes have been arranged in decreasing order, i.e., rq1 >
772 > 773 > ... , and f, = cL/vA ATAR is a frequency normalization.
2.1.2
Near Field
As the value of Df increases, more and more spatial modes of the field start coupling
appreciable power into the receiver. If we arrange the fractional power transfers of
the spatial modes in decreasing order and denote the fractional power transfer of the
ith mode by 7ij, then -
from the eigenvalue behavior of the classical free-space modal
decomposition theory [23, 24] -
the qj's depend upon the carrier frequency
f
as
shown in Fig. 2-3.
When Df
> 1, near-field propagation prevails. In this regime, there are Df spatial
modes which have
e7
~ 1, and all the other modes are insignificant [5].
21
2.2
Thermal Noise
In the absence of any extraneous noise sources other than pure loss, the environment
mode b (2.1) is in the vacuum state, which is the minimum 'noise' required by quantum
mechanics to preserve correct commutator brackets of annihilation operators of the
output modes. Different types of noise can be described by different initial states of
the environment mode 6. We define a thermal noise lossy bosonic channel to be a
channel that describes the effect of coupling the lossy channel to a thermal reservoir
at temperature T [7]. The thermal environment mode 6 is excited in the state given
by the density matrix
PT
=
-
exp
A
2
a,
2a)(d
(2.3)
where 1a) are the coherent state vectors and S is the mean photon number per mode
of the thermal state PT, given by the Planck's formula for the mean number of photons
per mode of a blackbody radiation source at thermal equilibrium at temperature T:
=
N 1r~~uu)
=
hf/JkBT
-1
24
where h and kB are the Planck's constant and Boltzmann's constant respectively.
2.3
Quantum Detectors for Single Mode Bosonic
States
An arbitrary quantum measurement on a single EM field mode can be described in
terms of a set of Hermitian operators {lb}, which satisfy
(positivity),
(2.5)
(completeness).
(2.6)
b ;> 0
nib
=
f
b
22
These operators thus comprise a Positive Operator Value Measure (POVM). When
field mode is in the state given by the density operator
/a,
the probability of an
outcome b is given by
(2.7)
Pr(bla) = Tr(I afb).
Some of the basic quantum measurement schemes for single mode (SM) bosonic
states that we will consider in this thesis are (a) Direct Detection, (b) Homodyne
detection, and (c) Heterodyne Detection. Their POVM descriptions are given below
[6].
2.3.1
Direct Detection
Direct detection (or photon counting) refers to a widely used method of optical
demodulation that responds to a short time averaged power of the optical field.
Quantum mechanically, it refers to a measurement done in the number-state basis
{in),n
= 0,1, 2,...}. The POVM corresponding to Direct Detection is given by
H
2.3.2
for n=0,1,2,...
= In)(ni,
(2.8)
Homodyne Detection
Homodyne Detection refers to a quantum detection scheme in which the incoming
field is mixed with a strong local oscillator (LO) field through a 50-50 beam-splitter
(BS). The LO field is spatially and temporally coherent with the incoming field. The
outputs of the BS are detected by two ideal photon counters.
The output of the
homodyne detector is the scaled difference of the output of the two photon counters.
Homodyne detection measures one quadrature of the field in a given direction in
phase-space, depending upon the phase of the LO-field. Homodyne detection can be
described by the POVM
nm
=
Iai)(oCiI,
23
for ai E R,
(2.9)
where 11,
are projection operators onto the 6-function normalized eigenstates lai)
of the quadrature component ai = Re(a), that is in phase with the LO field whose
phase, for convenience, has been set to zero.
2.3.3
Heterodyne Detection
Heterodyne detection refers to a simultaneous measurement of both the quadratures
of the optical field. It mixes the received signal with a strong LO, whose center
frequency is offset by a radio frequency
--
known as the intermediate frequency (IF)
from that of the signal, on a 50-50 BS. Quadrature detection on the IF difference
signal from photodetectors placed at the BS output ports then yields the desired
measurements. Heterodyne detection can be described by the POVM
1
EH = -c)(aL,
7r
(2.10)
for aE C,
where 1a) is a coherent state.
2.4
Classical Capacity of a Quantum Channel
Let us consider an information source that emits discrete independent, identicallydistributed (i.i.d.) classical symbols, indexed by i = 1,2,..., A.
The source emits
one letter at a time with a probability distribution PN, where PN(i) is the apriori
probability that the symbol i is emitted. Our goal is to transmit the output of this
source in the best possible manner over our quantum channel (2.1) using appropriate
quantum states for encoding and an optimum detection scheme at the receiver. In
particular, we wish to compute the capacity for transmitting classical information in
bits, per use of a lossy quantum channel.
Let us assume that we excite the field mode & (in one signalling interval), in the
quantum state ,i E 'R to encode the source-symbol i. Let us denote the state at the
output of the channel by <P(i) where the channel <P is a completely positive (CP) trace
preserving map. A block code of length n would consist of codewords that in general
24
could be an entangled state 3 E
At the output, the most general quantum
Ro(n
mechanical detection scheme can be described in terms of a POVM, which we shall
call a 'decision rule' and denote by X = {X 3 }. Such a generalized measurement can be
thought of as a measurement of a quantity X whose possible results of measurement
j
= 1, 2, ... , M, which can be treated as an index for a set of classical
output symbols.
The conditional probabilities that a particular output symbol is
are labelled by
received given that a particular input symbol was transmitted through the channel,
i.e. the classical transition probabilities [9, 11, 12] are given by
(2.11)
P(ji) = Tr(I(pA)X)
The classical mutual information and the quantum Holevo information for single
use of this channel are respectively given by [13]
pN()PJ
I(pN, (, X) = E
AH(PN
where S()
= -Tr(
log
(ZPN(i)i
=S
P(k)
pN(i)S(i
-E
(2.12)
(2.13)
log ,) is the Von Neumann entropy of the state 3. We define [13]
C (1) = sup sup I(pN, 4*", X)
{PNI
(2.14)
X
C,(@)= sup AH(pN, I
n),
(2.15)
{PN(i)}
where (DO' = (D 0 4D 0 ... 0 1 represents multiple uses of the channel, and PN is a
probability distribution over all states in H71. It can be shown that both of the above
quantities are superadditive and that their averages with respect to n, as n -+ oc exist
[13]. From Shannon theory we know that, at any rate less than C(I), using product
state inputs over many uses of the channel to encode the source letters, one can
achieve arbitrarily low error probability by using suitable classical coding techniques
25
[20]. We define CcQ to be the classical information capacity of a quantum channel 1
when the transmitter can encode messages using product state inputs and the receiver
can decode using measurements entangled over arbitrarily many uses of the channel.
The Holevo-Schumacher-Westmoreland (HSW) Theorem states that [13, 14].
(2.16)
CCQ = Ci(D).
The ultimate capacity of a quantum channel 1 to communicate classical information is the supremum of the CCQ of n uses of ID (i.e. (pon), divided by n. We call
this capacity the 'Classical Capacity' of a quantum channel, and denote it by CQQ
or simply by C. The subscript 'QQ' signifies the fact that input-states and decoding
can both be entangled over a large number of channel uses. So, the classical capacity
of a quantum channel is
C = CQQ = sup
n
2.5
n
(2.17)
Entanglement-Assisted Capacity
A majority of the striking advantages that quantum information theory enjoys over
its classical counterpart stem from the wonders that quantum entanglement [4] can
do to communication protocols. Teleportation and Superdense Coding [4] are two
such examples.
Surprisingly, it turns out that the classical information carrying
capacity of a quantum channel can be increased way beyond its Classical Capacity
(2.17), by using prior shared entanglement as a resource. The entanglement-assisted
classical capacity CE is defined to be the maximum asymptotic rate of reliable bit
transmission with the aid of unlimited amount of prior entanglement shared between
the transmitter and the receiver. The importance of CE lies in the fact that it gives
an upper bound to all the relevant capacities of the quantum channel, including the
quantum and classical (unassisted) capacities [21].
In can be shown [21] that the natural generalization of the concept of mutual
information between input and output of a channel, to the quantum case, is given
26
by the Quantum Mutual Information I(, ,) of a quantum channel (D, where 3 is the
input state, given by
I(,y)
=
S() + S(I'()) - S((W 0 T)PP)
(2.18)
where P is a purificationi of the input density matrix ,. It can be shown that the
entanglement-assisted capacity CE is given by
CE=
max I(1,D)
(2.19)
where the maximum is taken over all possible inputs / to the channel [21].
2.6
Wideband Channel Model
The study of quantum limits to wideband bosonic communication rates has been one
of the primary applications of quantum communication theory [15, 8]. The wideband
capacity of a power-constrained lossless bosonic channel has been shown to be proportional to the square root of the total input power P [8]. Until very recently [18], the
exact solution to the wideband classical capacity of the lossy bosonic channel was not
known. In this thesis, we will demonstrate the calculation of the wideband classical
capacities of the free-space channel (with and without thermal noise) for a variety
of cases, viz.
ultimate classical capacity, entanglement-assisted capacity, capacity
achieved by using a coherent state encoding with heterodyne or homodyne detection,
and capacity achieved by using an encoding based on an optimized set of quadrature
squeezed states and homodyne detection. For all these cases, the procedure we use
to compute the input-power-constrained wideband capacities from the respective SM
(bits per use) capacities is essentially the same. The procedure is illustrated below.
Let us consider an input-power-constrained frequency multiplexed channel, in
'Purification of a mixed state is a purely mathematical process [4]: one can show that given any
system A in a mixed state k^, it is always possible to introduce an additional reference system R
and a pure state in the combined state space JAR), such that this pure state reduces to yA when
one looks at system A alone, i.e. pA - Tr(IAR)(ARI).
27
which each frequency bin (of width b Hz) is in general a SM thermal noise channel characterized by a transmissivity qi and an input-output relation similar to (2.1).
In the following discussion, we shall consider communication using a single spatial
mode of the bosonic field at each frequency
f
E [fmin,
fmax],
where fmin and fm,
for
a real bosonic channel would primarily depend upon practical limitations on transmitter and receiver design, and on the optimum power allocation corresponding to
the given set of transmitter and receiver structures. We denote the center frequency
of the ith frequency bin by fi and use bfih for the average photon transmission rate in
the ith bin. The average input power constraint is then given by
P= b
(2.20)
hfiii.
In far field free-space propagation ,the transmissivity of the ith bin is ij
=
2
where A - ATAR/C 2 L 2 (2.2). Let us denote by Ri the average number of thermal
photons in the ith bin of the environment; Ni is given by (2.4) with
f
=
fi.
If we
denote the SM capacity (in bits per use) of the ith frequency bin by C(7i, fii, Si), then
the wideband capacity CWB can be obtained as a function of the average input power
constraint P, using a constrained maxization:
CwB(P) = MaxZC (qi(fi), fi,
(M)
(2.21)
subject to
b
hfig ; P.
(2.22)
The above maximization can be done using a Lagrange multiplier technique either analytically or numerically, depending upon the complexity of the SM capacity
formula C (rh(fi), fii, Si(fi)) for the particular kind of capacity in question.
Note that this formulation assumes that different frequency bins are employed
as parallel channels.
Because of superadditivity, it must be shown that optimum
capacity, for our channel, does not require entangling different frequency bins. Of
28
course, when seeking the capacity of specific transmitter-structure/receiver-structure
combinations, we are free to assume that these structures do not entangle different
frequency bins. In the subsequent chapters, we first work through the capacity calculations of all the different cases without thermal noise. We later summarize all the
thermal noise capacity results together in one chapter.
29
Chapter 3
Capacity using Number State
Inputs and Direct Detection
Receivers
Number states are eigenstates of the photon number operator N = &f&, where d is the
annihilation operator associated with a single mode of the bosonic field. The number
state In) is the eigenstate associated with the eigenvalue of n photons. Clearly, as
is a Hermitian operator, it is an observable, and its eigenstates
{In); n
=
N
1, 2, ..., oo}
form a basis of the entire state space N. A direct detection receiver (or an ideal photon
counter) is a device that performs a measurement of N with unit efficiency, on an
incident single mode (SM) field. If the state incident on such a receiver is the number
state In), then the result of measurement is n with certainty.
3.1
Number State Capacity: Lossless Channel
Consider a zero-temperature SM lossless (rj = 1) bosonic channel (2.1), with a maximum average number of photons h, at the input. It is well known that classical
capacity (2.17) can be achieved for this channel using photon number states and a
unit-efficiency direct detection receiver [15]. The probability distribution of number
states that achieves capacity is the Bose-Einstein distribution, given by
30
PN(n) =
1+h
1+h
(3.1)
for n = 0,1,2,...
_ ,n
where PN(n) is the probability of transmitting the state In)(nI in one use of the
channel. The capacity (in bits per use) of this SM number state channel is given by
[15]
C(") = g(h) =
"-1)
log 2 (1+
(3.2)
).
+ log 2 (1 +
We also know that the ultimate classical capacity (in bits per second) of the
wideband lossless channel with an average input-power constraint P (in Watts) is
given by [15]
(3.3)
V
Cultimate(P)
which can be achieved by number state encoding and direct detection [15].
In the
rest of this chapter, we shall investigate the impact of loss on the capacities of SM
and wideband channels with number state transmission and direct detection.
3.2
Number State Capacity: Lossy Channel
Let us transmit a pure number state PIN
=
In)(nj with probability pN(n) through
a zero-temperature SM lossy channel (2.1) with transmissivity 'q. It can be shown
[4], that the output is a mixed state given by POUT
= E'=o (nk(i
-
)n~kIk)(kI,
when we transmit In). If we detect this state using a unit-efficiency direct detection
receiver, the probability of getting m counts is (n)rp,(I
-
'
for m = 0, 1, 2,..., n.
So, the classical transition probabilities (2.11) for a single use of the channel can be
written as
for m > n
0
(2)r"(1
31
-
)
for m ;m.
3.2.1
SM Capacity
To get the SM capacity (in bits per use), we have to maximize the classical mutual
information I(M; N) (2.12) over all input probability distributions pN(r) subject to a
fixed value h of average photon-number at the input. This constrained maximization
problem can be solved numerically using an iterative algorithm, known as the Blahut-
Arimoto (BA) Algorithm [16] (See Appendix A for the details of the algorithm). For
the special case Tpi << 1 (low-power, high-loss), some approximate analytical results
can be obtained.
In this regime, the SM capacity (in bits per use) of the number
state channel is given by
C ~ H(7pi) where H(x)
H(7),
(3.5)
X log 2 (x) - (1 - X) log 2 (1 - X) is the binary entropy function. The
results of a Blahut-Arimoto simulation for SM capacity in this regime has been compared with the above approximation in Fig. 3-1.
Interestingly, on carrying out the BA algorithm [16] for the general case, we find
that the capacity achieving probability distributions have multiple peaks (See Fig.
3-2). This suggests that to achieve the best capacity using number state inputs, one
would almost preferentially use a set of optimum numbers of photons in each channel
use. A sample plot of the optimal probability distribution for 7 = 0.2 and i! = 2.62
is given in Fig. 3-2.
