Real-World Quantum Measurements: Fun With Photons and Atoms

1. 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

2. DRAMATIS PERSONÆ Toronto quantum optics & cold atoms group: Postdocs: Morgan Mitchell ( ICFO) Matt Partlow An-Ning Zhang Optics: Rob Adamson Kevin Resch(Zeilinger ) Lynden(Krister) Shalm Masoud Mohseni (Lidar) Xingxing Xing 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,...

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

4. 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.”

5. OUTLINEMeasurement: 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}

6. 1 Distinguishing the indistinguishable...

7. How to distinguish non-orthogonal states optimally vs. 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 to use an expanded (3D or more) system. To distinguish 3 states, we need 4D or more. Use generalized (POVM) quantum measurements. [see, e.g., Y. Sun, J. Bergou, and M. Hillery, Phys. Rev. A 66, 032315 (2002).]

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

9. 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 4D space produces: …and these states can be distinguished with certainty up to 55% of the time

10. Experimental schematic (ancilla)

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

12. 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)

13. 2 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?)

14. Conditional measurements(Aharonov, Albert, and Vaidman) Measurement of A Reconciliation: measure A "weakly." Poor resolution, but little disturbance. …. can be quite odd … AAV, PRL 60, 1351 ('88) Prepare a particle in |i> …try to "measure" some observable A… postselect the particle to be in |f> 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.

15. A+B A+B B+C Predicting the past... 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.

16. A + B = X+B+Y X Y B + C = X+B-Y 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?

17. X + B' = X+B+Y X Y X + 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.

18. The Rub

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

20. A Weak Measurement of A Initial State of Pointer Final Pointer Readout Hint=gApx System-pointer coupling x 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...

21. The 3-box problem: weak msmts 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. 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? 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)]

22. A Gedankenexperiment... e- e- e- e-

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

24. 2a Seeing without looking

25. " Quantum seeing in the dark " D C BS2 BS1 The bomb must be there... yet my photon never interacted with it. (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: Consider a collection of bombs so sensitive that a collision with any single particle (photon, electron, etc.) is guarranteed to trigger it. Suppose that certain of the bombs are defective, but differ in their behaviour in no way other than that they will not blow up when triggered. Is there any way to identify the working bombs (or some of them) without blowing them up? Bomb absent: Only detector C fires Bomb present: "boom!" 1/2 C 1/4 D 1/4

26. Hardy's Paradox(for Elitzur-Vaidman “interaction-free measurements”) D+ D- C+ C- BS2+ BS2- I+ I- O- O+ W BS1+ BS1- e- e+ D+ –> e- was in D- –> e+ was in D+D- –> ? But … if they were both in, they should have annihilated!

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

28. Weak Measurements in Hardy’s Paradox 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 Ideal Weak Values

29. 3 Quantum tomography: what & why? • 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! • 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

30. Quantum Information What's so great about it?

31. 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. The only hope for QI is quantum error correction. We must learn how to measure what the system is doing, and then correct it. across the Danube (...Another talk, or more!)

32. Density matrices and superoperators Two photons: HH, HV, VH, VV, or any superpositions. State has four coefficients. Density matrix has 4x4 = 16 coefficients. Superoperator has 16x16 = 256 coefficients.

33. 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... Spin state of Cs atoms (F=4), in two bases Polarisation state of 3 photons (GHZ state) Klose, Smith, Jessen, PRL 86 (21) 4721 (01) Resch, Walther, Zeilinger, PRL 94 (7) 070402 (05)

34. 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, S705 (2005)

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

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

37. The status of quantum cryptography

38. 4a Measurement as a tool: Post-selective operations for the construction of novel (and possibly useful) entangled states...

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

40. Trick #1 SPDC laser 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... |0> + e |2> + O(e2)  |3> + O(3) + O(2) + terms with <3 photons |0> +  |1> + O(2) * But we’re working on it (collab. with Rich Mirin’s quantum-dot group at NIST)

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

42. The basic optical scheme

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

44. 4b 4b Complete characterisation when you have incomplete information

45. Fundamentally Indistinguishablevs.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?

46. Quantum State Tomography Distinguishable Photon Hilbert Space Indistinguishable 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 ?

47. The Partial Density Matrix Inaccessible information Inaccessible information The answer: there are only 10 linearly independent parameters which are invariant under permutations of the particles. One example: 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 n3, rather than 4n) R.B.A. Adamson, L.K. Shalm, M.W. Mitchell, and A.M. Steinberg, quant-ph/0601134

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

49. The moral of the story Post-selected systems often exhibit surprising behaviour which can be probed using weak measurements. Post-selection can also enable us to generate novel “interactions” (KLM proposal for quantum computing), and for instance to produce useful entangled states. POVMs, or generalized quantum measurements, are in some ways more powerful than textbook-style projectors Quantum process tomography may be useful for characterizing and "correcting" quantum systems (ensemble measurements). 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: