Optical Communications at the Quantum Limit
Zachary Dutton
Raytheon BBN Technologies
Cambridge, MA
ICQIQC 2013
IISc
Bangolore, India
11 January, 2013
Quantum Information Processing (QuIP group)
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Founded at BBN in 2008
Now comprised of researchers in
superconducting electronics,
experimental quantum optics,
information and physics theory
Identify new discoveries in physics
laboratories that can deliver new
capabilities to computation,
communication, and sensing systems
Understand the impacts of these
discoveries on enhanced (or new)
capabilities and transition them to
engineered systems
Interests range from physical to
application layer
Partner with academic and industrial
partners to attack the difficult
challenges
Applications
(e.g. Entanglement enhanced LADAR;
secret key distribution, quantum
computation)
System tools
Devices
(e.g. quantum
repeater stations)
Components
(e.g. multi-qubit
chips,
entanglement
sources)
(e.g. quantum
functional language
and compilation )
Physical Theory
(e.g. non-standard
optical receivers)
Physical Technologies
(e.g. superconducting qubits,
superconducting photon
detectors, SPDC sources)
Application areas
• Quantum computation
– cQED based superconducting qubits and processing architectures
– Programming languages, error correction, and compilers for quantum
computation
– Low power high performance classical computation based on
superconducting logic
• Quantum communications
– Long distance quantum key distribution (QKD)
– Photon Information Efficient COMMunications (PIECOMM) fundamental
limits of power & spectral efficiency in communication
• Quantum enhanced sensing
– Quantum imaging fundamental limits of imaging power efficiency
– Quantum illumination using entanglement for improved target detection
BBN Laboratory for Bits and
Waves cryo-lab
• Research in superconducting circuit
based quantum computation
– In collaboration with IBM and university
partners we are designing a scalable cQED
qubit architecture based on the surface code
Surface code
structure
• Two 10 mk dilution refrigerators
Single-qubit Cliffords characterized with RB
Q1
Q2
Q3
Gate time
20 ns
16.7 ns
16.7 ns
Avg.
error
0.0035
0.0025
0.0030
Error per
generator
0.0022
0.0016
0.0019
3-qubit / 2-cavity sub-cell
E. Magesan, et. al PRL 109, 080505 (2012)
• Research on nonlinear optical effects at
microwave frequencies
– Coherent population trapping (CPT)
K. Murali, et. al PRL 93, 087003 (2004)
W. R. Kelly, et. al, PRL 104 163601 (2010)
Light is very good at carrying information
– Much higher bandwidth (higher data rates) than RF
– Long range (suitable for deep space communication)
Holevo limit
Homodyne
Heterodyne
– What are the limits of optical communication and how can we
reach them?
– It requires quantum theory to model a physical channel
Our “PIECOMM” team
BBN
MIT
• Saikat Guha (PI)
• Jeffrey H. Shapiro
– Nivedita Chandrasekaran
• Zachary Dutton
• Hari Krovi
• Monika Patel
• Jeff Chen
• Seth Lloyd
• Lizhong Zheng
– Hye Won Chung
• Gregory Wornell
– Yuval Kochman
– Ligong Wang
• Franco N. C. Wong
– Valentina Schettini
• Jonathan Habif
• Richard Lazarus
• Karl Berggren
– Francesco Bellei
– Hasan Korre
Discrimination of optical states below the
standard quantum limit (SQL)
Direct detection of optical states
• Direct detection (intensity detection): the “shot noise” limit
– Photon detection (arrival) process is a Poisson point process with
rate,
– Total number of “clicks” on the detector (k) has a Poisson
distribution
– The number of clicks detected has no information about phase ϕ
• The “phase space” picture
Im
Re
Coherent detection of optical states
• Coherent detection (Homodyne and Heterodyne detection)
measures both intensity and phase
– Homodyne detection measures one chosen quadrature (θ)
– Heterodyne detection measures two orthogonal quadratures
simultaneously but with twice as much noise on each measurement
• Quantum description of a classical laser pulse: coherent
state of light
Detection scheme matched to modulation
• On-off keying (OOK) modulation
Im
Direct detection
Re
• Binary phase shift keying (BPSK) modulation
Im
Homodyne detection
Re
• Quadrature phase shift keying (QPSK) modulation
Im
Heterodyne detection
Re
Quantum State Discrimination
• Direct, homodyne, and heterodyne detection define the
standard quantum limit (SQL)
• However, there exists a lower fundamental limit (Helstrom
bound) to discrimination error for non-orthogonal states
• Optimal (Minimum Probability of Error) discrimination
between two non-orthogonal pure states
– Binary pure states,
and
– MPE measurement: Minimize
such that
– Helstrom measurement
, with Prob.