A qualitative explanation of the appearance of these peaks in the optimum distribution is as follows. When photons are sent through the lossy channel in bunches
of known magnitudes, the effect of the loss is to spread each bunch by some amount
about its mean value. If the initial bunches are spaced out well enough, the photon counting receiver is able to resolve them with little or no ambiguity. Thus, we
should expect that the peaks of the probability distributions would space out more
and more as we increase the loss (i.e., decrease the transmissivity 71), as lesser loss
would imply lesser 'spread' of the peaks, and hence for the same level of 'ambiguity'
at the receiver, we could afford to have the peaks closer together. Though the above
32
5.000e-4
4.000e-4
WI
3.2OOe-4
-
1.000e-4
-
Approximation - Blahut Arimoto -0-
1000
U.
f
I-
0.000
0.020
0.080
0.060
0.040
I
I
I
I
I
0.100
Ii
I
0.120
i
1
0.140
Average Photon Number (nbar)
Figure 3-1: Capacity (in bits per use) of the SM lossy number state channel computed for very low average input photon numbers, (a) Numerically using the 'BlahutArimoto Algorithm' for r = 0.001 [blue solid line], and (b) Using approximation (3.5)
[black dashed line].
11111111
0
-10 -20 -30 -40 E
-50 0
-60 n-2
-70 -80 -90
0
I
20
A.
40
1
60
.
100
80
Photon number (n)
120
140
160
Figure 3-2: Capacity achieving probability distribution for r7 = 0.2 and A = 2.62
photons.
33
0.000
eta = 0.1
-10.000
_
-20.000
-
-30.000
-
-40.000
-
Ct
I/
-
~
~I
*\
.~
-50.000
If ~{ \\
V
/ X
*\
-60.000
0.000
-
eta = 0.2-eta =0.3
20.000
40.000
60.000
A
80.000
100.000
120.000
number of photons (n)
Figure 3-3: Capacity achieving probability distributions of photon numbers at the
input of the SM lossy bosonic channel for three different values of transmissivity
--- 77 = 0.1,0.2, and 0.3 (the black, blue and green lines respectively). All three
distributions correspond to the same average number constraint ft = 2.62. As the
transmissivity decreases, the peaks of the distribution spread out.
explanation is not completely rigorous, this intuitive idea has been illustrated by actual BA simulation results for the optimum distributions for 27
(Fig. 3-3).
=
0.1, 0.2, and 0.3.
As one increases the transmissivity to unity, the multiple peaks in the
optimum probability distribution gradually merge, and the distribution converges to
the Bose-Einstein distribution (3.1).
Finally, a plot of the capacity (in bits per use) CNS-DD(!) as a function of h for
various values of q is given in Fig. 3-4. As one would expect, for the same value of
ii, the capacity increases with increasing transmissivity 71.
3.2.2
Wideband Capacity
The wideband capacity of the number state channel can be obtained numerically
by maximizing the sum of the SM capacities (obtained using the BA algorithm)
34
2-
0.50
eta =
e
eta =
0.2
0.14
eta =
eta =
I
I
0
20
10
30
I
0.1
0.06
I
40
Average photon number
Figure 3-4: Capacity (in bits per use) CNS-DD(5i), of the SM lossy number state
channel computed numerically using the 'Blahut-Arimoto Algorithm', and plotted as
a function of h, for q = 0.2, 0.14, 0.1, and 0.06.
across the entire frequency range of transmission, subject to the average input power
constraint P (Watts), using the Lagrange Multiplier technique described in Section
2.6. We will consider two cases
(a) Constant high loss (frequency independent) and
low input power at all frequencies; and (b) Frequency independent loss with arbitrary
input power. The numerical analysis of the frequency dependent wideband far-field
free-space channel turns out to be too complicated for the number state inputs.
Constant High Loss and Low Input Power
Let us assume that the total input power P is so small that the mean photon numbers
fi
<
1, Vi, where i is a discrete index corresponding to a narrow frequency bin of
width b Hz centered at
fi
(see Section 2.6). Assume that the transmissivity 7 < 1
and is same for all frequencies. Carrying out the constrained maximization (2.21)
for this case, and incorporating appropriate approximations, we obtain the optimum
distribution of photons across frequencies as
35
[i+ AhI
W = exp
f1 ,
(3.6)
where A is the Lagrange multiplier.
Substituting this result into the the power constraint equation (2.22), and approximating the sums by integrals by assuming a vanishingly small bin width, we
obtain
P
=
fi exp (-[ + Ohfi])
bh
(3.8)
Xe
S
(3.7)
Ihf-'(3.9)
eh021
where 3 = Aln 2/7, and allowing the integration to be over all frequencies is permissible because the contribution of the optimum average photon number becomes
vanishingly small as f -- oc.
Substituting this relation into the capacity equation (3.5), and approximating the
sums by integrals by assuming a vanishingly small bin width, we obtain
C
=
(3.10)
[H(pfi) - fiH(77)]
b
Sb
[-
- (1-iij)
ln (nit)
In (1- 71i)
71ln 77+ i(1 - 7) ln(1 - 7)]
+
[-ri 1n1 - qhi ln hi + (1 - 7nh)
In
3
3hln 2 j 0
i In i
nh + ifn ln 77 - r77i (1 - I)]
(neglecting terms of second order in 71)
217
(1 + x)e-dx = flhln
2~
36
(3.11)
(3.12)
Eliminating the Lagrange multiplier coefficient f from the power equation and the
capacity equation, we obtain the wideband capacity CWB in bits/sec as a function of
input power P:
CWB-
2T2
(3.13)
P
Note that in the limit of low power and high loss, the wideband capacity of a
lossy channel with frequency independent transmissivity, achieved by number state
encoding and direct detection, is proportional to the transmissivity 77.
Constant Loss with Arbitrary Input Power
Now, let us consider the general case of arbitrary transmissivity 77, without any restriction on the input power constraint P. The analysis developed in this section
can be used to compute the wideband capacity with frequency independent loss for
any transmitter-detector combination for which the SM capacity (in bits per use) is
known.
Let us set up the Lagrangian J(ni, A) as follows:
J(hi, A) = b
C(, hi) + A(P - b
hfih),
where C(TI, hi) is the SM capacity in bits per use. Setting &J(nii,
(3.14)
A)/Ohi = 0, we
obtain
____
-
Oni
(3.15)
Ahfi.
The above equation can be inverted to express hi as a function of Afi. Making
the substitution z = Af, and converting the sums to integrals by going to the limit
of vanishingly small bin width, we can define h(z) as a function with a continuous
argument.
Let us define a function m(z) = zn(z).
Also, as we know C(rj, h(z)),
we may express the capacity as a function of z, say C,,(z). Now, going through the
algebra, one can show that the wideband capacity CwB satisfies
37
CWB,
P
fo Cizdz
1
(3.16)
f" m(z)dz
Given the SM capacity, the term in the square braces is just a positive constant
which can be evaluated easily. For rI = 0.2 for all frequencies, the capacity was calculated to be CwB,=0.2
CWB,Iossless.
=
0.5877/P/h, which is almost r7 times the lossless capacity
So, similar to the 'low-power high-loss' case, at r7 = 0.2 with arbitrary
input power P, the wideband number state capacity of the lossy channel (with frequency independent loss) is almost proportional to the transmissivity.
We later show that the ultimate classical capacity of the wideband lossy channel C =
V/CwB,ossIess,
which can be achieved by a coherent-state encoding.
one important conclusion is that transmissivity -
So,
at least for the case of frequency-independent
in contrast to the lossless case, number state encoding and direct
detection is not optimal for the lossy channel. Another important observation from
the above analysis is that, for frequency independent transmissivity, the wideband
classical capacity of any lossy channel can be expressed as a constant times VP/h,
where P is the constraint on total input power in Watts.
38
Chapter 4
Capacity using Coherent State
Inputs and Structured Receivers
Coherent states
{fla)}
are defined by the eigenvalue equation for the non-Hermitian
annihilation operator d,
ja)
with, in general, complex eigenvalues {a}.
coherent state
Ia)
(4.1)
= ala),
The mean complex amplitude of the
is given by (d) = a = ai + ia 2 = (el)
+ i(e 2 ), where ei and
&2
are the normalized quadrature operators. Coherent states are quadrature minimum
uncertainty states each of whose quadrature components are uncorrelated and have
variances
((A&i)
2
) = ((Aa2)2)
= 1/4. One can derive the number-state expansion of
a coherent state [15],
Ia) =
e-
/2
-In).
(4.2)
The two other properties of coherent states that we will use for our analysis are
the inner product of two coherent states and the overcompleteness property:
39
(3ce)
=
e(aO*-a*O)/2e-a-012 /2,
=
Jd
a)(a
(4.3)
(4.4)
7r
In this chapter, we will investigate the classical capacity of the SM lossy channel
and the wideband free-space optical channel in both far-field and near-field propagation regimes, achievable using coherent state encoding with homodyne or heterodyne
detection (see Section 2.3) at the receivers. The lossless capacities for coherent state
inputs have been dealt with in detail in [15], and are explained briefly in the next
section.
4.1
Coherent State Capacity: Lossless Channel
Let us transmit a coherent state 1, = 1a)(al through a zero-temperature SM lossless
(rq = 1) bosonic channel (2.1), with a probability measure pA(a)d2a. Here, A is a
complex random variable representing the input alphabet, which takes on the value
a, if the coherent state 1a) is transmitted in a single use of the channel. Thus, the
unconditional channel density operator (per use) takes the form
p = JPA(a)Pad2a,
(4.5)
and the average input power constraint can be written as
i = Tr( &'&) = Jd2aja2pA(a).
4.1.1
(4.6)
SM Capacity: Heterodyne Detection
As we discuss in Section 2.3, ideal heterodyne detection involves a simultaneous measurement of both quadratures of the field mode, and is described by the POVM [6]
40
U1
(4.7)
= 1-11)(I1.
7r
The conditional probability density to read out 3 = 1, + i0 2 at the output of the
heterodyne detector, given that
Ia) was
PBIA(f 3 I&e) =
transmitted is given by
(4.8)
Tr(&1 )
1(#a)12
(4.9)
7r
1
-
7r
exp (-13
a12),
-
(4.10)
where B is a complex random variable representing the output alphabet of the channel. From Eqn. (4.10), we conclude that this channel can be looked upon as two
independent and identical, parallel additive Gaussian noise channels (corresponding
to the two quadratures) with mean squared amplitude constraint of h/2 at the input,
and a noise variance of 1/2 for each of the two channels. We know from Shannon
theory that the capacity of an additive white Gaussian noise (AWGN) channel is
achieved by a Gaussian input probability density. So, the capacity achieving input
distribution is given by
PA (a) =
7rn
(4.11)
)
exp(-
which makes the unconditional channel density operator (4.5), a thermal state with
mean photon number i! (2.3). The SM capacity in bits per use of this channel can
be obtained using the classical Shannon formula [20], by adding the contributions of
the two quadratures:
Ccoherent--heterodyne(f)
+
g109
2 (
=
2
=
log2 (1 +
41
).
(4.12)
(4.13)
4.1.2
SM Capacity: Homodyne Detection
As we discussed in Section 2.3, ideal homodyne detection involves the measurement
of a single quadrature of the field mode, and for the case when the local oscillator
phase is oriented with the d, quadrature, is described by the POVM [6]
(4.14)
N13= If1)(1
The conditional probability density to read out 13,at the output of the homodyne
detector, given that
Ia)
was transmitted is given by
PBIAC(311a)
= I(Qu1a)1 2
-
27r
(1)
(exp
(1
)2
2(1)4
,
(4.15)
where we have assumed a is real.
From the above equation, we see that this channel is equivalent to an AWGN
channel with mean squared amplitude constraint of ft photons at the input, and noise
variance, of 1/4. The SM capacity (in bits per use) of the lossless channel for a
coherent state input with homodyne detection is thus given by
log 2 (
Ccoherent-homodyne(i)
-
+
)
1
log 2 (1 + 4Ai).
2
(4.17)
Note that, at high values of h, 'coherent-heterodyne' does better than 'coherenthomodyne', whereas at small values of h, 'coherent-homodyne' does better.
This
is because homodyne detection has less noise but less 'bandwidth' than heterodyne
detection. So, at low h values, where low-noise operation is more important than
'bandwidth', homodyne detection yields a higher capacity. The opposite situation
prevails at high h values, where 'bandwidth' is at a premium, so that heterodyne
detection yields a higher capacity.
42
4.1.3
Wideband Capacity: Heterodyne and Homodyne Detection
If P is the constraint on the maximum average input power in Watts (averaged
across frequencies), then the wideband capacities of the lossless bosonic channel using
coherent state encoding with homodyne and heterodyne detection can be calculated
using the constrained maximization procedure explained in Section 2.6. The various
known wideband capacities (in bits per second) of the lossless channel are summarized
below [15].
Cuitimate(P) =
r
2P
In 2
3h
-1
2P
Ccoherent -heterodyne(P)
Ccoherent-homodyne(P)
I
(4.18)
,
-
h
Note that interestingly, the capacity expressions for Ccoherent-heterodyne(P)
Ccoherent-homodyne(P)
(.9
9
(4.20)
and
are identical. Also, we observe that
Cuitimate(P)
Ccoherent-heterodyne (P)
7r
(4.21)
N/3
and thus the performance of heterodyne detection is consistently suboptimal at all
input power levels. In won't be out of context here to mention that, later in this
thesis, we prove a sufficient condition for the lossy wideband channel capacity with
heterodyne detection and coherent state encoding to approach the ultimate classical
capacity as the input power P -+ oc.
4.1.4
Wideband Capacity: Restricted Bandwidth
The wideband capacities for the lossless channel in the previous sub-section were
all calculated under the assumption that we have at our disposal photons at all
frequencies in the unbounded interval (0, oo), which is not the case in practice. Also
43
it turns out that, in the 'ultimate capacity' case, no matter what the magnitude of
the input power P is, the capacity achieving power allocation uses non-zero power at
all frequencies
f
E (0, oc). Whereas, in the case of either heterodyne or homodyne
detection, it can be shown that for a given input power level P, the best distribution
populates only up to a certain cut-off frequency
fe,
though the cut-off frequencies
for homodyne and heterodyne are different for a given P. Interestingly, as noted
in the previous sub-section, after we solve the constrained maximization problem
by integrating the respective optimal power allocation and SM capacity expressions
between
f
E (0, fc], the wideband capacity expressions for
Ccoherent-homodyne(P)
Ccoherent heterodyne(P)
and
are found to be identical.
What happens to the homodyne and heterodyne capacities if we now impose a
pair of lower and an upper cut-off frequencies of transmission? Interestingly, it turns
out that in this case homodyne and heterodyne wideband capacities are no longer
the same. Homodyne performs better at lower values of P, whereas heterodyne wins
at higher values of P. Let us work out the wideband capacities for these two cases in
detail to see why this happens.
The analysis for heterodyne and homodyne capacities are similar. So, we will
first work out the heterodyne capacity in detail, then state the homodyne result and
compare the two cases. Maximizing the overall information rate for the wideband
lossless channel with coherent state inputs and heterodyne detection
ln(1 + fij),
C=
(4.22)
subject to total input power P (2.20), we obtain the optimum intensity (W/Hz)
distribution as
Ih - P
b
-
hfi,* = 1o - hfi,
(4.23)
where Io = (1/A In 2) and A is a Lagrange multiplier. Replacing the sums by integrals, and imposing Kuhn-Tucker like conditions, we can write the optimal intensity
variation as
44
1(f)
=
{l'o
- hf, if > 0
(4.24)
otherwise
0,
with input power P given by
fmax
P=
J I(f)df.
(4.25)
fmin
The allocation of intensity across different frequencies in the transmission band
can be looked upon as a 'water-filling' solution to an infinite set of classical parallel
additive white Gaussian noise (AWGN) channels [2] (Fig. 4-1(a)). For a given value of
1
o, due to the lower cut-off frequency constraint, and the optimal intensity expression,
f
non-zero power is allocated only in the region
frequency cut-off
fm ,
is greater than
intensity I(f) in the region
f
where
f,
= 1 0 /h. Integrating the optimal
E [fmin, fc], we get,
hf)df = - (fc 2 - 2fcfmin + fmin 2 ).
f=Ioh (Io -
P
f,
E [fmin, fc], as long as the upper
(4.26)
2
fmin
From this result it follows that
fc =
2P + fmin.
(4.27)
4.7
As the input power P is increased from zero upwards,
f m in,
until it hits the upper cut-off frequency
f m ..
fc
increases starting from
The value of P for which this
happens will be termed the 'critical power' for heterodyne capacity, which we denote
as Pat .
In order to simplify the notation, let us normalize the input power by hf2x to
obtain a dimensionless quantity P' =
P/hfm2..
Also, let us define a normalized
wideband capacity C' = C/fm., and normalize all frequencies by fmax and denote
them by
f'
f/f
m ,.
It is easy to deduce from Eqn. (4.27), that
p'
(1
45
-
f
.i
(4.28)
het
hf
hf
I0
I
I
I
I
I
I
I
I
I
I
I
I
I
I
0
Z
Z:
~T-~-
--
fmin
TZ
7__:_
/
f
f
max
f
fmin f max
10
10
fch
h
(b)
(a)
Figure 4-1: Optimal allocation of intensity across the transmission bandwidth to
achieve maximum capacity on a lossless channel using coherent-state inputs and heterodyne detection. (a) f m a fc, (b) f m a < fc.
When P' is increased beyond P't,
fc
keeps increasing beyond fm , but the power
'fills up' vertically in the region f E [fmin, f mn], as shown in Fig. 4-1(b).
The above equations can be solved analytically to compute the normalized wideband capacity
Ccoherent-heterodyne(P')
as a function of the normalized input power P'.
For P' < P'cr, we have
Ccoherent-heterodyne(PI)
and fm.
where f'= f/fm, = v2P + fn
Ccoherent-heterodyne
where f m
r
< fe, and
fc. For P' >P'i, we get
fc
f min
In 2
f' is given
f',=
(4.29)
m
2P/ - fnin
=
( Jf)
fMm
(4.30)
n
by
1(1 + f' in) + (1
P'
-
(4.31)
If a similar analysis is carried out for the homodyne capacity case, one can easily
46
compute the normalized wideband capacity
Ccoherent-homodyne(P').
For this case, the
critical power is given by
(1
crit
P hom
-
(4.32)
2.
8
For P' < picntm, we have
('f
1m(inf'
V
Ccoherent-homodyne(P')
In 2
where
f' =
8P' + f' 1i, and fm
Ccoherent-homodyne(P)
where f m , < fc, and
=
>
212
fc. For P
I('
(4.33)
ki))(4
inin
2
Pr
fnin)+
,
In f
3
g
frnin
m-
f)4)
f' is given by
1
f
4P'______
= 2 (1 + fif)
2
+
4,'
(4.35)
(1 - fmin)
Now, if we use these four boxed expressions to plot the normalized capacities
coherent-homodyne(P')
and Ccoherent-heterodyne(P'), as a function of
P' = P/hfm.
(see
Fig. 4-2), we find that homodyne detection does better at lower input power levels
whereas there is a value of power
Pcross-over, beyond
which heterodyne detection takes
over. As explained in Section 4.1.2, this crossover occurs because of the different noise
and bandwidth characteristics of heterodyne and homodyne detection. As expected
though, by imposing these additional constraints by forcing lower and upper cutoff frequencies, the overall performance in either case deteriorates from the optimal
homodyne (or heterodyne) performance given by C = (1/In 2)
4.2
2P/h (4.19),(4.20).
Coherent State Capacity: Lossy Channel
Let us transmit a coherent state Ia) (aI with a probability measure pA(a)d 2a, through
a zero-temperature SM lossy channel (2.1) with transmissivity r7. It can be easily
47
2.006
,,
do
fJ
I
-.
~.-
I
I
*
I-
410
1.00
'p
-
H cross-oer
lei
N
0.500
.
-
Homodyne -- P < P-crit-hom
Homodyne -- P > P-crit-horn
-Heterodyne -- P < P-crit-het
Heterodyne P > P-crit-het)---
Cht
/p
peltt
Het
Hom
Honr--Het
0.306
0.00
0.300
ILI
I0
-
(no bandwudth restriction).
*I
oe
1.200
-
.
1.5ON
P' (normalized input power)
Figure 4-2: This figure illustrates the effect of forced lower and upper cut-off frequencies (fmin and fm), on the coherent state homodyne and heterodyne capacities.
Given values of fmin and fm., the critical power levels (the input power at which the
critical frequency f, hits the upper-cut off frequency fm, and beyond which power
'fills up' only between fmin and f m ) have been denoted by Ph" and Ph t respectively
for the two cases. In this figure, we plot the normalized capacities C' = C/fmax vs. a
normalized (dimensionless) input power P'=-P/hfm. We normalize all frequencies
by f m . and denote them by f' =_ f/f ma. The 'dash-dotted' (black) line represents
the capacity achievable using either homodyne or heterodyne with unrestricted use
of bandwidth (Eqn. (4.20)), i.e. C = (1/ln2)V2P/h bits per second. The 'solid'
(green) line, and the 'dashed' (blue) line represent homodyne and heterodyne capacities respectively for any set of cut-off frequencies satisfying fi = 0.2. For this value
picrit
of fin, P'ho = 0.16, and P'ht = 0.32. Different capacity expressions hold true for
power levels below and above the respective critical powers (Shown by the thick and
the thin lines in the figure respectively).
-ci
48
shown that when a coherent state
= ca (aI is sent through this channel, the
/IN
output is also a coherent state whose mean is contracted towards the origin of the
phase space by the factor /i, i.e. POUT
=I
V 177a)
(Vja. If this output state is detected
using homodyne (or heterodyne) measurement, we can use arguments similar to those
in the previous section to show that the channel in this case can be represented by
one (or a set of two parallel) AWGN channel(s); and hence capacity is achieved by a
Gaussian distribution of coherent states at the input.
4.2.1
SM Capacity: Homodyne and Heterodyne Detection
The SM capacities of the lossy channel with transmissivity 71 for coherent state encoding with homodyne and heterodyne detection respectively are given by
Ccoherent-hornodyne(7, h)
=
Ccoherent-heterodyne (7, A)
=
log 2 (1+ 477A)
log 2 (1 +
I
and
(4.36)
(4.37)
which can be easily deduced from the results in [15] by observing that we can think
of the above as the capacity of a lossless channel with an input 1,/ija) (.f/-aI and a
power constraint Tr(a al
qa)(Aaj)
; an.
Note again that, at high values of h, 'coherent-heterodyne' does better than
'coherent-homodyne' whereas at small values of A, 'coherent-homodyne' does better.
4.2.2
Wideband Capacity
Wideband capacities achievable using coherent state inputs can be calculated from
the corresponding SM capacities using the technique illustrated in Section 2.6. In
this section, we will consider the wideband capacity calculation for four cases -
(a)
Frequency independent loss, (b) Free-space channel in the far-field regime, (c) FreeSpace channel in the near-field regime, and (d) Single spatial mode propagation of
49
the Free-space channel - Ultra wideband.
Frequency Independent Loss
As is evident from comparing Eqn. (4.36) and Eqn. (4.37), the calculations for the
case of homodyne detection will be very similar to that of heterodyne detection. So
once again we shall go through the heterodyne detection case, and thereafter state
the corresponding final results for the homodyne detection capacity.
Let us assume that rI
=
r,Vj. Maximizing the overall information rate
(4.38)
ln(1 + qfi),
C =b
subject to total input power P (2.20), we get the optimum intensity (W/Hz) distribution as
Pb
Is - - = hf fi? = Io -
hf-
*,(4.39)
where lo = (1/A In 2) and A is a Lagrange multiplier. Replacing the sums by integrals
over all positive frequencies, and imposing Kuhn-Tucker like conditions, we can write
the intensity variation as
L
I0 -
Iif
0,
if > 0
(4.40)
otherwise
with input power P given by
fmax
(4.41)
P= JI(f)df.
fmin
The allocation of intensity across different frequencies in the transmission band
can be looked upon as a 'water-filling' solution to an infinite set of classical parallel
additive white Gaussian noise (AWGN) channels [2] (Fig. 4-3(a)).
The above equations can be solved analytically
case -
exactly as we did for the lossless
to obtain the wideband capacity in bits per second. The wideband capacities
50
h7
hf
1
)7
0
'io-
P
7
fmin
fmax
1o77
4
f
fmin f max
f
4
h
(a)
10
fch
(b)
Figure 4-3: Optimal allocation of intensity across the transmission bandwidth to
achieve maximum capacity on a channel with frequency independent transmissivity
7, using coherent-state inputs and heterodyne detection. (a) f m a > fe, (b) f m a < fc.
for the lossy channel in the ideal case, when we calculate the capacities under the
assumption that we have at our disposal photons, at all frequencies of transmission
(i.e. no lower and upper cut-off frequencies), are given by
Ccoherent-heterodyne
Ccoherent-homodyne(17
P)
=
P)
=
2njP
1
127h'
1
1n2
and
2 P
h
(4.42)
(4.43)
So, the performance of homodyne and heterodyne in this ideal case are the same.
Next, let us impose a set of lower and upper cut-off frequencies given by
fmin
and
fm. respectively. Carrying out our analysis along similar lines to the lossless case,
and using the same normalization as in Section 4.1.4, the critical power level for
heterodyne capacity is found to be
(
For p' ; p'crt, we have
51
2
(4.44)
Ccoherent -heterodyne (77, P')
where
f'
=
2rP' +
fmin
-
(4.45)
in
fmj., and f m , > fc. For P > P' , we get
Ccoherent-heterodyne
where f mr
( 2rP'
I
=
P
frn)
-
=q
In 2
fc
in
+
m in/
-
(fmm
)
(4.46)
.
< f, and f' is given by
f',=
1(1
77P'
+ f'in) +
(4.47)
.
If a similar analysis is carried out for the homodyne capacity case, one can easily
compute the normalized wideband capacity
CCoherent-homodyne(P').
For this case, the
critical power is given by
)
(1 -
p/critf
hor8
(4.48)
For P' < P'/[cA, we have
Ccoherent-homodyne(17,
where
f'
= V8rP' +
fmin,
P')
=
2T/Pl - 1 fmin I n
(
'2
mm
> pycrit
and fmax
fc. For P' >
fr)
Ccoherent-homodyne(77, P')
where f mr
In 2
i+
oP'
n fc
-
w
(4.49)
n c
e
we get
finn
(/1)
(4.50)
< fc, and f' is given by
f'C =
1(1
+ fnin) +
4 7P'
-f)
(4.51)
So all the wideband results pertaining to the case of frequency independent loss
can be obtained from the corresponding expressions of the lossless case, by replacing
52
P with TIP. Also, imposing cut-off frequencies has the effect of twisting apart the
homodyne and the heterodyne curves so that homodyne does better than heterodyne
in the power-limited regime and vice versa in the bandwidth-limited (high-power)
regime.
Free-Space Channel: Far-Field Propagation
The next case we consider is the free-space quantum optical channel (see Section 2.1),
with an additional assumption that in the entire frequency range of transmission, the
propagation remains well within the far-field regime, i.e.
[fmin, fm,,].
DJ = q(f) <
1, Vf E
In this case we can assume that at any given frequency in the above
range, only one spatial mode of the field couples appreciable power into the receiver
aperture with fractional power transfer being almost equal in magnitude to the freespace Fresnel number at that frequency D1 , and the rest of the spatial modes of the
field can be neglected
[5].
Let us denote by i, as we did in Section 2.6, a discrete frequency index, so that
2
the transmissivity of the ith frequency bin can be written as 7i = Afi , where A =
ATAR/C 2 L 2 is a geometrical parameter of the channel. We define a constant K = h/A
which has dimensions of power. On carrying out the analysis along similar lines as
the frequency independent loss case, the optimum intensity distribution for coherent
state encoding and heterodyne detection is found to be
I(f) =
{
I0 - L'
if > 0
'
0,
(4.52)
otherwise
where Io = (1/A In 2) and A is a Lagrange multiplier. In this case, we see that capacity
is achieved with preferential use of high-frequency photons. So, it might seem that
we can achieve as much capacity as we would like to, by using higher and higher
frequencies. But this is not true, because the highest frequency of our transmission
band must adhere to the far-field condition for the single-spatial-mode assumption to
remain valid. Eqn. (4.52) suggests that we normalize all the quantities with respect
to fmna,
the maximum allowed frequency of our transmission band. Let us define
53
k
1
f
f
f'
11
Figure 4-4: Optimal allocation of intensity across the transmission bandwidth to
achieve maximum capacity on a channel with frequency dependent transmissivity
r7 cc f2 , using coherent-state inputs and heterodyne detection.
I' -_ - -
I(f)fma
I'(f')
where
f' =
f/fma,
mT
0,
K
p
if > 0
-w(4.53)
otherwise
and 10' = Iofmax/K. Let us define a dimensionless (normalized)
power P' = P/K, where P is the total average input power in Watts, and a normalized
capacity C' = C/f m a, where C is the wideband capacity of the channel in bits/sec.
It is now straightforward to show that
C'=
P'=
where
f'in
=
In 2
ln(1 + I'(f')f')df',
and
(4.54)
"i
I'(f')df',
/1|in
(4.55)
fmin/fmax.
Analogous to the channel with frequency-independent loss, we can look upon the
above power-allocation as similar to the classical idea of 'water-filling' for parallel
AWGN channels. Interestingly, in this case, the 'water' fills up starting from the
54
highest allowed frequency towards lower frequencies.
Depending upon the value of
f',5 P' has a critical value Pc'rit, after which the 'water' fills up vertically in the
region defined by
f' E
[fVrin, 11.
Carrying out the integrations above in the low-power
regime, when P' < Pcrit, i.e. 1/6 >
f,
we find that the normalized capacity C' is
given by
C'coherent--heterodyne(P')
where
=
In 16
In 2 (I01
-
1+
(4.56)
,
16 is a dimensionless parameter, which is obtained from the power constraint
equation Eqn. (2.20), that takes the following form in this case:
P' = 1/ - 1 - nI6.
In the high-power regime, when P' > Pcrit 1 i.e.
1
C'coherent-heterodyne(PI)
P'
=
I;(1
<
(4.57)
fin, the corresponding
( n IV - 1 + f'in
-
fa) +
- frin
equations
IOf'in)),
and(4.58)
(4.59)
ln(fl'i).
In Fig. 4-5, the normalized capacity C' = C/fm, has been plotted vs. normalized
power P' = P/K for coherent-state inputs and heterodyne detection, for different values of
frin.
We find that the performance degrades steadily, at high powers, as
f min
is
increased from zero, because of the increasingly stringent bandwidth constraint. The
interesting thing to note in this case is that the far-field capacity is directly proportional to the maximum frequency of the transmission band, fm,.
This should not
be interpreted as a possibility to have infinite capacity by increasing
fm .
without
bound, because the single-spatial-mode approximation requires that even the highest
frequency photon be sufficiently deep into the far-field (Dfmax < 1). The calculations
for the case with coherent-state inputs and homodyne detection can be performed
along similar lines. In Fig. 4-6, we plot C' vs. P' for both homodyne and heterodyne
55
65-4-
U
2
1
0
0
20
40
60
Normalized Power P'
80
100
Figure 4-5: Dependence of C' on P' for coherent-state inputs and heterodyne detection, for different values of fai.. The blue (solid) line is for fjni = 0, the red (dotted)
line is for flnin = 0.1 and the green (dashed) line is for fani = 0.2. Observe that all the
three curves coincide in the low power regime, because when P' < PC'rit, C' does not
depend on f.in. When P' > Pcrit, performance of the channel deteriorates because of
restricting ourselves to a non-zero fn.in.
56
2-
1l.5
1
N0.
0
0
0
2
6
4
Normalized Power, P'
8
Figure 4-6: Dependence of C' on P' for coherent-state inputs with heterodyne and
= 0. The blue (solid) line is for heterodyne detection,
homodyne detection, for f
and the red (dashed) line is for homodyne detection.
detection with
f'ai.
= 0. We see, as before, that homodyne detection performs better
in the low-power (power-limited) regime, because the lower noise-variance associated
with homodyne detection overwhelms the bandwidth advantage of heterodyne detection. But, in the high-power (bandwidth-limited) regime, the bandwidth advantage
of heterodyne detection makes it outperform homodyne detection.
Free-Space Channel: Near-Field Propagation
Let us again consider the free-space optical channel that we did in the previous subsection, but now with the assumption that the entire frequency range of transmission
remains well within the near-field propagation regime, i.e. D
>
1, Vf E [fm"i, fmaxj.
It is well known that [5] in this regime at any given frequency of transmission f,
there are approximately Df spatial modes of the bosonic field, each one coupling
power perfectly (without loss, q = 1) to the receiver aperture. The remaining spatial
modes have insignificant contribution to the overall power transfer, and hence may be
neglected. The wideband capacity calculation for this case can be set up as follows.
57
Let us assume that the frequency axis in the region
f
E [f
m in,
f
m ax]
into small bins of width b Hz, and denote the center frequency of the
is divided up
jth
bin by
f.
A = ATAR/C 2 L 2 is again a geometrical parameter of the channel (Fig. 2-1). The
free-space Fresnel number for the frequency
fi
is given by Df,
are Df, independent spatial modes of the field at frequency
fi,
=
Afi 2 . So, there
each of which couple
power perfectly into the receiver aperture. Without loss of generality, assume that
each spatial mode at the frequency
f, is excited in
a state with the same mean photon
number denoted by iii. So, the heterodyne detection capacity C, of this wideband
channel is given by
C = max b
(Afi2)log 2 (1+
i)
,
(4.60)
subject to the maximum average input power constraint P, which now takes the form
P=
(4.61)
(Af2 )hfib.
Solving this constrained maximization problem along similar lines as the SM lossless case, we obtain the optimum intensity (W/Hz) distribution
P-= (Afi 2 )hfi*
= Afi 2 (Jo -
hfi),
(4.62)
b
where Io = (1/A in 2) and A is a Lagrange multiplier. Replacing the sums by integrals, and imposing Kuhn-Tucker like conditions as before, we can write the optimal
intensity variation as
- hf), if > 0
Af2(I
0,
(4.63)
otherwise
with input power P given by
P
J I(f)df.
fmin
58
(4.64)
For a given value of Io, due to the lower cut-off frequency constraint, and the opti-
f E [fmin, fc],
fc, where f, = 10 /h.
mal intensity expression, non-zero power is allocated only in the region
as long as the upper frequency cut-off fmax is greater than
Integrating the optimal intensity I(f) in the region
P=
fc=Io/h
In
Af 2 (Io
j
fai
Ah
- hf)df = 12
12
f E [fm in, LI,
(fe4 - 4fe f m in3
From the above equation, we see that as expected,
substitute
f,
=
f m in
fc
we get,
+ 3fmin 4 ).
= fmin,
(4.65)
for P = 0. Let us
+ g(fmin, P), where it can be easily shown that g(fmin, P) is a
function satisfying
2
2
g(fmin, P)4 + (4fmin)g(fmin, P) 3 + (6fmin )g(fmin, P) =
12P
Ah
(4.66)
It is easy to observe that the expression on the left hand side is a strictly increasing
function of g(fnin, P) for fm in > 0, in the region g(fmin, P) > 0. Also, g(fnin, 0)
= 0,
and g(fmin, P) increases as P increases. So, given a value of fmin, and P > 0, there
exists a unique value of g(fmin, P), which corresponds to a unique value of
As the input power P is increased from zero upwards,
it hits the upper cut-off frequency
f m ,.
fc
f.
increases from fmin, until
The value of P for which this happens will
be termed the 'critical power' for heterodyne capacity, which we denote as P
.r~t
To simplify the expressions in the calculations that follow, let us denote by P'
P/(Ahfm), a normalized (and dimensionless) input power. We define a normalized
and normalize all frequencies by the upper cut-off frequency
capacity C' - C/Afm,
fmax, i.e.
f' =
f/fmax.
Note that the normalization we use for C and P here,
Also, note that with the above
is different from what we used in Section 4.1.4.
normalization, Eqn. (4.65) and Eqn. (4.66) take the following form:
p,=
12P'
l('
12
=
=
g(fnin,
- 4f'f m'
_
P')
i
3
+ 3f' i 4),
')+(4fin)g'(fmnin,
59
(4.67)
PF
')+(6fin2
in,2,(.)
where f' = f.'i + g'(f.'in, P'). So, following an argument similar to the one we used
f 'in
before, we can prove that for a given set of values of
and P' > 0, there exists a
unique value of f'.
It is easy to deduce from Eqn. (4.65) that
p/crit
Pht=
(1 -
fi'in3 )
3
When P' is increased beyond this value,
power only 'fills up' in the region
f
_
(1 -
f, keeps
f(469)
increasing beyond fma, but the
E [fmin, fma]. The above equations can be solved
analytically to compute the normalized wideband capacity
P'
=
fc/fma,
For
< fc, and
(4.70)
we get
Coherent-heterodyne(P
f ma
1 + 3 In f/
is obtained for a given value of P' as explained above, and
fma > fc. For P' > Pt'
where
"3
91 2
Ccoherent-heterodyne (P')
f'
CCoherent-heterodyne(P').
we have
h
P',
where
(.9
4
f'
=
912
91n2
"3 )
+ 3lnf' - 3ffni 3 ln
in
(4.71)
fMin.
is given by
12P' + 3(1 - fin
4(1- fm(in.7
(4.72)
If a similar analysis is carried out for the homodyne capacity case, one can easily
compute the normalized wideband capacity
CCoherent-homodyne(P').
For this case, the
normalized critical power is given by
crit
Fhor'
For F' < p',,i
_
(1-
3
fin
(1 -
fiin 4)
16
12
we have
60
(4.73)
Ccoherent-homodyne(P')
where f m ,
1
=
3
3
+
3I
fl)
(474
> fe, and f' is calculated following a similar procedure to that of the
heterodyne case. The relationship between normalized input power P' and f' in this
case is given by
1 (f
P
48
C
-
4f
3fina
+ 3fm'i4).
(4.75)
In
(4.76)
For P' > p'crit we get
Ccoherent-homodyne(P)
1
3
[)
-2
where fma.a < fc, and f' is given by
=
48P' + 3(1 4(1 - f'
)
3
)
Now, we use all the above four boxed expressions for to plot the (normalized)
capacities
Ccoherent-homodyne(P')
and Ccoherent-heterodyne(P), as functions of P' (see Fig.
4-7). We find that, similar to the lossless case, in the wideband free-space channel in
the near-field regime, homodyne detection does better at lower input power levels and
there is a value of power
Pcross-over,
beyond which heterodyne takes over as having
higher capacity.
Free-Space Channel: Single Spatial Mode - Ultra Wideband
In the previous two sections, we calculated the coherent state capacities of the freespace channel, first assuming that the entire transmission bandwidth is in the farfield (Df
<
1) propagation regime, and next assuming that the entire transmission
bandwidth is in the near-field (D > 1) regime. For both these cases, the expressions
for wideband capacity achievable using coherent state encoding with either homodyne
or heterodyne detection could be calculated explicitly. What happens if there is no
61
IsI
0.400
'
-
~0.300
.200
Homodyne -- P < P-crit-hom
Homodyne P > P-rit-hom
Heterodyne
0.000
--
P > P-crit-het
I
0.000
0.100
0.300
0.200
0.500
0.400
(normalized input power)
0.600
P
ma=
Ahf~
Figure 4-7: This figure illustrates the coherent state homodyne and heterodyne capacities for the free-space optical channel in the near-field regime. All the frequencies
in the transmission range f E [fmin, fmaxl satisfy Df > 1. In this curve, a normalized
capacity C' = C/Afm. has been plotted against a dimensionless normalized input
2 2
power P' = P/(Ahf,), where A = ATAR/C L is a geometrical parameter of the
channel (Fig. 2-1). Given values of f min and fm a, the critical power levels (the input power at which the critical frequency f, hits the upper-cut off frequency f ma,
and beyond which power 'fills up' only between fmin and fma) have been denoted
het respectively for the two cases. The 'solid' (black and blue) line,
b pithond
m~ and pc
and the 'dashed' (red and green) line represent homodyne and heterodyne capacities
respectively for any set of cut-off frequencies satisfying f' i = 0.3. For this value
= 0.019, and
of f'in, the values of the normalized critical input power are: p,
h't = 0.076 respectively. Different capacity expressions hold true for power levels
below and above the respective critical powers (Shown by the thick and the thin lines
in the figure respectively).
62
restriction on the bandwidth of communication over the input power limited freespace channel? For example, consider an ultra wideband satellite to satellite freespace communication link with transmitter and receiver apertures of areas AT ~~400
cm 2 and AR
%
1 In 2 , separated by L = 400 km of free-space.
Assume that the
frequency of transmission range from that associated with a wavelength of a few tens
of microns (which would typically be deep into the near-field regime - Df ~~0.001),
to that associated with a wavelength of several hundred nanometers (which could
be considered well into the near field regime
Df
~ 25).
In such a case, any
capacity calculation would have to take into consideration the actual variation of
modal transmissivity with frequency of transmission over a broad range of frequencies
(see Fig. 2-3).
In Section 2.1.2, we observed that as the value of Df increases from the near-field
regime, more and more spatial modes of the field start coupling appreciable amount
of power into the receiver. If we arrange the fractional power transfers of the spatial
modes in decreasing order and denote the fractional power transfer (transmissivity)
of the ith mode by 77i, then
-
from the eigenvalue behavior of the classical free-space
modal decomposition theory [23, 24] -- the rhi's depend upon the carrier frequency
f
as shown in Fig. 2-3.
To calculate the general capacity of the ultra wideband free-space channel, we must
sum the information capacities of all the contributing spatial modes at a frequency,
taking into account the actual values of modal transmissivity for each one of them.
As there is no good analytical expression for their functional dependence on frequency
of transmission, it is seemingly very difficult to come up with an explicit formula for
capacity. So, what we shall do is to find the capacity of the channel assuming that
only a single spatial mode propagates at all frequencies. As capacity is additive, an
analysis very similar to this can be used in conjunction with the relative magnitudes of
transmissivity for different spatial modes to compute the true wideband capacity. In
this section, we will calculate the capacity under the assumption of single spatial mode
transmission. For our analysis, we will assume that we have photons at our disposal
at all frequencies, i.e. for any f E (0, oo). We justify later why this assumption is
63
reasonable.
Setting up the calculation in the usual manner, we have to maximize the sum of
SM capacities C(r(f), A) across the entire frequency band of interest. For heterodyne
detection capacity, we have to maximize
C = max C(q(f,), hi) = max b
log 2 (1 + 1(fi)i)]
,
(4.78)
subject to the maximum average input power constraint
P=
hfibi.
(4.79)
Because of the nonlinear frequency dependence of transmissivity, we have to do
this maximization numerically (using the Lagrange multiplier method). The point to
note here is the following. At very low frequencies this channel looks like the far-field
channel we analyzed earlier, in which q oc f 2 . So in that region, we might expect that
the optimal intensity allocation uses high frequency photons preferentially, and that
the intensity goes to zero at low frequencies. At higher frequencies, this channel is
closer to the lossless wideband channel we considered earlier, for which we know that,
the optimal intensity allocation goes to zero at very high frequencies. So, in the ultra
wideband case, we would expect the intensity allocation to vanish both for very low
and very high frequencies. This is why our assumption of
f
E (0, oc) is reasonable.
As an example of our calculation, we illustrate the results of one of our numerical
evaluations for the ultra wideband capacities of homodyne and heterodyne detection,
using the set of geometrical parameters listed in the first paragraph of this sub-section
(Fig. 4-8). In this case, the frequency range of transmission that has been used, is
f
E [0 Hz, 3000 THz], corresponding to wavelengths in the range [300 nm, cc).
General Loss Model: A Wideband Analysis
Let us now analyze the wideband capacity calculation for the coherent state channel
with homodyne and heterodyne detection for a general frequency dependent loss. It
64
.-7-0
00
1 -1
P (input power in Watts)
161
Figure 4-8: An example of ultra wideband capacities for coherent state encoding
with homodyne or heterodyne detection, for the free-space optical channel under the
assumption of a single spatial mode transmission.
can be shown' that, for any frequency dependent transmissivity 77(f),
Ccoherent-homodyne
coherent-heterodyne(4
(-
To compute the general wideband capacity using coherent states and heterodyne
detection, we have to perform the constrained maximization (4.78). The optimum
frequency allocation of intensity I(f) is normalized by the Planck's constant to obtain
I
where
h
1' - 9(A) if > 0
0,
otherwise
A))
and g(f) =
n'
is a Lagrange multiplier,
f/(f)
(-1
is a function that is no less
than f, Vf E (0, oo). Let us denote the total input power normalized by the Planck's
'This can be shown by comparing the two constrained maximization problems and realizing that
by suitably scaling the capacity and the input power, one maximization problem can be represented
in terms of the other. This simple idea was proved by B.J. Yen, and S. Guha, Research Laboratory
of Electronics, MIT
65
constant by P' = P/h. lo' is obtained from the power constraint equation:
P' =
if
(Io' - g(f)) df,
EA(P')
(4.82)
where A(P') is a subset of (0, oc) in which the integrand is non-negative. This power
allocation can again be looked upon as a classical 'water-filling' solution, where the
power P' 'fills' up into the g(f)-curve, starting at the global minimum of g(f). Increasing the value of P'increases the value of Io'. The optimal power allocation for
the rl transmissivity shown in Fig. 2-3, has been plotted in Fig. 4-9. The capacity
C is found to be:
Ccoherent-heterodyne
(P')
-
In 2
InnI
2
+
[n(P/
n(A(P'))
k
A(P')
g(f)df)
JA (P')
Ing(f)df
-
ni(A(P'))]
/J
(4.83)
where p(A(P')) is the measure of the set A(P') in Hz. For the lossless case, g(f) =
f
and A(P') = [0, V2P']. Substituting this A(P') into Eqn. (4.83), one readily obtains
the lossless heterodyne capacity C = v2P'/ln 2 (4.19).
This analysis suggests an
interesting way to look at the heterodyne (or homodyne) detection capacity and an
obvious numerical algorithmic approach to evaluate the wideband capacity for an
arbitrary frequency dependent transmissivity.
66
0P
-
g~gf))-/-
A(P')
frequency (f)
Figure 4-9: The optimal power allocation for the wideband coherent state channel
with heterodyne detection depicted as 'water-filling', for frequency dependent transmissivity shown for r/1 in Fig. 2-3.
67
Chapter 5
Capacity using Squeezed State
Inputs and Homodyne Detection
We saw in the previous chapter that coherent states are minimum uncertainty product
states, whose quadrature components have equal variances. A coherent state 1a) is
sometimes represented on the phase plane by a quantum 'error-circle', representing the
vacuum noise, whose center is at the tip of the vector representing the mean complex
amplitude a, and radius being equal to the quadrature uncertainty
((A& 1 )2)1/
2
1/2. Squeezed states are also quadrature minimum uncertainty product states, but
their quadrature components have unequal variances. To define quadrature-squeezed
states, we need to introduce the squeeze operator [15]
S(r, #) = exp
(2
-
where r is called the 'squeezing parameter' and
-t5ei)
4 determines
,
(5.1)
the phase of the squeez-
ing. The squeeze operator transforms the modal annihilation operator according to
S(r, q)d [S(r, 0) t = d cosh r + &te2 io sinh r.
(5.2)
A quadrature-squeezed state
Ia)(,,) = D(6, a)S(r, #) 0)
68
(5.3)
is obtained by 'squeezing' the vacuum state and then displacing it.
D(d, a) =
As choosing 0 amounts to a rota-
exp(aet - a*&) is the displacement operator.
tion in the phase-plane, we can always arrange to set q = 0, which we do henceforth.
The mean complex amplitude of the squeezed state Qa(r,,) is a, similar to a coherent
state. The quadrature components are uncorrelated and have variances
((,A&)2)
_l -2r,
(5.4)
_e2r.
(5.5)
4
((A=2)2)
4
These states thus constitute the entire class of quadrature minimum uncertainty
product states. If r > 0, the first quadrature will have a variance reduced below the
vacuum level, whereas the second will have a variance increased above the vacuum
level. We usually recognize et as the 'squeezed quadrature' and &2as the 'amplified
quadrature'. A squeezed state can be represented on the phase plane much in the same
way as a coherent state, except that the 'error-circle' is replaced by an 'error-ellipse'.
5.1
Squeezed State Capacity: Lossless Channel
Let us transmit a squeezed state Pa, = Ial)(r,O)(r,O)(Ci
Iwith
a probability measure
PA1 (al)dal, at each use of the SM channel, where a, is real and r > 0. The uncon-
ditioned channel density operator per use, is given by
P=
JPA (a)a
daj,
(5.6)
and the mean photon number constraint takes the form
i! = Tr(&f&)
=
where
69
.2 + sinh 2 (r),
(5.7)
01
= J a2PA(ai)dai
(5.8)
is the second moment of the distribution PA, (a,). The important thing to note here
is that the sinh 2 (r) contribution to n represents the excess excitation of the amplified
quadrature, which through the power constraint, limits the degree of squeezing.
SM Capacity
5.1.1
As we know [5], ideal homodyne detection measures a quadrature component, and
hence for measurement along the di quadrature is described by the POVM
n2x
where
{
= lx1)(Xil,
(5.9)
xi) } are the eigenstates of &1. The conditional probability density to read x1
at the output given that a1 was transmitted, is given by
PX1IA 1 (x 1 Iai)
-
(5.10)
1 2t)
Tr(,
(5.11)
I(Xil l)(r,O)12
1
27r(
exp
e-2r)
(x1 -ai)2~
- X.
2(e-2r)
(5.12)
Similar to the case of coherent state channel, the channel noise is additive and
Gaussian. So, the mutual information is maximized by a Gaussian input probability
density
a12)
e
PA1 (a,) =
/2r
2
exp
2/.2
(5.13)
,
and the SM capacity (in bits per use) is given by
C(fI, r) =
log 2 (+
_
=
log 2 [1 + 4e 2 r(
70
-
sinh 2 r)].
(5.14)
A further maximization with respect to the squeeze parameter r yields the SM
capacity per use of the squeezed state channel:
Csqueezed-homodyne
5.1.2
(h) = 1og
2
(5.15)
(1 + 2h).
Wideband Capacity
As is evident from the expression of the SM capacity of the lossless squeezed state
channel, the wideband capacity calculation closely parallels that of the coherent state
heterodyne detection channel. The wideband capacity (in bits per second) in the case
when we assume that there is no restriction on the available bandwidth, is given by
Csqueezed-homodyne(P)
(5.16)
h
=
where P is the total input power constraint at the input. It can again be similarly
shown that if we impose additional restraint on the available bandwidth by imposing
a pair of lower and upper cut-off frequencies
fmin
and fma,
the wideband capacity
deteriorates from the ideal case. Carrying out our analysis along similar lines to the
coherent state channel, the normalized critical power level for squeezed state coherent
detection capacity is found to be
p crit Ici= _ pcrit
sq
hfm2a
sq
1-
fI
_fmi
4
(5.17)
)2
(-7
For P' <pcrit , we have
-
C queezedhomodyne
where f' = V47M
-
f(P/inin
(5.18)
+ f'in , and fma > fc. For P' > pcrit we get
Csqueezed-homodyne(PI)
1
[(1
-
fi)+
71
Infc
-
fmin ln
(5.19)
where
f m ax
< fe, and
f' is given
f'
by
1
= - (1 + fmi) +
2 -(1
2P'
,.(.0
- fmin)
The above expressions have been plotted in Fig. 5-1, in their respective regions
'
of validity for an arbitrary pair of cut-off frequencies satisfying
=
0.2. Squeezed
state encoding with homodyne detection consistently performs better than coherent
state encoding and homodyne detection, which is expected because whatever performance is achieved by a coherent state encoding can also be achieved using a squeezed
state encoding (using squeeze parameter r = 0 all the time).
The figure shows
comparative performance of squeezed state encoding and homodyne detection with
respect to that of coherent state encoding with homodyne and heterodyne detection.
5.2
Squeezed State Capacity: Lossy Channel
Let us transmit a squeezed state &, = Ial)(r,o)(r,o)(ail with an apriori probability
measure PA, (al)dai, at each use of the zero-temperature SM lossy channel of transmissivity 17, where a 1 is real and r > 0. There is a class of quantum states known
as 'Gaussian States', that can be characterized solely by specifying their first and
second moments, (d), (dt d), and (z5 2 ), because their Wigner characteristic function
is of Gaussian form. All squeezed states are Gaussian states. It can be shown that if
the input modes & and b in the SM lossy channel ( = V77e +
'1 -
17b)
are excited in
Gaussian states, then the output mode e is also in a Gaussian state. The mean and
the variance of the first quadrature of the output state are respectively given by
(&1)
=
(1
(5.21)
and
\Vce,
-
+ (1
-
1),
(5.22)
where the second term in the variance is the (T = 0) vacuum noise contributed by
the b mode.
72
2.000
-
Coe -
~
~ Li----
(
/-Coherent-Hom
-
Coherent-Hor ---P < P-crit-hom
C.- ---P<> P-crit-hom
- Coherent-Het ---P < P-cnt-het
Coherent-Het ---P >P-crit-het
Squeezed-Hom ---P < P-crit-sq
Squeezed-Horn
---
P >P-crit-sq
Squeezed-Hom (no bandwidth restriction)
Coherent-[Horn
6.010
1.300
=
Hed] (no bandwidth restriction)-0.900
0.608
---
1.200
-
1.50
P (normaIized input power)
Figure 5-1: This figure shows comparative performance of squeezed state encoding and
homodyne detection with respect to that of coherent state encoding with homodyne
and heterodyne detection. The 'solid' lines (red, green and grey) represent coherent
state heterodyne, coherent state homodyne, and squeezed state homodyne capacities
respectively for an arbitrary set of cut-off frequencies satisfying fmj. = 0.2. Different
capacity expressions hold true for power levels below and above the respective critical
powers (Shown by the thick and the thin lines in the figure respectively). For the above
=
= 0.32, P't
value of Ain, the values of the normalized cut-off power are: P't
0.08, and P'
= 0.16. The thick dashed (blue) line is the best achievable performance
of the coherent state channel (with either heterodyne or homodyne detection), and
is given by C' = (1/ In 2)v/2-.
The thick dotted (red) line is the best achievable
4 .
performance of the squeezed state channel, and is given by C' = (1/ ln 2)vri
73
5.2.1
SM Capacity
The conditional probability density to read x, at the output given that a 1 was transmitted, is given by
pXiIAi(XlIOZ1)
(5.23)
Tr(/S1iXXi)I
=
exp
rj)
([-gg-2
27
i -.
2(1 [1 -,q+
(-
(5.24)
(7-ae2
7e-2r
Because the channel noise is additive and Gaussian, the mutual information is maximized for a Gaussian input density (5.13), and the SM capacity is given by
C(7, i, r) = -log 2
2
1+
1 1-77+ e-2r
4
(5.25)
.sinh2r
r7
A further maximization with respect to the squeezing parameter r yields the SM
capacity (in bits per use):
Csqueezed-homodyne(77,
ii) =
-log
2
2
+ 2
1+
(1
(x(77, h) + x(77, f))
+ X(, h) 1
,
(5.26)
where the function x(ij, h) is given by
X(,
5.2.2
)=
7
+ 2h)12 -1
(I+
.(5.27)
Wideband Capacity
It is evident from the SM formula for the capacity of the squeezed state channel that
calculation of the wideband capacity analytically is very difficult. So, to compute
the wideband capacity of the squeezed state channel, we perform the constrained
maximization problem numerically using the Lagrange multiplier method.
We illustrate one example of our numerical calculation of squeezed state capacity
74
l17Ul
0
~1
U
Ut
.0
N
N
U
.0
U
Coherent-Homodyne
Cohrent-Hetrodyne-- Squeezed-Homodyne - - - - -
I
/
Q
1I 5
i
. . . . . .. .
.
. . . . . ..
i
10-2
1-4
134
10
P (Input power in Watts)
Figure 5-2: An example of ultra wideband capacities for squeezed state channel and
coherent state encoding with homodyne or heterodyne detection, for the free-space
optical channel under the assumption of a single spatial mode transmission.
to show how it typically compares with the coherent state capacities. This example
has been done for the ultra wideband single-spatial-mode capacity for the free-space
channel parameters we used in Fig. 4-8. As expected, squeezed state capacity is
consistently higher than the coherent state homodyne detection capacity, and heterodyne detection eventually outperforms both the homodyne detection schemes in the
bandwidth-limited (high-power) regime.
75
Chapter 6
The Ultimate Classical Capacity of
the Lossy Channel
In the previous chapters, we calculated and compared classical information capacities
of the single mode and wideband lossy bosonic channel, using various combinations
of encoding schemes and receiver structures. In this chapter, we discuss very recent
progress made by collaborative efforts of several people in, and associated with the
research group of Prof. Jeffrey H. Shapiro, at RLE, MIT [18], in the direction of
finding the ultimate information capacity (2.17) of the lossy bosonic channel. We will
perform the calculation of ultimate wideband capacity for the free-space channel in
the far-field, as an example. We also show that if certain conditions are satisfied, one
can achieve the ultimate wideband capacity of a lossy channel asymptotically in the
high power regime, using coherent states and heterodyne detection.
We stated earlier that the ultimate classical capacity of a quantum channel (2.17)
is the maximum of the 'Holevo Information' at the output of the channel, maximized
over all possible sets of input states and apriori input probability distributions. Also,
we mentioned that the ultimate classical capacities of the SM and the wideband
lossless channels are known [15], and that a number-state encoding along with a
direct detection receiver can be used to achieve these capacities (3.3).
76
6.1
Ultimate Capacity: SM and Wideband Lossy
Channels
Recently, it was shown that the ultimate classical capacity can be explicitly calculated
for the T = 0 SM and wideband lossy bosonic channels [18].
The SM capacity of a
lossy channel with transmissivity 71 is given by
Cuitimate(7, ii) = g("i),
(6.1)
where g(.) is as defined in (3.2).
To evaluate the wideband capacity for a specific case, one has to perform the
maximization
C = max
where
mi is
g(lifii),
(6.2)
the transmissivity of the ith mode, and the maximum is evaluated subject
to the usual average input power constraint (2.20). It can also be argued that this
capacity can be achieved using a single use of the channel by using a random code
on coherent states, factored over frequency modes [18].
This means that for this
channel neither non-classical states for encoding, nor entangled codewords (either
entangled over successive channel uses, or entangled over modes) are necessary to
achieve capacity. It might be possible, however, that one could achieve capacity using
quantum encodings, and that such encodings might have lower error probabilities for
finite length block codes than those of the capacity-achieving coherent state encoding.
The wideband capacity for a frequency independent modal transmissivity q, is given
by
c.
Cultimate
-1
1n 2
2nP(63
3h.
(6.3)
Now that we know the lossy channel capacities for so many cases, including the
ultimate classical capacity, it is a good time to put them all all together in one
77
3210
Coherent-..Homodyne
Coherent--Heterodyne
2DO -Squeezed--- Homodyne
Ultimate Capacity
Number-State.--Direct-Detection
---
-
-
-
-
.
iaoo 1200
Q
0.600
0000
3000
6000
9000
12000
150O0
180O
Input average photon number constraint (nbar)
Figure 6-1: Single Mode Capacities (in bits per use of the channel) for various cases.
The transmissivity of the SM channel q = 0.16.
place and compare. In Fig. 6-1, we plot all the SM capacities of the lossy bosonic
channel with transmissivity 7 = 0.16. All the plots are evaluations of the respective
analytic SM capacity expressions as functions of the average input photon number
A, except the plot corresponding to the case of number-state and direct detection,
which has been obtained by using a numerical Blahut-Arimito procedure. We notice
that as the transmissivity is so low, homodyne detection with optimized squeezed
states performs only marginally better than coherent states and homodyne detection,
though eventually at high photon numbers, coherent states and heterodyne detection
outperform both of them. The solid (green) line in the figure corresponds to the SM
ultimate capacity of channel, given by g(0.16A).
