How much can you tell about a state?
process tomography (continued)
& optimal state discrimination
• Optical-latice state & process tomography (cont.)
• Discrimination of non-orthogonal states
• "Best guess" approach
• Unambiguous discrimination
• POVMs versus projective measurements
• A linear-optical experiment
• State-filtering (discrimination of mixed states)
9 Dec 2003
(next-to-last lecture!)
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]
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)
Nonclassical dip
Smooth gaussian
Hard to tell what...
Can we see something more
interesting?
"Number states" (energy eigenstates) other than the vacuum are
nonclassical – the Q function has a dip, and the Wigner function goes negative.
We do our best to prepare an excited state, succeeding on approximately
80% of the atoms.
If we wait until |g> and |e> decohere, we achieve a phase-independent state –
the delay is irrelevant, and W is azimuthally symmetric.
The observed dip in Q comes about because a translation in any direction
in phase space increases the population in |g>.
The Q distribution with a dip?
Experiment
Theory
Can we find a clearer signature?
Theory: W(x,p) for 80%
excitation
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
W can be extracted from Q by a deconvolution, but this
leads to so much extra noise that often nothing is recognizable...
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.
Easier to just reconstruct 
(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.
TOMOGRAPHY 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.
Tomography References
Your favorite quantum optics text -- Loudon, Walls/Milburn,Milonni, etc. -for introduction to phase-space methods in quantum optics. Kim's book
Phase Space Methods in Quantum Mechanics. Leonhardt's Measuring the
Quantum State of Light.
Theory:
Single-photon tomography:
Wigner, Phys. Rev. 40, 749 (1932)
Kwiat......
Hillery et al., Phys. Rep. 106, 121 (1984)
Ancilla-assisted tomography:
Altepeter et al., .....
Early tomography experiments:
Tomography on quantum fields:
Smithey et al, PRL 70, 1244 (1993)
Lvovsky et al., PRL 87, 050402 (2001)
(light modes)
Dunn et al., Phys. Rev. Lett. 74, 884 (1995)
(molecules)
Measurement of negative Wigner functions:
Nogues et al, Phys. Rev. A 62, 054101 (2000)
(cavity QED)
Leibfried et al, PRL 77, 4281 (1996)
(trapped ion)
Two-photon process tomography:
Mitchell et al., PRL. 91, 120402 (2003)
Applications of process tomography:
Weinstein et al., .....
(in NMR experiment)
Boulant et al., quant-ph/0211046
(interpreting superoperators)
White et al., quant-ph/0308115
(for 2-photon gates)
Can one distinguish between
nonorthogonal states?
H-polarized photon
45o-polarized photon
• Single instances of non-orthogonal quantum states cannot be
distinguished with certainty. Obviously, ensembles can.
• This is one of the central features of quantum information
which leads to secure (eavesdrop-proof) communications.
• Crucial element: we must learn how to distinguish quantum
states as well as possible -- and we must know how well a
potential eavesdropper could do.
What's the best way to tell these apart?
|a
|"A"

|b
(if they occur with equal
a priori probability)
|"B"
Error rate = (1- sin )/2
0 if <a|b> = 0 (ideal measurement)
1/2 if <a|b> = 1 (pure guessing)
BUT: can we ever tell for sure?
Some interaction would take input states |a> and |b>
to "meter states" |"A"> and |"B">, which we could distinguish perfectly.
But unitary interactions preserve overlap:
Theory: how to distinguish nonorthogonal states optimally
Step 1:
Repeat the letters "POVM" over and over.
Step 2:
Ask Janos, Mark, and Yuqing for help.
[or see, e.g., Y. Sun, J. Bergou, and M. Hillery, Phys.
Rev. A 66, 032315 (2002).]
The view from the laboratory:
A measurement of a two-state system can only
yield two possible results.
If the measurement isn't guaranteed to succeed, there
are three possible results: (1), (2), and ("I don't know").
Therefore, to discriminate between two non-orth.
states, we need three measurement outcomes –
no 2D operator has 3 different eigenstates, though.
