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!) How much can you tell about a state? process tomography (continued) & optimal state discrimination

Wait… Quantum state reconstruction Shift…  x [Now, we can also perform translation directly in both x and p] Measure ground state population (former for HO only; latter requires only symmetry)

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? Theory Experiment Can we find a clearer signature?

Theory: W(x,p) for 80% excitation 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" [P g -P e ](x,p) for a mostly-excited incoherent mixture

Easier to just reconstruct  (for a 2-state lattice, with c 0 |0> + c 1 |1>) left in ground band tunnels out during adiabatic lowering (escaped during preparation) initial statedisplaceddelayed & displaced |c 0 | 2 |c 0 + c 1 | 2 |c 0 + i c 1 | 2 |c 1 | 2

input density matrices output density matrices Time-evolution of some states

Extracting a superoperator: prepare a complete set of input states and measure each output

Observed Bloch sphere Bloch sphere predicted from truncated harmonic-oscillator plus decoherence as measured previously. 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) 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 (n 2 measurements for a density matrix). A superoperator determines the time-evolution of a density matrix (including decoherence), and requires n 4 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. Single-photon tomography: Kwiat Ancilla-assisted tomography: Altepeter et al.,..... Tomography on quantum fields: Lvovsky et al., PRL 87, (2001) Two-photon process tomography: Mitchell et al., PRL. 91, (2003) Applications of process tomography: Weinstein et al.,..... (in NMR experiment) Boulant et al., quant-ph/ (interpreting superoperators) White et al., quant-ph/ (for 2-photon gates) Theory: Wigner, Phys. Rev. 40, 749 (1932) Hillery et al., Phys. Rep. 106, 121 (1984) Early tomography experiments: Smithey et al, PRL 70, 1244 (1993) (light modes) Dunn et al., Phys. Rev. Lett. 74, 884 (1995) (molecules) Measurement of negative Wigner functions: Nogues et al, Phys. Rev. A 62, (2000) (cavity QED) Leibfried et al, PRL 77, 4281 (1996) (trapped ion)

Can one distinguish between nonorthogonal states? 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  |b   Error rate = (1- sin  )/2 0 if = 0 (ideal measurement) 1/2 if = 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: |"A"  |"B"  (if they occur with equal a priori probability)

Theory: how to distinguish non- orthogonal states optimally Step 1: Repeat the letters "POVM" over and over. 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. Step 2: Ask Janos, Mark, and Yuqing for help. [or see, e.g., Y. Sun, J. Bergou, and M. Hillery, Phys. Rev. A 66, (2002).]

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? |a  |b   Assuming, as always, equal probability of |a> or |b>, we choose in which basis to measure randomly. The success probability is then: |a  |b  |"A"  |"B"  The only way to be sure "A" means a is to be sure it doesn't mean b...

POVM von Neumann measurement How do they compare? At 0, the von Neumann strategy has a discontinuity-- only then can you succeed regardless of measurement choice. At = 0.707, the von Neumann strategy succeeds 25% of the time, while the optimum is 29.3%.

But a unitary transformation in a 4D space produces: …and these states can be distinguished with certainty up to 55% of the time 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.)

Experimental schematic (ancilla)

A 14-path interferometer for arbitrary 2-qubit unitaries...

Success! The correct state was identified 55% of the time-- Much better than the 33% maximum for standard measurements. "I don't know" "Definitely 3" "Definitely 2" "Definitely 1" 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 A. Chefles, Phys. Lett. A 239, 339 (1998) D. Dieks, Phys. Lett. A 126, 303 (1998) A. Peres, Phys. Lett. A 128, 19 (1988) A. Chefles and S. M. Barnett, Phys. Lett. A 250, 223 (1998) Y. Sun, M. Hillery, and J. A. Bergou, Phys. Rev. A 64, (2001) 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, (2002) J. A. Bergou, U. Herzog, and M. Hillery, Phys. Rev. Lett. 90, (2003) 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) S. M. Barnett and E. Riis, J. Mod. Opt. 44, 1061 (1997) B. Huttner et al., Phys. Rev. A 54, 3783 (1996) R. B. M. Clarke et al., Phys Rev A 63, (2001) R. B. M. Clarke et al., Phys Rev A 64, (2001) T. Rudolph, R. W. Spekkens, and P. S. Turner, Phys. Rev. A 68, (2003) M. Takeoka, M. Ban, and M. Sasaki, Phys. Rev. A 68, (2003).