6.2
Free-Space Channel: Far-Field Propagation
In this section, we will consider the calculation of wideband capacity of the free-space
optical communication channel in the far-field regime. Interestingly it turns out that
78
exactly the same normalization of capacity and power used in the far-field analysis
of the coherent state channel can be used for the ultimate capacity.
Starting our wideband analysis with a frequency axis divided up into small bins of
2
size b Hz each, we denote the transmissivity of the ith frequency bin as 77i = Afi , where
ATAR/C 2L 2 is a geometrical parameter of the channel.
A
K
=
We define a constant
h/A which has dimensions of power. On carrying out the analysis along similar
lines as the coherent state heterodyne detection case, the optimal number allocation
is found to be
n.*
2
=Afi
64
(6.4)
(exp (A In 2)K)
Aff2
where A is the Lagrange multiplier. The superscript (*) signifies that this is the optimum distribution of mean photon numbers across all frequencies. Also, the optimal
allocation of intensity (energy per mode) across the frequencies is given by
i = hfish*
=
(.
fiexp
(6.5)
-A
In2)
Let us define a new parameter lo = (1/A In 2) which we will treat as a Lagrange
multiplier.
If we now let our frequency bin width b become very small, so that
we can treat the frequency
f
as a continuous parameter, the optimal mean number
distribution W*(f), and the optimal intensity allocation 1(f) are respectively given
by:
=
n(f)
exp(_K_)
expk
"
,
-
if > 0
(6.6)
otherwise
0,
and
K
I(f)
=
,
0,
Note that
1(f) > 0 +-l
if > 0
(6.7)
KIf-1
otherwise
Io > 0. Also note that I(f) -- + o, as
79
f
-+
oc. So, to keep
this far-field analysis valid, we must impose an upper frequency cutoff fmax that satisfies the far-field condition Dmax
< 1. Eqn. (6.7) suggests that we normalize all the
quantities with respect to fmax, the maximum allowed frequency of our transmission
band. Let us define
I~f~fmx
IYf
where
f' = f/fmax, and Io' =
emoi'_1
e /07
Kf)fa
K
(0,
_
,if
> 06.
otherwise
68
Iofmax/K. Let us define a dimensionless (normalized)
power P' = P/K, where P is the total average input power in Watts, and a normalized
capacity C' = C/f
m a,
where C is the wideband capacity of the channel in bits/sec.
It is now straightforward to show that the normalized Capacity C' is given by
In
+
C'ultimate(P') = ln
,o,
df1 ,
(6.9)
where the parameter 10' is computed from the power constraint equation
=
J
elIo 4'' -
df'
(6.10)
The normalized capacity C' = C/fmax has been plotted vs. the normalized power
P' = P/K, along with the rates achievable using coherent-state encoding with heterodyne and homodyne detection receivers, in Fig. 6-2. It is observed that the classical
capacity of the far-field optical channel is proportional to the maximum frequency of
transmission
fmax. Again, as in the coherent-state channel, this does not imply a pos-
sibility of infinite capacity because this result depends on the single-spatial-mode approximation and the quadratic frequency dependence of modal transmissivity, which
requires that the highest frequency photon resides sufficiently deep into the far-field
(Dm.
< 1).
80
10- .............
................
...
8M
E
z0
00
250
500
750
1000
Normalized Power, P'
Figure 6-2: Dependence of C' on P' for the ultimate classical capacity compared to
C' for coherent-state inputs with heterodyne and homodyne detection. All curves
assume I = 0. The blue (solid) line is the ultimate classical capacity, the red
(dotted) line is for heterodyne detection, and the green (dashed) line is for homodyne
detection.
6.3
Asymptotic Optimality of Heterodyne Detection
Interestingly, it is observed in some of the cases we have considered so far, that the
performance of coherent-state encoding with heterodyne detection asymptotically approaches the optimal capacity in the limit of large input power P. In this context, we
will first consider the free-space channel in the far-field regime and show analytically
that one can achieve ultimate capacity for this channel using coherent states and heterodyne detection in the limit of high input power. After that, we will demonstrate
a sufficient condition for this to hold true for any general wideband bosonic channel.
81
6.3.1
Heterodyne Detection Optimality: Free-Space Far-Field
Propagation
Using the normalization from the previous section, it can be shown that the normalized capacity for coherent state encoding and heterodyne detection, C' is given
by
C'=
in2
inIs -1 +
'01
(6.11)
,
where I is a dimensionless parameter obtained from the power constraint equation,
that takes the following form:
P' = IO/-
1 - In I/.
(6.12)
Now, let us compare the performance of the coherent state heterodyne detection
case with the ultimate capacity, in the high power regime. For simplicity, we will only
consider the case of
fmi
=
0.
In the ultimate capacity case, from Eqn. (6.10), we infer that high values of P'
will be obtained by high values of the parameter 1o'. For 10'
P'/
- (o'f') df'.
fo P
-
oc,
(6.13)
So, for large values of Io', Io' ~ P'. We see from Eqn. (6.12) that the same holds
for coherent state - heterodyne detection case as well (as in Io' and 1 are negligible in
comparison to Io' for large 1o'. So, for large values of normalized input power P',
82
C'ultimate (P)
of+
C'coherent-heterodyne (P)
f In
in I0 - 1 +
df
1 + f ln (Io'f') df'
In Io' - 11111j'-i(6.15)
1 + (In Io' - 1)
In lo' - 1
ln(P')
ln(P') - 1
1.
.4)
(-5
(6.16)
(6.18)
From the above, and the definitions of normalized capacity and normalized power,
we conclude that as P -- 00,
ln(P/K)
ln(P/K) - 1
Cuitimate(P)
Ccoherent -heterodyne(P)
-
1.
(6.19)
So, unlike the case of the lossless channel and the case of frequency independent
transmissivity, in which cases the above ratio is found to be
ir/v/5 independent
of the
input power P, we see that in the case of the far-field free-space channel coherent state
encoding and heterodyne detection is indeed asymptotically optimal, in the limit of
large input power.
6.3.2
A Sufficient Condition for Asymptotic Optimality of
Coherent State Encoding and Heterodyne Detection
We saw that in the case of frequency independent loss the capacity achieved by coherent state encoding and heterodyne detection does not approach the ultimate capacity
in the high power limit, whereas it does in the case of the far-field free-space channel.
One important difference in these two cases is that the optimal intensity distribution
across frequencies I(f), goes to zero at high frequencies for the frequency independent
loss, whereas 1(f) -+ Io as f -- oc for the far-field free-space channel. Because of this
behavior of the optimal intensity distribution for the free-space channel, we have to
83
impose an upper cut-off frequency fma, in our capacity calculation (In a real optical
communication system, the upper cut-off frequency will be decided by the available
hardware and other resources).
As the total input power P -+ 00, the optimum mean photon number at every
frequency W*(f, P) -*
oc.
Note that one can obtain
*(f, P) by substituting the
Lagrange multiplier A in the expression for h*(f,A) in terms of the input power P
by solving the power constraint equation.
The introduction of the upper cut-off
frequency causes the above convergence to become 'uniform'. In the absence of an
upper cut-off frequency, however high the power level P might be, there wouldn't
be any single N, such that h*(f, P) > N, Vf E (0, oo), which is the condition of
uniform convergence.
It can be rigorously shown that, for any arbitrary frequency
dependence of modal transmissivity, a uniform convergence condition can be set up as
a sufficient condition for the asymptotic optimality of heterodyne detection. Consider
the following lemma [19]:
Lemma 1 (SGY) For wideband quantum communication with an arbitraryfrequency
dependence of modal transmissivity r(f), and a given frequency band for communica-
tion F
(fmin, fm.),
if the ratio of optimum mean photon numbers for coherent-state
heterodyne detection case hi*e(f, P) to that of ultimate capacity case h*lt(f, P), uniformly converges to unity as P
as P ->
--
oc, and *l(
f, P) uniformly converges to infinity
oc, then coherent state encoding with heterodyne detection asymptotically
approaches optimum performance in the limit of high average input power P. In
particular,if
hiet(f, P) uniformly 1,VfEY
ilt(f, P)
as P -> o,
(6.20)
and
(f
*
P uniformly
, F)
fE
oo,Vf E
-
as P -+ 00, then given any e > 0, 3Po, s.t. VP > PO,
84
(6.21)
1 < Chet(P) < 1 + C.
Chet(P)-
Proof -
(6.22)
The left hand inequality of Eqn. (6.22) has already been proved in [18].
Here, we will prove the right-hand side of the inequality. Given e > 0, we want to
show that there exists a Po s.t. VP > PO, the right hand side inequality of Eqn. (6.22)
is satisfied.
Choose any 6 > 0. Eqn. (6.20) implies that, given any 6 > 0, ]P*, s.t. VP > P*
and Vf E F,
1-6 <
het' P) <1 + 6.
nUlt (f, P)
(6.23)
Define
-
log(1 + (1 - J)Wu;t (f,P*))
log(1 + h*lt(f, P*))
As log(1 + x) is an increasing function of x for x > 0, E"(6, f) > 0, Vf E F and for
all 6 > 0. Define a function
f (x) = log(1 + (1 - 6)x) - (1 - E"(6, f)) log(1 + X).
(6.25)
It is easy to see that h*It (f,P*) is a zero of f(x), and that f(x) is an increasing
function of x, for all E"(6, f) > 0. Now, we use Eqn. (6.21) to conclude that given
any N however large, ]P**, s.t.
N=
P > P** => i4(f,
P) > N, Vf E F.
Choose
*It (f,P*). We conclude from Eqn. (6.24), and the fact that f(x) is an increasing
function of x, that VP > P**,
log(1 + (1 - 6)h*t(f, P)) > (1 - e"(6, f)) log(1 + h*t(f, P)).
(6.26)
Let us define
E"(6) = sup (E"(6, f)),
f
85
(6.27)
so that we may now rewrite Eqn. (6.26) as, VP > P**,
log(1 + (1 - 6)*it(f, P)) > (1 - E"(6)) log(1 + init(f, P)).
(6.28)
We know from the previous sections that:
Cult(P)
_
Chet (P)
g9(h*It(f,Ip)) df
fm
(6.29)
f f*na log (1 + haet (f, p)) d
From Eqn. (6.29) and Eqn. (6.23), and the fact that log(1 + x) is an increasing
function of x, we conclude that for all P > P*,
g (h*It(f P))df
<Cuit(P)
(f, P))df Chet(P)
+ 6)i*1l(fIP)d
fmax
Imin log(1 + (1
ffax
ff
-
g (f*It (f,P))df
nmmn
fj-x log(1 + (1
-
6)It(f, P))df
(6.30)
Next, we define
P*** = max(P*, P**)
(6.31)
Using Eqn. (6.28) and the right hand inequality of Eqn. (6.30), we conclude that for
all P > P***,
Cult (P)fX(i*(f,
Chet (P) ~ (1 -
P))df
log (1 +i*I(f,P))df
I"(6)) ff"a
We know that the function g(x)/ log(x) -
(6.32)
1 from above, as x -+ oo. So, given any
E',7 x*, s.t. VX > X*
<
-
g
log(1+ X)
< 1 + 6'.
(6.33)
Choose c' = (1 + e)(1 - E"(6)) - 1, and denote the corresponding x* = M. From
the uniform convergence of i*l(f, P) (Eqn. (6.21)), we can say that given this M,
P****, Is.t. VP > P****, in*(f, P) > M. Therefore VP > P****,
86
1<
'~P))
f
-< 1 + [(1 + W)( - E"(6)) - 11 .
(6.34)
Po = max(P****, P***)
(6.35)
-log(1 + i1*1t(fI P))
Finally, we define
so that for all P > Po and Vf E T,
P)) ; [log(1 + n*t(f, P))] (1 + f)(1 - E"(6)).
g(A1*a(f,
Substituting Eqn. (6.36) into Eqn. (6.32), and using the fact that Cult(P)
(6.36)
Chet(P)
[18], we conclude that
< Cuit(P) < 1 +
Chet(P) -
which is the statement we set out to prove.
87
(6.37)
Chapter 7
Capacity of a Lossy Channel with
Thermal Noise
In the previous chapters, we observed that free-space transmission of a single spatial
mode of the bosonic field through circular apertures can be treated as an interaction
of the input field mode & with an environment mode b through a beam splitter of
transmissivity 1, where q is a function of some geometrical parameters and the center
frequency
f,
of the carrier (2.1).
In the absence of any extraneous noise sources
other than pure loss, the environment mode 6 is in the vacuum state, which is the
minimum 'noise' required by quantum mechanics to preserve correct commutator
brackets of annihilation operators of the output modes (or in other words, to satisfy
the Heisenberg uncertainty relation at the output). Different types of noise can be
described by different initial states of the environment mode 6. In this chapter, we will
define a thermal noise lossy bosonic channel, which describes the effect of coupling
the lossy channel to a thermal reservoir at temperature T [7]. The ultimate classical
capacity for this channel is not known yet, though there have been some very recent
advances done in this direction by several people in, and associated with the research
group of Prof. Jeffrey H. Shapiro, at RLE, MIT. This chapter will summarize some of
that work. We will also show that wideband and SM capacities of the thermal noise
lossy bosonic channel can be calculated explicitly (either analytically or numerically)
for all the quantum encodings and receiver structures considered earlier in this thesis.
88
Background
7.1
A thermal noise lossy bosonic channel at temperature T > 0 (referred to as the
thermal noise channel from now on to save on nomenclature), is defined to be a single
mode (SM) bosonic channel (2.1), in which the environment mode 6 is excited in a
zero-mean isotropic Gaussian state
PT -- 7FN
pT,
given by the density matrix:
a) (aId 2 a,
exp ( _IN
(7.1)
where 1a) is a coherent state. N is the mean photon number of the thermal state PT,
given by the Planck's blackbody formula
N
= Tr~p
) =
ehf/kBT
-1
where h and kB are the Planck's constant and the Boltzmann's constant respectively,
f
and
is the frequency of excitation of the mode.
The thermal state PT has the
diagonal representation:
1
In)(nI,
PT = IV,_
(7.3)
where In) are the eigenvectors of the number operator btb, corresponding to eigenvalues n = 0,1
.It
is interesting to note that this channel can be further decomposed
into a lossy bosonic channel without thermal noise (2.1), followed by what we define
below as a classical Gaussian noise channel D(1@)_
(see Fig. 7-1).
A classical Gaussian noise channel is characterized by the completely positive
(CP) trace preserving map
(N(P)
1
?N which transforms an input state 3 into the state
given by the expression:
4N(p)
=1
D(a) D(a)d2G,
7rN
where
89
(7.4)
b(p,,N)
Environment (in thermal state)
a
c=J)&+ A-27
Output
Input
77
(a) Thermal Noise Channel (with loss)
III
Environment (in vacuum state)
a
da
Input
(b) Pure lossy channel
+
D(-7;
C
Classical Gaussian noise channel
Figure 7-1: The thermal noise channel can be decomposed into a lossy bosonic channel
without thermal noise (2.1), followed by a classical Gaussian noise channel <D(_,)
(7.4).
D(ce) = exp(atd - a*&)
(7.5)
is the unitary displacement operator in the phase space. This thermal noise model
can be applied to our free-space quantum optical channel, now with a background
blackbody noise source at thermal equilibrium at temperature T. The thermal noise
channel as explained above can also be used to describe single mode (SM) quantum
optical communication through a lossy optical fiber or a waveguide in the presence
of thermal noise.
In the power constrained wideband version of the thermal noise channel, a much
simpler version of the above model occurs in the case of 'white thermal noise', in which
we have one fixed value of mean number of thermal photons N for all frequencies of
transmission. This case, though not too realistic, is easier to handle analytically for
capacity calculations.
Additional N(f) possibilities might also be considered.
For
example, the noise introduced by an optical amplifier at any particular frequency
will have density operator of the form (7.1), but its average photon number -
90
as a
function of frequency - will not follow Planck's law, nor will it be a constant.
7.2
Ultimate Classical Capacity: Recent Advances
Consider a SM lossy thermal noise channel of transmissivity q, at temperature T. Let
there be a mean photon number constraint of i! at the input. Let us denote the mean
photon number of the thermal environment by N, which is given by Eqn.(7.2). Let us
denote the completely positive (CP) map describing the action of this thermal noise
channel on an input state, by EN. Also, we use {}
and {p} to denote the set of
input states forming the input alphabet of the channel, and their apriori probability
distribution respectively. It was shown earlier this year [7], that the capacity of this
channel C(r, ii, N), can be lower-bounded by
C(7, ii, N)
g(nii + (1 - T)N) - g((1 - 77)N).
(7.6)
Proceeding along similar lines to the ultimate capacity proof in [18], one can
upper-bound the SM capacity as shown below.
C(q
,f,
N)
=max
[S(EN(ZPi))
[pIn1
<
33
max S(&SN(pjp))
-
(7.7)
ZPjS(,E(N))
[i
- min [S(zE(
71
P j
(78
(7.9)
where all the above maximizations are done under the mean input photon number
constraint h = E pTr(Wdf&p). As the thermal environment state P/T is a zeromean state, the mean photon number at the output of the SM channel is given by
rqih + (1 - q)N. As all the output states of this channel will have this mean photon
number, the first term on the right-hand side of the last inequality (7.9) can be further
upper-bounded by max,3 S(p), with the constraint Tr(dt&3) < "n + (1 - 77)N. But,
91
we know that the maximum entropy subject to a mean number constraint fl, is given
by g(ii). So, we now have:
C(, ii, N) < g(jn + (1 - i)N) - min [S(EN())].
(7.10)
The next thing we need to do is to find the second term in Eqn. (7.9), i.e. the
minimum output entropy of a SM thermal noise channel. We stated in the previous
section that the thermal noise channel EN can be decomposed into a composition
of two channels, viz.
F N
where S' represents the SM lossy channel
- 4 (1-9)N
without thermal noise, and I(1-n)N is the classical Gaussian noise channel defined in
the previous section. The minimum output entropy of S1 is zero, because any coherent
state at the input produces a coherent state at the output, which has zero entropy by
virtue of it being a pure state. In particular, the vacuum state input 10) sent through
I
produces a vacuum state at the output, which clearly has zero entropy. Also, the
minimum output entropy of ((1-,)N
is less than or equal to that of £,, because in
general there need not exist any input state to S that would produce at the output
of 6', the minimum output entropy achieving input state for the classical channel
4
(1-o)N
(see Fig. 7-1). Thus, we have
min [S(E,())] > min [S(Q(1-7)N(5))] ,
(7.11)
which along with (7.10), implies
C( 1 , h, N)
g(7ii + (1 - 71)N) - min [S(<P(1-n)N())]
(7-12)
Clearly, when they act on the vacuum input state 10), both the channels
4I(1-q)N produce the thermal output state of mean photon number N, given by
92
and
SEN (10)(01)
(7.13)
4(1-n)N 1)(1
=
I
' (
(1-r/I)N +1
Y--
S)
(1 - ))N(IO)n)(nO
(7.14)
(1-r/I)N+1I
exp
(1
2a.
-a)(ad
(7.15)
A coherent input state Ia) produces a displaced version of the above thermal state,
whose mean complex amplitude is equal to ,/a,
from that of the vacuum-state input, i.e. g((1 that the minimum output entropy of
D(1-,)N
but whose entropy is unchanged
7 )N).
So, if we are able to show
is achieved by a coherent state input,
that would imply that the minimum output entropies of both these channels are
the same, and given by g((1 - 7)N). This argument follows from the decomposition
1,
= D(1-,)N
o , , and the fact that any coherent state can be obtained at the output
of S by a suitable coherent state input.
From the concavity of Von Neumann entropy S(), we may infer that the minimum
output entropy of both these channels can be achieved by pure input states. Also, it
can be shown that displacing a state in the phase plane does not change its output
entropy. Hence, we may limit our search (for input states that yield the minimum
output entropy) to zero-mean pure states. The purpose of introduction of the classical
Gaussian noise channel
'1 N
was that this channel looks much simpler than the general
SM thermal noise channel F, and to find the minimum output entropy of E
suffices to prove that the minimum output entropy of
'I(1-7)N
,
it
is achieved by coherent
state inputs.
Conjecture 1 (GGLMSYY)
The minimum output entropy of the SM thermal noise
channel SN is achieved by coherent input states, and hence is given by
min [S(EN())]
g ((1 - r)N)
(7.16)
Though the above conjecture has not been proved yet, several partial results have
been derived. If this conjecture is proved to be true, then from Equations (7.9) and
93
(7.6), we would be able to conclude that the bits per use capacity of the SM thermal
noise channel 6', is equal to g(pi + (1 - q)N) - g((1 - q)N).
Some intuition behind the preceding conjecture can be explained as follows. If
the input to the thermal noise channel is an isotropic state (i.e. circularly symmetric
in the phase plane), then the output is also an isotropic state, the reason being that
the anti-normally ordered characteristic function of the output of a beam-splitter for
uncorrelated input states can be decomposed into the product of the anti-normally
ordered characteristic functions of the input states; and the product of two isotropic
functions is isotropic as well. It can be shown that if we limit ourselves to zero-mean
Gaussian input states, the one that achieves minimum output entropy is the vacuum
state, which clearly is isotropic. So it seems that one might expect the minimum
output entropy of S' to be achieved for an isotropic state at the input. If this belief
is true, then we have numerical evidence to show that in the class of the all isotropic
input states, the vacuum state has the least output entropy. We stated earlier that
from the concavity of Von Neumann entropy, we could conclude that the minimum
output entropy of the channel can be achieved by pure input states. It can be shown
analytically, that the number states, which include the vacuum state, are the only
states which are pure as well as isotropic. Also, we know that if one state majorizes
another state, its entropy is smaller, i.e.
,1 >- P2 =4
S(
2
) < S(
2
). So, if we can show
that the output state corresponding to a 'lower' number state majorizes the output
state corresponding to a 'higher' one, then we would be able to conclude that among
all number states, the vacuum state 10) has the least output entropy; and hence would
be able to assert that, in the class of all isotropic states, the vacuum state input yields
the least output entropy, which we know is equal to g((1 -q)N).
7.2.1
Number States: Majorization Result
In this sub-section, we will try to demonstrate the assertion we made towards the end
of the last section regarding the majorization of output states corresponding to input
number states.
94
Conjecture 2 Given two photon number states In1)(nil and |n 2 )(n 21, n 1 < n 2 ->
E N(Ini)(nil)
>- S,'(1n2)(n2l)
In the section on the capacity of the thermal noise channel
Evidence (Numerical)-
with number state inputs and direct detection (later in this chapter), we show that
the output of the thermal noise channel for an input number state In) (nI is given by
sN(
n)(n1)
Pn,mIm) (ml,
=
(7.17)
m=O
where the coefficients Pn,m are given by
m + n\ (1
Pl
m
m
=
2
)m+n(l + N)nN m
(1 + (1 - r7)N)m+n+l
--
m-n; -(i
x) F[_
7(1
with F[a, 3; -y; z] -
F(a, ;;z)
2 F 1 (G,
= 1+
+ n), (N -
(1l + N))(1 + (1 - r)N)1
- 77)2N(1 + N)
(718)
3; -y; z) being the hypergeometric function
7.-1
z + a(a+1)0(0 +1)z2
-y(- + 1) - 2!
a(& + 1)(a + 2)/3(3 + 1)(0 + 2) 3 +
+(y + 1)(7 + 2) -3!
(7.19)
From the definition of majorizationwe have,
m
m
if
Ipni,k
E Pn2,k,
k=O
k=O
Then, S N(Ini)(nil) >In Fig. 7-2, we plot EM_ 0
,
Vm - [0,Oo3,
SN(In2)(n2l).
(7.20)
as functions of m for several values of n, for a
thermal noise channel with r/ = 0.8 and N = 5 photons. As expected, we find that
EkAO
Pnk
-
1, as m -*
oc, because Pn,k's are probability distributions. More-
over, we observe that as n increases, the curves shift downwards. Thus, at any given
95
I1860
(r
DEEM)
-
-
I/
0600 -
/
/
/
*1*
I
/
I
I
I
/
I
k 0
OAE4
02M0
I
I#
I
//
/
n90
2'
0Da00.00
2000
4000
80
60M10
100A
M
Figure 7-2: A plot of E'U 0 Pn,k for a SM thermal noise channel with transmissivity
The dark
r/ = 0.8, and N = 5 photons, as a function of m; for n = 0, 6, 12, 18,.
(black) lines correspond to the values n = 0, 30, 60, 90.
value of m,
_
e Pn,k decreases with increasing n, supporting our majorization con-
jecture. Given the nature of the expression for the eigenvalues pn,m of the output
states, it seems difficult to construct a general analytical proof of the majorization
conjecture. However, numerous calculations with different values of transmissivity
and mean thermal photon numbers support our conjecture.
As a result of our numerical work, we deem it highly likely that output states
corresponding to 'smaller' number-state inputs majorize (and thus have smaller values
of Von Neumann entropy than) the output states for 'larger' number-state inputs (Fig.
7-3). Hence, we assert from our earlier discussion that -
in the class of all isotropic
input states, the vacuum state input yields the least output entropy for the SM channel
E'
which we know to be equal to g((1 - r)N).
96
0.
-
-21=,00=
1100 S-n=4,
0.300
,N=5
-
n=3, N=5---N=5 -------
0.200
----
-n
00
OAOO
.0800
1.000
eta (transmissivity)
Figure 7-3: A plot of output entropies of the SM thermal noise channel as a function
of q for number states inputs In)(n1, corresponding to n = 0, 1, 2, 3 and 4. The value
of N = 5. We see that output entropy is minimum for the vacuum state 10) (01, which
is shown in the figure by the dark black line. The results for n > 0 were obtained
by numerical evaluation of the p.,m from (7.18); the n = 0 entropy result so obtained
matches the analytic result g((1 - 7)N).
97
7.3
Capacity using known Transmitter-Receiver Structures
We saw in the last section that the ultimate classical capacity of the lossy thermal
channel is as yet an unsolved problem. Nevertheless, we have intuitive reasons and
evidence to believe that the SM capacity of the thermal noise channel is given by
CQft, N) = g(rpi + (1 - q)N) - g((1 - q)N).
In this section, we will evaluate
capacities of the thermal noise channel E,N that can be achieved by some of the
standard encoding schemes and receiver structures.
7.3.1
Number States and Direct Detection
Let us transmit a pure number state
AIN
=
In)(n
with probability pN(n) through
a SM lossy thermal noise channel with transmissivity q and thermal mean photon
number N. Let us denote the channel by the Heisenberg evolution (see Fig. 7-1(a)):
6 = AT& + /1 -ab
(7.21)
where &and 6 are the annihilation operators of the input and output modes respectively, and b denotes the thermal environment. The initial states of the input modes
are given by
pa
b
=
In)(n
00 Nn
+ 1)n+1 In)bb(nI
n=O (N
(7.22)
(7.23)
The first step in finding the capacity is to find the output state corresponding to
the input state In) (nI and express it in the number state basis. Using this representation, we can read off the transition probabilities which can be plugged into a Blahut
Arimoto algorithm routine to obtain the optimum input distribution PN(n) and the
capacity.
98
We can find the anti-normally ordered characteristic function of the output state
as follows [22]:
(
* )
=
(e -C*eC
(7.24)
)
/hae 5 Vat) (e-* \beC
=
(e-*
=
xi(VP
=
(e~KI(n e
=
(e-71K2
*,V()x(
1-*,
C( e -VC(* l))
(7.25)
(v/FZ-Ibt)
(x (
(7.26)
-1 ()
1 - g(*, /1 -
(nIleC te-NvC*arln)) (e-(1+N)(12(1
(7.27)
(7.28)
-)
where we have used the fact that the input to the channel is independent of the
thermal environment. In the penultimate step above, we have expressed the antinormally ordered characteristic function of a in terms of its normally ordered form.
Also, it can be shown easily, that the anti-normally ordered characteristic function
of a thermal state with mean photon number N is given by e-(1+N)1|2 . To calculate
(nIe 0a7
e-vC*61n), we express the exponent operator in its Taylor series, and use
the 'annihilation' property of d, to obtain
(njeOI 6t e-01 *61
(n- kI|/)))
S(
2)k
k=O
(/-1(*)k
n | n - k)
(7.29)
(7.30)
! ()
(7.31)
=Cn(- 1(12),7
where the Cn(x) are the Laguerre polynomials. Now, we find the output state by the
operator inverse Fourier transform.
99
=
(7.32)
((*, ()e--C eC*6
X
J
(
_-(I+N)1(12(l -'),Cn
7Ce- 771(2
)-(
_ q(2
te(*
(7.33)
Because the input states are diagonal in the number state basis (7.22) (7.23), it
turns out that the output state of the channel is also diagonal in the number state
basis, as can easily be verified by evaluating (mifIm') for m
$
m' by doing the
integration in polar coordinates. The mh diagonal matrix element of fc is given by
(mI,3cIm)
=
(M~firIir2f7rr
=-
=
J
d 2x
- N-y)(|
(7.34)
TI~2'mjj)(-4
0-0+-1)(2L
re-(1+N-N)r2
]d]
e-(1+N-N)x
2(,r
2)dr
Cn(7x)2m(x)dx.
Note that if we substitute 77 = 1 into the above expression, we get (mIcIm)
(7.35)
(7-36)
= 6mn,
by virtue of the orthonormality relation for the Laguerre polynomials, which complies
with the lossless channel. The lossy-case integral can be evaluated explicitly in terms
of the Hypergeometric function [10]. The final expression for (mfi.c1m) is given by
M
m + n (1- )m+n(1 + N)"N
( m ) (1 + (1 - i)N)m+n+1
x F [m, -ni; -(in + n, (N - 7 (1 + N))(1 + (1 - 7 )N)]
(1 - -) 2 N(1 + N)
(737)
Thus, the output state of the channel is given by
00
fc =
Z (mjicIm)Im)(mT,
(7.38)
m=0
hence, the transition probabilities (2.11) for a single use of the channel, when a unity
100
efficiency direct detection receiver is used at the output, can be written as
PMIN~mn
c
-Tll)
(7.39)
To get the SM capacity (in bits per use) of the thermal noise channel using number
states and direct detection, we have to maximize the classical mutual information
I(M; N) (2.12) over all input probability distributions PN(n) subject to a fixed value
A of mean photon number at the input. This constrained maximization problem can
be solved numerically using the Blahut Arimoto (BA) Algorithm [16] (See Appendix
A for the details of the algorithm). Interestingly, on carrying out the BA algorithm
[16], we find that the capacity achieving probability distributions have multiple peaks
similar to the pure lossy case (See Fig. 7-4). This again suggests that to achieve the
best capacity using number state inputs one would almost preferentially use a set of
optimum numbers of photons in each channel use. In the pure lossy case, when the
transmissivity of the channel is increased towards unity, the peaks disappear and the
optimum distribution converges to the exponential Bose-Einstein distribution (linear
on a log scale). We have numerically evaluated the optimum distributions for the pure
lossy case (,q
-
0.8, A
=
3.7), and the thermal noise channel (,q = 0.8, A = 3.7, N = 5).