How to describe a measurement with
3 outcomes in a 2D space?
Generalized quantum measurements may be described by POVMs, or
"positive operator-valued measures"... recall:
There is no limitation on the number of operators in this sum.
Why extra states?
We want a nonunitary transformation to take non-orthogonal a and b
to orthogonal "A" and "B".
This can be accomplished by measurement – i.e., by throwing out events.
How does this compare with
projective measurement?
|b
|"A"
|a

|b
The only way to be sure
"A" means a is to be sure
it doesn't mean b...
|"B"
|a
Assuming, as always, equal probability of |a> or |b>, we choose in
which basis to measure randomly.
The success probability is then:
How do they compare?
POVM
von Neumann
measurement
At 0, the von Neumann strategy has a discontinuity-only then can you succeed regardless of measurement choice.
At <a|b> = 0.707, the von Neumann strategy succeeds 25% of the time,
while the optimum is 29.3%.
The advantage is higher in higher dim.
Consider these three non-orthogonal states:
Projective measurements can distinguish these states
with certainty no more than 1/3 of the time.
(No more than one member of an orthonormal basis is orthogonal
to two of the above states, so only one pair may be ruled out.)
But a unitary transformation in a 4D space produces:
…and these states can be distinguished with certainty
up to 55% of the time
Experimental schematic
(ancilla)
A 14-path interferometer for
arbitrary 2-qubit unitaries...
Success!
"Definitely 3"
"Definitely 2"
"Definitely 1"
"I don't know"
The correct state was identified 55% of the time-Much better than the 33% maximum for standard measurements.
Further interesting result: mixed states may also be discriminated, contrary
to earlier wisdom.
STATE-DISCRIMINATION
SUMMARY
Non-orthogonal states may be distinguished with certainty
("unambiguously") if a finite rate of "inconclusives" is tolerated.
The optimal (lowest) inconclusive rate is the absolute value of the
overlap between the states (in 2D), and cannot be achieved by any
projective measurement.
POVMs, implementable by coupling to a larger Hilbert space,
can achieve this optimum. In optics, they may be realized with
optical multiports (interferometers).
We successfully distinguish among 3 non-orthogonal states 55%
of the time, where standard quantum measurements are limited
to 33%.
More recent observation: "state filtering" or discrimination of
mixed states is also possible.
State-discrimination References
C. W. Helstrom, Quantum Detection and Estimation
Theory (Academic Press, New York, 1976)
I. D. Ivanovic, Phys. Lett. A \23} 257 (1987).
A. Chefles and S. M. Barnett, J. Mod. Opt. 45, 1295 (1998)
A. Chefles, Phys. Lett. A 239, 339 (1998)
S. M. Barnett and E. Riis, J. Mod. Opt. 44, 1061 (1997)
D. Dieks, Phys. Lett. A 126, 303 (1998)
B. Huttner et al., Phys. Rev. A 54, 3783 (1996)
A. Peres, Phys. Lett. A 128, 19 (1988)
R. B. M. Clarke et al., Phys Rev A 63, 040305 (2001)
A. Chefles and S. M. Barnett, Phys. Lett. A 250, 223 (1998)
R. B. M. Clarke et al., Phys Rev A 64, 012303 (2001)
Y. Sun, M. Hillery, and J. A. Bergou, Phys. Rev. A 64, 022311
(2001)
T. Rudolph, R. W. Spekkens, and P. S. Turner, Phys. Rev.
A 68, 0101301 (2003)
M. Takeoka, M. Ban, and M. Sasaki, Phys. Rev. A 68,
012307 (2003).
J. A. Bergou, M. Hillery, and Y. Sun, J. Mod. Opt. 47, 487
(2000)
Y. Sun, J. A. Bergou, and M. Hillery, Phys. Rev. A 66, 032315
(2002)
J. A. Bergou, U. Herzog, and M. Hillery, Phys. Rev. Lett. 90,
257901 (2003)