and
– Minimum error probability
Helstrom(1976)
The Dolinar receiver
• Can achieve the binary Helstrom
bound
• Utilizes real time classical feedback
and nulling
• Can beat the SQL limit (coherent
and DD)
• Original demonstration beat DD limit
(with QE corrected) on binary phase
shift keyed (BPSK) input
Cook, Martin, Geremia Nature (2007)
Dolinar (1976)
• Recently NIST-Gaithersburg
demonstrated a QPSK version
(Boundarant receiver) with an
amplitude slicing technique
(Takeoka receiver)
– 13 dB below SQL (6 dB below perfect
detector SQL)
Becerra, et. al, Nature Phonics
(in press)
The “Generalized Kennedy” Receiver
click
“off”
say “off”
say “on”
“on”
no click
• Utilizes optimized (but constant) nulling
• Recently demonstrated by NICT and unambiguously beats SQL
Exact
nulling
Tsujino, PRL
(2011)
arXiv:1103.5592
Pe (OOK)
DD
GK
Helstrom
Np(dB)
PPM demodulation using the
Conditional Pulse Nulling (CPN) receiver
PPM Pulses
SPD
Decision
Nulling
1
1
k
k
DD pulse 2, 3,
2
1
1 4
2
k
Null pulse-1
k
2
DD
pulse
3,
4
and DD
2
3
Null pulse-2
4
3
DD
pulse 4
and DD
Null pulse-3 3
4
and DD
3
4
Nulling “Decision Tree”
Reaches sub-SQL error demodulation
of a codeword of multiple symbols (joint
detection receiver – JDR)
Dolinar, MIT Ph.D. Thesis 1976, TDA Progress Report, 42-72, 1982
Guha, Habif, Takeoka, J. Mod. Optics, Vol. 58, Nos. 3–4, 10–20, 257–265, 2011
Decoder
System Diagram of a CPN Receiver
Chen, Habif, Dutton, Lazarus & Guha, Nature Photonics (2012), Xiv:1111.4017
Experimental results
• First experimentally tested perfect nulling
• Beat direct detection limit by 1.3 dB at Np =0.65
• At higher Np the CPN receiver begins to degrade, in
agreement with a model with a mode-mismatch of
0.05
• We believe this mis-match could be improved by an
order of magnitute, which would yield 6 dB
improvement at Np =2
DD
CPN
Helstrom
Dashed: ideal
Solid: model
b
a
a 
q
b 

Np
(1  d ) N p e
N null e
iq
iq
N 0 | a  b |  N P D
2
 [ 2 (1  cos q )  D T  D P ] N P 
[( 1  d  1)( 1  cos q )  ( 2  d  2 1  d )] N P
DTT
T
Optimal nulling CPN
• Just as the Generalized Kennedy
receiver (with optimal nulling
amplitude) can improve upon direct
detection, the CPN can be improved
by optimal nulling
– This effect is pronounced at low Np<1.0
– For Np=0.65, this ideally increases the
improvement over DD from 1.3 dB to 2.2
dB.
– For this data set, 3% mode mis-match
model gave best agreement with data
We demonstrated 2.1 dB
improvement in Pe over DD by
optimizing the null amplitude
Chen, Habif, Dutton, Lazarus & Guha, Nature Photonics (2012), Xiv:1111.4017
Achieving the Helstrom limit generally
• The CPN receiver approaches the Helstrom limit in a
special case
– PPM + high photon number
– Recent work has extended to more generally to high photon
number case Nair & Guha, arXiv:1212.2048
– Is implemented simply with linear optics and classical feedback
• Is there a prescription to do this for all regimes?
– Yes – though a clear implementation is not yet apparent
– da Silva’s approach solves the long standing problem of
minimum probability of error (MPE) measurement in
discriminating an arbitrary number of coherent states
da Silva, Guha and Dutton, QCMC 2012, arXiv: 1208.5758 (2012)
The slicing receiver
Infinite dimensional coherent states can be “compressed” into finite dimensional qubits
U is a highly nonlinear operation
Coherent states can be “sliced” into multiple small amplitude pieces (living in the
|0> and |1> sub-space) then compressed into a qubit register
The {Ui} are determined by
unitary compression:
Ul |hj>|mj,l> = |f>|mj,l>
|{hj>} are
hypotheses
Binary example: BPSK receiver
The compressed qubit
da Silva, Guha and Dutton, QCMC’12, ArXiv: 1208.5758 (2012)
Slicing receiver performance
• The final step is then a projective measurement of the
register
• Note that this is essentially a small continuous variable
optical quantum computer
The receiver is seen here to work for 2 and 3 state discrimination
and is fully generalizable
0.6
0.5
0.1
0.05
0.1
Pe
Pe
0.01
0.005
0.001
0.0005
0.05
0.0001
0.00005
0.00001
0.01
n = 2, 10, 30, 100 slices
0.1
0.2
0.3
0.4 0.5
Α
1
n = 2, 10, 30, 100 slices
0.005
2
3
0.1
0.2
0.3 0.4 0.5
1
Α
2
3
4 5
Fundamental limits of capacity
Connecting of sub-SQL to capacity
• We have recently identified fundamental
limits of photon efficiency (bits per photon)
for classical optical communications
– Quantum mechanics (Holevo bounds) must be
employed to calculate
– Important for power contrained systems such as
deep space (e.g. lunar and Mars) systems
• Classical coherent states (laser pulses) are
sufficient to reach this capacity
– This is good news, since any other state would
be destroyed during transmission through the
atmosphere
– One needs non-standard receivers
– Joint detecton receivers (JDRs) are necessary
Giovannetti, Guha, Lloyd,
Maccone, & Shapiro (2004)
Shapiro, Guha, Erkmen (2005)
Guha, PRL (2011)
Reliable communication over a noisy
channel
Claude Shannon
“Father of information theory”
1916-2001
Classical communication over a
classical channel
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Classical information transmission via classical symbols
noisy channel
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Noise does not preclude error-free digital communication
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Error-free communication can be accomplished if
 the data rate is below the channel capacity,
 appropriate error-correction coding is employed
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Channel capacity is the maximum mutual information
Shannon (1948)
Shannon’s intuition
•
Coding is essentially a way to drive down errors via redundancy
–
–
e.g. a rate r=0.33, the repetition code would send “000” and “111” to communicate one bit
during three pulse slots using a binary modulation alphabet
Chooses 2k of the 2n possible codewords to form a rate r=k/n binary code
[100100101]
(decoding)
[101101001]
Codebook: a pruned set of
2nR binary sequences
(Code rate, R = k/n)
- Sending nR bits of information over n channel uses: R bits per channel use
- Shannon [1948]: As long as R < C, there exists a (n,k,d) codebook with n
large enough that the probability of codeword decoding error goes to zero as
n goes to infinity.
Concatenated coding and JDRs
• Traditional low-complexity codes rely on concatenation of inner (binary)
codes (e.g. BCH) and outer (non-binary) codes (e.g. Reed-Solomon).
• Our JDR results (Cn > C1) are all Shannon capacity results, suggesting a
concatenated coding approach where the JDR acts on the inner code
inner super-channel: Shannon capacity: Cn > C1
(N,R)
outer
encoder
(n,r)
inner
encoder
Modulator
Physical
channel
Optical
Receiver
Demodu
-lator
Detec
-tor
Inner
decoder
Outer
decoder
Joint-detection receiver (JDR)
Guha, Dutton, Shapiro
arXiv:1102.1963v1 [quant-ph]
Proc. ISIT 2011
• In the CPN case we can use Reed-Solomon codes as outer codes over
the PPM inner code
•
•
•
•
Code parameters are the block length n and information bits per block k
PPM rate rPPM=(log2M)/M
The total rate of the concatenated code is r=(k/n) rPPM
We have shown that the lower error rates of CPN can improve coding latency
The Holevo capacity limit
Alexander Holevo
Steklov Mathematical Institute,
Moscow, Russia
Classical communication over a
quantum channel
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Classical information transmission via quantum states
 The “channel” is determined by the receiver measurement itself. How do we
know which receiver will get the highest capacity?
xmtr
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quantum
channel
rcvr
Quantum entropy bound
Forney (1963),
Gordon (1964),
Holevo (1973),
Yuen (1993)
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Channel capacity is the maximum of Holevo information (over
prior probabilities, transmitter states, and POVMs)
Holevo (1998),
Schumacher,
Westmoreland (1997)
Giovannetti, Guha, Lloyd, Maccone,
Shapiro, & Yuen (2004)
29
Limits of photon efficiency
• Photon information efficiency (PIE) is the bits per photon
• Achieving the Holevo limit will require optical codes and JDRs
Holevo
OOK + DD
BPSK+Dolinar
BPSK+homodyne
Quantum polar codes can reach
Holevo
Wilde, Guha, arXiv: 1109.2591, IEEE 2012
Guha, Wilde, arXiv: 1202.0533, ISIT, 2012
DD limit also can be
reached by BPSK with
“Green Machine” JDR
Guha, PRL (2011)
Guha, Dutton, Shapiro arXiv:1102.1963v1 Proc. ISIT (2011)
Shannon versus Holevo Capacity Limits
Guha, PRL (2011)
• Pure-loss bosonic channel (shown for BPSK alphabet)
superadditive capacity
Any projective
measurement
Any projective
measurement
Any projective
measurement
•
With a joint quantum measurement, for any R<C∞, there is a code for which the optimal
quantum measurement minimizes the Pr(error) of discriminating the codewords
•
Optimal measurement can be implemented as unitary transformation (beamsplitters,
phase-shifters, squeezing, photon counting or Kerr nonlinearities) on the (optical) codeword
followed by a sequence of projective measurements on the single-symbol state spaces
Slicing JDR to achieve Holevo limit
Performing the MPE measurement, treating the JDR (inner) code block as a waveform
will allow us to reach the Helstrom limit
1
2
3
4
N
N qubit unitary gate
Circuit requires O(2N) single and two qubit gates [Solovay-Kitaev theorem]
Can this circuit be simplified for codes that have certain symmetry properties?
Conclusions and Ongoing research
• We have demonstrated the first joint detection receiver
– CPN reduces error rate of PPM demodulation (as compared to
SQL)
– This reduced error rate reduces coding latency (though does not
increase capacity)
• We have solved the long-standing problem of MPE
discrimination of an arbitrary number of coherent states
– This can be applied to reach the Holevo capacity by treating long
codeword blocks as waveforms to be discrminated with a JDR
– This is an interesting application of a small quantum computer!
• The Holevo limit can be reached with coherent states
– However, it requires new JDR receivers and optimal coding
– Quantum polar codes can reach the Holevo limit
– Still working on explicit optical implementations for the slicing or
other optimal receivers
Back-up
The Binary Symmetric Channel: capacity
• Binary symmetric channel: two inputs and two outputs
– Used in early days to model the telegraph channel
0
0
B
A
1
• Capacity
– Capacity attained for equal prior probability on A
– Binary entropy function
1
CPN vs. DD coding latency comparison
•
•
•
•
•
We took our experimental and ideal theory data for both DD and CPN systems
to compare coding overhead and latency
At ns=-5 dB (pulse energy Np = 1.25, M=4) we had
Pe(DD)
Peras(DD)
Pe(CPN)
Peras(CPN)
Ideally
0.0
0.289
0.082
0.011
Experiment
0.004
0.287
0.092
0.052
We then calculated the RS block length required for coded error rate <10-10
At low rates, the higher number of erasures helps DD’s outer-coding, despite the
higher uncoded error rate
At higher rates (near capacity) the CPN greatly outperforms DD
105
104
DD
nmin 103
CPN
102
101
RS rate (n/k)