It can be seen from plots of these distributions (Fig. 7-4), that in the pure lossy case
there are no peaks at this value of transmissivity and the distribution is close to linear,
whereas for the thermal noise lossy channel the distribution still has multiple peaks.
For the optimum distribution of the thermal noise channel to converge sufficiently
close to the Bose-Einstein distribution,
T1
has to be even closer to unity.
To obtain the SM capacity of the thermal noise channel, we have to execute the
BA algorithm multiple times for different h constraints. A sample result is shown in
Fig. 7-5 for 77 = 0.8, and N = 5 photons.
101
0300
I
Lossy chann e(eta=0.8) --no thennal noise
Lossy chann 1 (eta=0.8) -- with thermal noise (N-5)
-101300
-
-20N
-
--
a4
~30.00-
N
i
N
0300
10a00
20300
30.000
50.000
40.001
A.
:
e
e
e
-40O
60.000
70.300
-I
1
9020W
801300
Number of transmitted photons (n)
Figure 7-4: A plot of optimum apriori probability distributions (in dB) for a pure
SM lossy channel (q = 0.8, h = 3.7), and a SM thermal noise channel (q = 0.8,ii =
3.7, N = 5). In the pure lossy case, there are no peaks at this value of transmissivity
and the distribution is close to linear, whereas for the thermal noise lossy channel,
the distribution still has multiple peaks. For the optimum distribution of the thermal
noise channel to converge sufficiently close to the Bose-Einstein distribution, n has to
be even closer to unity.
IA41N
1.200
i
-
1100 -
0.
0.300 -
I
0.600 0A0
-
0.200
-
'
0.000
0.000
21300
41300
61000
8100
101100
1
12AM0
Mean input photon number (nbar)
Figure 7-5: A sample calculation of the number-state direct-detection capacity (using
the Blahut-Arimoto algorithm) of a SM lossy thermal noise channel with transmissivity
i
= 0.8, and mean thermal photon number N = 5.
102
7.3.2
Coherent States and Homodyne or Heterodyne Detection
To evaluate the capacities of the thermal noise channel with coherent state inputs, it
is useful to think in terms of the decomposition of the thermal noise channel E' into
a pure lossy channel g followed by a classical Gaussian noise channel
7-1). If a coherent state
Ia)(al
is input to the classical channel
4
N,
'I(1-n)N
(Fig.
the output is a
displaced thermal state given by
exp
1
7r N
1- N)
I!)(O3d
N
2
0.
(7.40)
It can be easily shown that if heterodyne detection is done on a displaced thermal
given by (7.40), where a = a 1 +ja 2, the output is a pair of Gaussian random variables
with mean and covariance function given by
ail
J
)(aj) ~N(
-- Heterodyne -
N+1
((a2
With the help of the decomposition
gN
S
0
,(2
0
0(1-q)N,
(7.41)
2-
and the above result,
(7.41), we can use reasoning similar to the one we gave in Chapter 4, to evaluate the
SM heterodyne detection capacity in bits per channel use, with the following result:
Ccoherent
-heterodyne(n ,if, N)
=
2
=
log2
log 2
1
+
1
+
(l
7/2
(7.42)
l).
(7.43)
(1 - 71)N + 1
The homodyne capacity can be computed using arguments similar to the ones in
Chapter 4, along with the fact that the quadrature variance of homodyne measurement on the thermal state (7.40), is equal to (2N+1)/4. This can be shown by finding
the distribution of the homodyne detection outcome by integrating out the second
103
quadrature of the Wigner function of the thermal state. The homodyne capacity (in
bits per channel use) is thus given by
Ccoherent-homodyne(7,
=
h, N)
1
-log
2
=
7.3.3
10g2
2
1
1
+ (2(1-'q)N + 1)/4)
(7.44)
+
(7.45)
N
.1
2(1 - rq)N + 1
Squeezed States and Homodyne Detection
Let us transmit a squeezed state &
= Iai)(r,)(r,O) (a1 with an apriori probability
measure PA1 (ai)dai, at each use of the SM lossy thermal noise channel
8
N.
All
squeezed states are Gaussian states. It can be shown, via characteristic functions,
that if the input modes & and b in the SM lossy channel ( = V/&
+
l - rjb) are
excited in Gaussian states, then the output mode a is also in a Gaussian state. The
first quadrature mean and the variance of the output state are respectively given by
(&1)
=
Vai,
(1
(A12)
(7.46)
and
2N + 1
4
+ (14-
)
(7.47)
The conditional probability density to read x1 at the output given that a 1 was transmitted, is therefore given by
PX1IA 1 (Xi a i )
(7.48)
= Tr(pc1Xi)(XiI)
1
/27r
(~{(1 -
rj)(2N + 1) +
exp
e-2r])
-
2 (1 [(1 - q)(2N + 1) +,,,-2r
(7.49)
Since the channel noise is additive and Gaussian, the mutual information is max104
imized for a Gaussian input density (5.13), and the SM capacity is given by
C(r, ii, N, r)
=
1
- log 2
2
(1+
1
4
[
h
-- sinh 2 r
sn) 2
(1-rq)(2N+1) +
q;
(7.50)
r
e-2,r
A further maximization with respect to the squeezing parameter r yields the SM
capacity (in bits per use):
Csqueezed-homodyne(r7,
4fi + 2 - (x(7/, h, N) + x(I, h, N)- 1 )
1 10o
2
2
h,N)
(1-rq)(2N+1)
+ X(TI,
N)
(, .1
(7.51)
where the function x(rq, h, N) is given by
x(r/, h, N) =
(1- r)(2N + 1))
+2(1
-
)(2N + 1)
(1 - r)(2N + 1) +/+2)
(7.52)
7.4
Wideband Results and Discussion
We saw that the capacity of the SM lossy thermal noise channel can be evaluated
for standard encoding schemes and receiver structures. The wideband capacities can
be found by proceeding exactly in the same manner as we did for the pure lossy
channel. The only difference in this case is that there is now an additional frequency
dependence of the thermal mean photon number per mode (7.2) that one has to take
into consideration while doing the Lagrange multiplier maximization. This makes the
calculation of thermal noise capacities very involved analytically. Before turning to
wideband calculations, however, we introduce one more SM capacity measure, and
then compare all our capacity results for the SM lossy thermal noise channel.
It was shown recently [7], that the 'entanglement assisted classical capacity' of a
105
3100
B,-
2500
2.000
Ultimate capacity (???)
Coherent-Heterodyne
Coherent-Homodyne
-
Squeezed-Homodyne
- -- -
------
------Entanglement assisted
Number state Direct detection
0.500
OyC10
0.000
6100
41100
200
101100
Sam
12.00
Input mean photon number (nbar)
Figure 7-6: A sample simulation result of all the different capacities of a SM lossy
thermal noise channel we considered so far in the thesis. The channel S has transmissivity r = 0.8, and mean thermal photon number N = 5.
SM lossy thermal noise channel EN is
CEntanglement -asst.(I,
i, N)
= g(i) + g(N') - g(
+n
D -N'
2
1
D-
+ N'- 1
2
'
(7.53)
/(N + N'+ 1)
where N' _ ih + (1 - q)N, and DE
2
-4]N(N
+ 1). As noted in
Chapter 2, this capacity provides an upper bound on the ultimate capacity of the SM
lossy thermal noise channel.
Now, we are in a position to plot all the SM capacities of the thermal noise
channel. For the case q = 0.8 and N
=
5, we have plotted the different capacities
in Fig. 7-6. The plot with the '?'-sign pertains to what we think the ultimate (unassisted) classical capacity is.
We note that the entanglement-assisted capacity is
indeed an upper bound to all the other SM capacities. Also, as one would expect, the
conjectured ultimate capacity is greater than all the structured transmitter-receiver
capacities, and is less than the entanglement-assisted capacity.
106
The wideband capacities can be found by performing the constrained maximization numerically. We consider two cases. The first case is 'white Gaussian noise',
where we consider the free-space optical channel with single spatial mode propagation
(ultra wideband) with same geometrical parameters and bandwidth as we considered
in Chapter 4, and assume that across the entire frequency band the mean thermal
photon number per mode N = 5 remains constant (Fig. 7-7). The second case we
consider (Fig. 7-8), is the free-space channel with single spatial mode propagation
with the same parameters as above, but with a frequency dependent thermal mean
photon number distribution (7.2) given by the Planck's formula. In both the above
cases, it can be seen that the relative behavior of the various wideband capacities
are the same. Coherent State encoding and heterodyne detection is smallest in the
power limited regime, and catches up with both homodyne detection capacities (with
coherent state and squeezed state encoding respectively) in the bandwidth limited
(high power) regime. The (conjectured) wideband ultimate capacity is greater than
all the structured transmitter-receiver capacities at all values of input power.
107
2 B0Oe17
Ultimate (???)
1e00.17
-
.1.200,17
-
.e
'1
J8130016
4D0.
-
Squeezed-Homodyne
Coherent-Homodyne ------Coherent-Heterodyne - -- - Entanglement Assisted ------
-
-----------------
1
QMO
8.000
6.00
4.00
2.00
0.000
10-000
Input power (Watts)
Figure 7-7: Ultra-wideband capacity of a lossy thermal noise channel - Free-space
2
optical communication, single spatial mode propagation - AT = 400 cm and AR = 1
cm 2, separated by L = 400 km of free-space; with frequencies of transmission ranging
from that associated with a wavelength of 30 microns (which is deep into the far-field
regime), to that associated with a wavelength of 300 nanometers (which is well within
the near field regime). Mean thermal photon number N = 5 is assumed to remain
constant across the entire range of frequencies.
2OO0e16
00*
--
4.oooe15
I
0.00
0.000
0.050
0.100
0.150
Ultimate Capacity (???)
Squeezed-Homodyne - Coherent-Homodyne -----Coherent-Heterodyne - - - I
I
0 .200
0.250
-
0.300
Input power P (Watts)
Figure 7-8: Capacities of a wideband lossy thermal noise channel - - Free-space optical
communication, single spatial mode propagation - all parameters same as in Fig.
7-7, except that the mean thermal photon number N is now assumed to vary with
frequency of transmission f according to Eqn. (7.2)
108
Chapter 8
Conclusions and Scope for Future
Work
The fields of quantum information and quantum communication have grown tremendously over the last two decades, and there is a lot of interesting research that is
currently going on in these fields. Quantum Information theory sets theoretical limits
on achievable information rates that can be sent reliably over quantum channels.
In this thesis, we studied a broad class of classical information capacities of singlemode and wideband lossy bosonic channels. A lossy bosonic channel is a general model
of a quantum communication channel, which can be used for physical modelling of
communication over free-space, waveguides, lossy optical fibres, etc. Our focus in this
thesis was the free-space quantum-optical communication channel. We evaluated the
ultimate classical capacity of the broadband free-space optical channel both in the
far-field and near-field propagation regimes and compared it with achievable rates
using known transmitter and receiver structures. We found that, for the free-space
channel operating in the far-field regime, the performance of coherent-state encoding
and heterodyne detection asymptotically approaches the far-field capacity in the limit
of high input power.
We also generalized this fact by proving a set of sufficient
conditions, for heterodyne detection capacity to approach the ultimate capacity in
the high-power limit. We found the capacities achievable using structured transmitter
encoding and detection schemes, for the single-mode and the wideband lossy bosonic
109
channels with additive thermal noise. We also evaluated the wideband entanglementassisted capacity of the free-space channel.
The ultimate classical capacity of the thermal noise lossy bosonic channel is yet
to be found. We have conjectured this capacity, but the proof is not yet done. The
capacity of the general multimode bosonic channel is not known. It would be interesting to know how atmospheric effects like turbulence, fog, and dust affects the
capacity of classical communication using quantized light. Another interesting thing
to know would be the kind of receiver structures that could achieve the ultimate capacities for the lossy bosonic channel with and without thermal noise. Not much work
has been done yet on the quantum information capacity of quantum channels, which
could also be interesting to pursue, because it is conceivable that with the advent
of robust quantum information hardware in the future, with quantum information
running through networks of quantum computers, a good understanding of quantum
information capacities might become necessary.
110
Appendix A
Blahut-Arimoto Algorithm
Consider the following problem [2]: given two convex sets A and B in R,
we would
like to find the minimum distance between them
dmin
min d(a, b)
acA,bcB
where d(a, b) is the Euclidean distance between a and b.
(A.1)
An intuitively obvious
algorithm to do this would be to take any point x E A, and find the y E B that is
closest to it. Then fix this y and find the closest point in A. Repeating this process,
the distance can only decrease. It has been shown that [17], if the sets are convex
and if the distance satisfies certain conditions, then this alternating minimization
algorithm indeed converges to the absolute minimum. In particular, if the sets are
sets of probability distributions and the distance measure is the relative Shannon
entropy, then the algorithm does converge to the minimum relative entropy between
the two sets of distributions. This is the central idea on which the famous BlahutArimoto (BA) algorithm is based.
Given a channel transition probability matrix
P(ji), this algorithm can be used to find the channel capacity under possibly more
than one linear constraint by finding the optimum input probability distribution.
This algorithm works by iteratively improving the estimates of the input and the
conditional distributions.
Let pi denote the input distribution of a classical channel, and let P(jli) denote
111
the transition probabilities.
Suppose there is a linear constraint at the input of
the channel given by E, pjej
; E. Then the steps of the BA algorithm to find
the optimum (capacity achieving) input distribution p* are illustrated in Fig. A1.
If ej =
j,
Vj then the linear constraint becomes a constraint on the mean of
the input distribution. s is an independent parameter that has to be chosen before
executing the algorithm. The final value of the linear constraint E that the algorithm
eventually converges to depends upon the selection of the value of s. Unfortunately,
there is no very good general way to estimate the value of s which yields a desired
value of E. At each iteration, IL and Iu are universal lower and upper bounds to
C(E) - sE. The iterations are terminated when the absolute difference between Iu
and IL becomes smaller than some pre-decided e. This algorithm works very well to
find capacity achieving distributions for the lossy single mode number state channel,
with or without thermal noise.
112
START
Assign {pj(O)}
r =0
ci =exp
P(i)10 log
P(I
-) se
zp '"P(j Ii)
(max
S= l org(+
p , L"
= p , ,.g~
STO
'.c i
I
F
pi(r+1) =
C,
i(r)
Figure A-1: Blahut-Arimoto Algorithm
113
STOP
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