RealWorld Quantum Measurements Fun With Photons and Atoms

Real-World Quantum Measurements: Fun With Photons and Atoms Aephraim M. Steinberg Centre for Q. Info. & Q. Control Institute for Optical Sciences Dept. of Physics, U. of Toronto CAP 2006, Brock University

DRAMATIS PERSONÆ Toronto quantum optics & cold atoms group: Postdocs: Morgan Mitchell ( ICFO) Matt Partlow Optics: Rob Adamson Lynden(Krister) Shalm Xingxing Xing An-Ning Zhang Kevin Resch( Zeilinger ����� ) Masoud Mohseni ( Lidar) Jeff Lundeen ( Walmsley) Atoms: Jalani Fox (. . . Hinds) Stefan Myrskog ( Thywissen) Ana Jofre( Helmerson) Mirco Siercke Samansa Maneshi Chris Ellenor Rockson Chang Chao Zhuang Current ug’s: Shannon Wang, Ray Gao, Sabrina Liao, Max Touzel, Ardavan Darabi Some helpful theorists: Daniel Lidar, János Bergou, Pete Turner, John Sipe, Paul Brumer, Howard Wiseman, Michael Spanner, . . .

Quantum Computer Scientists The 3 quantum computer scientists: see nothing (must avoid "collapse"!) hear nothing (same story) say nothing (if any one admits thing is never going to work, that's the end of our funding!)

OUTLINE The grand unified theory of physics talks: “Never underestimate the pleasure people get from being told something they already know. ” Beyond the standard model: “If you don’t have time to explain something well, you might as well explain lots of things poorly. ”

OUTLINE Measurement: this is not your father’s observable! • {Forget about projection / von Neumann} • “Generalized” quantum measurement • Weak measurement (postselected quantum systems) • “Interaction-free” measurement, . . . • Quantum state & process tomography • Measurement as a novel interaction (quantum logic) • Quantum-enhanced measurement • Tomography given incomplete information • {and many more}

1 Distinguishing the indistinguishable. . .

How to distinguish non-orthogonal states optimally vs. The view from the laboratory: Use generalized (POVM) quantum measurements. A measurement of a two-state system can only [see, yield e. g. , Y. J. Bergou, and M. Hillery, Phys. two. Sun, possible results. Rev. A 66, 032315 isn't (2002). ] If the measurement 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 to use an expanded (3 D or more) system. To distinguish 3 states, we need 4 D or more.

The geometric picture 1 2 Q Q 90 o 1 Two non-orthogonal vectors The same vectors rotated so their projections onto x-y are orthogonal (The z-axis is “inconclusive”)

A test case 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 4 D 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. M. Mohseni, A. M. Steinberg, and J. Bergou, Phys. Rev. Lett. 93, 200403 (2004)

Can we talk about what goes on behind closed doors? (“Postselection” is the big new buzzword in QIP. . . but how should one describe post-selected states? )

Conditional measurements (Aharonov, Albert, and Vaidman) AAV, PRL 60, 1351 ('88) Prepare a particle in |i> …try to "measure" some observable A… postselect the particle to be in |f> Measurement of A Does <A> depend more on i or f, or equally on both? Clever answer: both, as Schrödinger time-reversible. Conventional answer: i, because of collapse. Reconciliation: measure A "weakly. " Poor resolution, but little disturbance. …. can be quite odd …

Predicting the past. . . A+B B+C What are the odds that the particle was in a given box (e. g. , box B)? It had to be in B, with 100% certainty.

Consider some redefinitions. . . In QM, there's no difference between a box and any other state (e. g. , a superposition of boxes). What if A is really X + Y and C is really X - Y? A+B = X+B+Y X Y B+C= X+B-Y

A redefinition of the redefinition. . . So: the very same logic leads us to conclude the particle was definitely in box X. X + B' = X+B+Y X + C' = X+B-Y

The Rub

A (von Neumann) Quantum Measurement of A Initial State of Pointer Final Pointer Readout Hint=g. Apx x System-pointer coupling x Well-resolved states System and pointer become entangled Decoherence / "collapse" Large back-action

A Weak Measurement of A Initial State of Pointer Final Pointer Readout Hint=g. Apx x System-pointer coupling x Poor resolution on each shot. Negligible back-action (system & pointer separable) Mean pointer shift is given by <A>wk. Need not lie within spectrum of A, or even be real. . .

The 3 -box problem: weak msmts Prepare a particle in a symmetric superposition of three boxes: A+B+C. Look to find it in this other superposition: A+B-C. Ask: between preparation and detection, what was the probability that it was in A? B? C? PA = < |A><A| >wk = (1/3) / (1/3) = 1 PB = < |B><B| >wk = (1/3) / (1/3) = 1 PC = < |C><C|>wk = (-1/3) / (1/3) = -1. Questions: were these postselected particles really all in A and all in B? can this negative "weak probability" be observed? [Aharonov & Vaidman, J. Phys. A 24, 2315 ('91)]

A Gedankenexperiment. . . ee- e- e-

A negative weak value for Prob(C) Perform weak msmt on rail C. A+B–C (neg. shift!) Rail C (pos. shift) Post-select either A, B, C, or A+B–C. Compare "pointer states" (vertical profiles). Rails A and B (no shift) K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Lett. A 324, 125 (2004).

2 a Seeing without looking

" Quantum seeing in the dark " (AKA: The Elitzur-Vaidman bomb experiment) A. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993) P. G. Kwiat, H. Weinfurter, and A. Zeilinger, Sci. Am. (Nov. , 1996) Problem: D C Consider a collection of bombs so sensitive that a collision with any single particle (photon, electron, etc. ) Bomb absent: is guarranteed to trigger it. Only detector C fires BS 2 that certain of Suppose the bombs are defective, but differ in their behaviour in no way other than that Bomb present: they will not blow up when triggered. "boom!" 1/2 bombs (or Is there any way to identify the working C up? 1/4 some of them) without blowing them BS 1 D 1/4 The bomb must be there. . . yet my photon never interacted with it.

Hardy's Paradox (for Elitzur-Vaidman “interaction-free measurements”) C+ D+ D- COutcome Prob D+ –> e- was in D+–> ande+ C-was 1/16 BS 2+ BS 2 I+ I- O+ W BS 1+ e+ OBS 1 e- D- and C+ 1/16 D+D- –> ? C+ and C- 9/16 D+ … andif D But they were - 1/16 both in, they 4/16 Explosion should have annihilated!

But what can we say about where the particles were or weren't, once D+ & D– fire? Probabilities e- in e+ in 0 e- out 1 1 e+ out 1 -1 0 In fact, this is precisely what Aharonov et al. ’s weak measurement formalism predicts for any sufficiently gentle attempt to “observe” these probabilities. . .

Weak Measurements in Hardy’s Paradox Ideal Weak Values N(I-) N(O ) N(I+) N(O+) 0 1 1 0 1 0 Experimental Weak Values (“Probabilities”? ) N(I-) N(O ) N(I+) 0. 243± 0. 068 0. 663± 0. 083 0. 882± 0. 015 N(O+) 0. 721± 0. 074 0. 758± 0. 083 0. 087± 0. 021 0. 925± 0. 024 0. 039± 0. 023

3 Quantum tomography: what & why? 1. 2. 3. 4. 5. Characterize unknown quantum states & processes Compare experimentally designed states & processes to design goals Extract quantities such as fidelity / purity / tangle Have enough information to extract any quantities defined in the future! 1. • or, for instance, show that no Bell-inequality could be violated Learn about imperfections / errors in order to figure out how to • improve the design to reduce imperfections • optimize quantum-error correction protocols for the system

Quantum Information What's so great about it?

What makes a computer quantum? (One partial answer. . . ) We need to understand the nature of quantum information itself. How to characterize and compare quantum states? How to most fully describe their evolution in a given system? How to manipulate them? The danger of errors & decoherence grows exponentially with system size. across the Danube The only hope for QI is quantum error correction. We must learn how to measure what the system is doing, and then correct it.

Density matrices and superoperators Two photons: HH, HV, VH, VV, or any superpositions. State has four coefficients. Density matrix has 4 x 4 = 16 coefficients. Superoperator has 16 x 16 = 256 coefficients.

