Quantum State and Process
Tomography: measuring mixed
states
•Now for something different:
• Instead of "weak" measurements, let's try to measure everything.
• Instead of a state overdetermined by preparation and postselection, let's
consider states which may be incompletely defined.
•Description of mixed states
• Density matrices
• Wigner functions
• Superoperators
• Brief nod to "traditional" (!) quantum state tomography
• Two-photon state & process tomography
• Optical-latice state & process tomography
2 Dec 2003
Recall: mixed states are described by
density matrices, not by wave
functions.
Diagonal elements = probabilities
Off-diagonal elements = "coherences"
(provide info. about relative phase)
Can't determine the coefficients by measuring a single system - to extract all this
information, we must study a large ensemble of identical particles.
How to extract coefficients?
Measure the expectation value of various operators A ;
each one provides a given linear combination of the matrix elements.
To measure all n2 elements for an n-dimensional system, should make
n2 different (linearly independent) measurements.
What about continuous variables?
The Wigner function is one of several phase-space distributions which
can play the same role as the density matrix:
It is not a probability (since what is the probability of an event which
can never be tested?), but it is the unique "quasi-probability distribution"
satisfying:
It may be negative, but P(x) and
P(p) never are... [in fact, W is
almost always negative somewhere.]
Like the density matrix, it can be used to extract any expectation value:
(In principle, an infinite number of observables must be measured
to extract all the infinite number of points in W(x,p).)
How does one measure these things?
Signal (weak)
Local
oscillator (strong)
|Es + |ELO| eif |2
|ELO|2 + 2 |ELO| Re Es cos f
- 2 |ELO| Im Es sin f + ...
Heterodyning allows one to measure Re (Es eiffor various f
thus we can extract integrals along every possible angle...
Much work over the last 10-20 years applying algorithms from
medical imaging to extract Wigner functions, e.g., of light...
E.g.: Wigner function of a photon
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
Lvovsky et al., Physical Review Letters 87 , 050402 (2001)
Circularly symmetric: no phase-dependence when you homodyne.
Dip in each marginal at 0 -- the only way this can be is:
negative quasiprobability at E1=E2=0.
Dip at middle is related to the Hong-Ou-Mandel dip, and its highphoton-number analog: put another way, it's our old discussion of
interference of number states, and how the photons tend to bunch.
But with quantum information in mind, let's think about something
different: polarisation states of photon pairs, and how they evolve...
Density matrices and superoperators
()
( )
One photon: H or V.
State: two coefficients
CH
CV
Density matrix: 2x2=4 coefficients
CHH CVH
CHV
CVV
Measure
intensity of horizontal
intensity of vertical
intensity of 45o
intensity of RH circular.
Propagator (superoperator): 4x4 = 16 coefficients.
Two photons: HH, HV, VH, HV, or any superpositions.
State has four coefficients.
Density matrix has 4x4 = 16 coefficients.
Superoperator has 16x16 = 256 coefficients.
Two-photon Process Tomography
Two waveplates per photon
for state preparation
HWP
QWP
HWP
Detector A
PBS
QWP
SPDC source
"Black Box" 50/50
Beamsplitter
QWP
HWP
QWP
PBS
HWP
Detector B
Argon Ion Laser
Two waveplates per
photon for state analysis
Hong-Ou-Mandel Interference
r
r
+
t
t
How often will both detectors fire together?
r2+t2 = 0; total destructive interference.
...iff the processes (& thus photons) indistinguishable.
If the photons have same polarisation, no coincidences.
Only in the singlet state |HV> – |VH> are the two photons
guaranteed to be orthogonal.
This interferometer is a "Bell-state filter," needed
for quantum teleportation and other applications.
Our Goal: use process tomography to test this filter.
“Measuring” the superoperator
Coincidencences
Output DM
}
}
}
}
16
input
states
Input
HH
HV
etc.
VV
16 analyzer settings
VH
“Measuring” the superoperator
Input
Superoperator
Output DM
HH
HV
VV
VH
etc.
Input
Output
Testing the superoperator
LL = input state
Predicted
Nphotons = 297 ± 14
Testing the superoperator
LL = input state
Predicted
Nphotons = 297 ± 14
Detector A
HWP
QWP
HWP
PBS
QWP
"Black Box" 50/50
Beamsplitter
BBO two-crystal
downconversion
source.
QWP
HWP
Argon Ion Laser
QWP
PBS
HWP
Detector B
Observed
Nphotons = 314
So, How's Our Singlet State Filter?
Bell singlet state:  = (HV-VH)/√2
1/2 -1/2
-1/2 1/2
Observed  , but a different maximally entangled state:
Model of real-world
beamsplitter
Singlet
filter
multi-layer dielectric
AR coating
45° “unpolarized” 50/50
dielectric beamsplitter
at 702 nm (CVI Laser)
f
f
birefringent element
+
singlet-state filter
+
birefringent element
Necessary correction determined from leading Kraus operator...
Superoperator provides information
needed to correct & diagnose operation
Measured superoperator,
in Bell-state basis:
The ideal filter would have a
single peak.
Leading Kraus operator allows
us to determine unitary error.
Superoperator after transformation
to correct polarisation rotations:
Dominated by a single peak;
residuals allow us to estimate
degree of decoherence and
other errors.
FUTURE: more efficient extraction of information for better correction of errors
iterative search for optimal encodings in presence of collective noise;...
Tomography in Optical Lattices
Rb atom trapped in one of the quantum levels
of a periodic potential formed by standing
light field (30GHz detuning, 10s of mK depth)
Complete characterisation of
process on arbitrary inputs?
Setup for lattice with adjustable position & velocity
First task: measuring state
populations
Time-resolved quantum states
Quantum state reconstruction
p
p
t
Dx
x
Wait…
x
Shift…
p
Dx
x
Measure ground
state population
Q(0,0) = Pg
W(0,0) =  (-1)n Pn
(former for HO only; latter requires only symmetry)
[Now, we can also perform translation directly in both x and p]
Recapturing atoms after setting
final vs midterm, both adjusted to 70 +/- 15
them
intotooscillation...
70 +/- 15
both adjusted
final vs midterm,
Series1
...or failing to recapture them
final vs midterm, both adjusted to 70 +/- 15
if both
you're
impatient
15
to 70 +/adjustedtoo
final vs midterm,
Series1
Oscillations in lattice wells
[essentially a measure of Q(r,) at fixed r-recall, r is set by size of shift and  by length of delay]
Extracted phase-space distributions
(Q rather than W in this case)
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
Data:"W-like" [Pg-Pe](x,p) for
a mostly-excited incoherent mixture
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
(For 2-level subspace, can also
choose 4 particular measurements
and directly extract density matrix)
Theory: W(x,p) for 80%
excitation
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
Atomic state measurement
(for a 2-state lattice, with c0|0> + c1|1>)
initial state
displaced
delayed & displaced
left in
ground band
tunnels out
during adiabatic
lowering
(escaped during
preparation)
|c0|2
|c1|2
|c0 + c1 |2
|c0 + i c1 |2
Extracting a superoperator:
prepare a complete set of input states and measure each output
Superoperator for resonant drive
Operation: x (resonantly couple 0 and 1 by modulating lattice periodically)
Measure superoperator to diagnose single-qubit operation
(and in future, to correct for errors and decoherence)
Observed Bloch sphere
Bloch sphere predicted from
truncated harmonic-oscillator
plus decoherence as measured previously.
Upcoming goals: generate tailored pulse sequences to preserve coherence;
determine whether decoherence is Markovian; et cetera.
SUMMARY
Any pure or mixed state may be represented by a density matrix or
phase-space distribution (e.g., Wigner function).
These can be reconstructed by making repeated measurements in
various bases (n2 measurements for a density matrix).
A superoperator determines the time-evolution of a density matrix
(including decoherence), and requires n4 measurements.
Elements in quantum-information systems can be characterized
by performing such measurements.
More work needs to be done on
(a) optimizing the extraction of useful information
(b) determining how to use the resulting superoperators.
References
Your favorite quantum optics text -- Loudon, Walls/Milburn, Meystre/Sargent, Milonni,
Scully/Zubairy, etc. -- for introduction to quantum optics & phase-space methods.
Schleich's Quantum Optics in Phase Space.
Leonhardt's Measuring the Quantum State of Light.
Theory:
Single-photon process tomography:
Wigner, Phys. Rev. 40, 749 (1932)
White et al., PRA 65, 012301 (2002)
Hillery et al., Phys. Rep. 106, 121 (1984)
James et al., PRA 65, 052312 (2001)
Ancilla-assisted photon-polarisation tomography:
Early tomography experiments:
Altepeter et al., PRL 90, 193601 (2003)
Smithey et al, PRL 70, 1244 (1993)
Phase-space tomography on single-photon fields:
(light modes)
Dunn et al., Phys. Rev. Lett. 74, 884 (1995)
(molecules)
Lvovsky et al., PRL 87, 050402 (2001)
Two-photon process tomography:
Mitchell et al., PRL. 91, 120402 (2003)
Measurement of negative Wigner functions:
Applications of process tomography:
Nogues et al, Phys. Rev. A 62, 054101 (2000)
Weinstein et al., PRL 86, 1889 (2001)
(cavity QED)
Leibfried et al, PRL 77, 4281 (1996)
(trapped ion)
(in NMR experiment)
Boulant et al., quant-ph/0211046
(interpreting superoperators)
White et al., quant-ph/0308115
(for 2-photon gates)