Some density matrices. . . Much work on reconstruction of optical density matrices in the Kwiat group; theory advances due to Hradil & others, James & others, etc. . . ; now a routine tool for characterizing new states, for testing gates or purification protocols, for testing hypothetical Bell Inequalities, etc. . . Polarisation state of 3 photons (GHZ state) Resch, Walther, Zeilinger, PRL 94 (7) 070402 (05) Spin state of Cs atoms (F=4), in two bases Klose, Smith, Jessen, PRL 86 (21) 4721 (01)

Wigner function of atoms’ vibrational quantum state in optical lattice J. F. Kanem, S. Maneshi, S. H. Myrskog, and A. M. Steinberg, J. Opt. B. 7, S 705 (2005)

Superoperator provides information needed to correct & diagnose operation Measured superoperator, in Bell-state basis: (expt) Leading Kraus operator allows us to determine unitary error. Superoperator after transformation to correct polarisation rotations: (predicted) Residuals allow us to estimate degree of decoherence and other errors. M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, Phys. Rev. Lett. 91 , 120402 (2003)

QPT of QFT Weinstein et al. , J. Chem. Phys. 121, 6117 (2004) To the trained eye, this is a Fourier transform. . .

The status of quantum cryptography

4 a Measurement as a tool: Post-selective operations for the construction of novel (and possibly useful) entangled states. . .

Highly number-entangled states ("low-noon" experiment). States such as |n, 0> + |0, n> ("noon" states) have been proposed for high-resolution interferometry – related to "spin-squeezed" states. Important factorisation: + = A "noon" state A really odd beast: one 0 o photon, one 120 o photon, and one 240 o photon. . . but of course, you can't tell them apart, let alone combine them into one mode! Theory: H. Lee et al. , Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002)

Trick #1 Okay, we don't even have single-photon sources*. But we can produce pairs of photons in down-conversion, and very weak coherent states from a laser, such that if we detect three photons, we can be pretty sure we got only one from the laser and only two from the down-conversion. . . SPDC |0> + |2> + O( 2) laser * But |0> + |1> + O( 2) |3> + O( 3) + O( 2) + terms with <3 photons we’re working on it (collab. with Rich Mirin’s quantum-dot group at NIST)

Postselective nonlinearity How to combine three non-orthogonal photons into one spatial mode? "mode-mashing" Yes, it's that easy! If you see three photons out one port, then they all went out that port.

The basic optical scheme

It works! Singles: Coincidences: Triple coincidences: Triples (bg subtracted): M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, Nature 429, 161 (2004)

4 b 4 b Complete characterisation when you have incomplete information

Fundamentally Indistinguishable vs. Experimentally Indistinguishable But what if when we combine our photons, there is some residual distinguishing information: some (fs) time difference, some small spectral difference, some chirp, . . . ? This will clearly degrade the state – but how do we characterize this if all we can measure is polarisation?

Quantum State Tomography Indistinguishable Photon Hilbert Space ? Distinguishable Photon Hilbert Space Yu. I. Bogdanov, et al Phys. Rev. Lett. 93, 230503 (2004) If we’re not sure whether or not the particles are distinguishable, do we work in 3 -dimensional or 4 -dimensional Hilbert space? If the latter, can we make all the necessary measurements, given that we don’t know how to tell the particles apart ?

The Partial Density Matrix The answer: there are only 10 linearly independent parameters which are invariant under permutations of the particles. One example: Inaccessible information The sections of the density matrix labelled “inaccessible” correspond to information about the ordering of photons with respect to inaccessible degrees of freedom. (For n photons, the # of parameters scales as n 3, rather than 4 n) R. B. A. Adamson, L. K. Shalm, M. W. Mitchell, and A. M. Steinberg, quant-ph/0601134

Experimental Results No Distinguishing Info When distinguishing information is introduced the HV-VH component increases without affecting the state in the symmetric space H H + V V Mixture of 45 – 45 and – 45 45

The moral of the story 1. Post-selected systems often exhibit surprising behaviour which can be probed using weak measurements. 2. Post-selection can also enable us to generate novel “interactions” (KLM proposal for quantum computing), and for instance to produce useful entangled states. 3. POVMs, or generalized quantum measurements, are in some ways more powerful than textbook-style projectors 4. Quantum process tomography may be useful for characterizing and "correcting" quantum systems (ensemble measurements). 5. A modified sort of tomography is possible on “practically indistinguishable” particles Predicted Wigner-Poincaré function for a variety of “triphoton states” we are starting to produce: