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

2. DRAMATIS PERSONÆ Toronto quantum optics & cold atoms group: Postdocs: Morgan Mitchell (  ICFO) Matt PartlowAn-Ning Zhang Optics: Rob AdamsonKevin Resch(  Zeilinger   ) Lynden(Krister) ShalmMasoud Mohseni (  Lidar) Xingxing XingJeff Lundeen (  Walmsley) Atoms: Jalani Fox (...  Hinds)Stefan Myrskog (  Thywissen) Ana Jofre(  Helmerson)Mirco Siercke Samansa ManeshiChris Ellenor Rockson ChangChao 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. 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 the end of our funding!) Quantum Computer Scientists

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. 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}

6. Distinguishing the indistinguishable... 1

7. How to distinguish non-orthogonal states optimally Use generalized (POVM) quantum measurements. [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 to use an expanded (3D or more) system. To distinguish 3 states, we need 4D or more. vs.

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

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

10. Experimental schematic (ancilla)

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

12. 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" M. Mohseni, A.M. Steinberg, and J. Bergou, Phys. Rev. Lett. 93, 200403 (2004)

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

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

16. 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 B + C = X+B-Y XY

17. A redefinition of the redefinition... X + B' = X+B+Y X + C' = X+B-Y XY 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 Well-resolved states System and pointer become entangled Decoherence / "collapse" Large back-action Initial State of Pointer x x H int =gAp x System-pointer coupling Final Pointer Readout

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

21. 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? Questions: were these postselected particles really all in A and all in B? can this negative "weak probability" be observed? P A = wk = (1/3) / (1/3) = 1 P B = wk = (1/3) / (1/3) = 1 P C = wk = (-1/3) / (1/3) =  1. [Aharonov & Vaidman, J. Phys. A 24, 2315 ('91)]

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

23. Rails A and B (no shift) Rail C (pos. shift) A+B–C (neg. 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. Data for P A, P B, and P C... Rail C Rails A and B WEAKSTRONG

25. Seeing without looking 2a2a

26. 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? " 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) BS1 BS2 D C Bomb absent: Only detector C fires Bomb present: "boom!"1/2 C1/4 D1/4 The bomb must be there... yet my photon never interacted with it.

27. BS1 - e-e- BS2 - O-O- C-C- D-D- I-I- BS1 + BS2 + I+I+ e+e+ O+O+ D+D+ C+C+ W OutcomeProb D + and C - 1/16 D - and C + 1/16 C + and C - 9/16 D + and D - 1/16 Explosion4/16 Hardy's Paradox (for Elitzur-Vaidman “interaction-free measurements”) D- –> e+ was in D+D- –> ? But … if they were both in, they should have annihilated! D+ –> e- was in

28. Probabilitiese- ine- out e+ in1 e+ out0 10 01 1 11 But what can we say about where the particles were or weren't, once D+ & D– fire? In fact, this is precisely what Aharonov et al.’s weak measurement formalism predicts for any sufficiently gentle attempt to “observe” these probabilities...

29. N(I - )N(O  ) N(I + ) 011 N(O + ) 1 11 0 10  0.039±0.023 0.925±0.024 0.087±0.021  0.758±0.083 0.721±0.074N(O + ) 0.882±0.0150.663±0.0830.243±0.068N(I + ) N(O  ) N(I - ) Experimental Weak Values (“Probabilities”?) Ideal Weak Values Weak Measurements in Hardy’s Paradox

30. Quantum tomography: what & why? 3 1.Characterize unknown quantum states & processes 2.Compare experimentally designed states & processes to design goals 3.Extract quantities such as fidelity / purity / tangle 4.Have enough information to extract any quantities defined in the future! or, for instance, show that no Bell-inequality could be violated 5.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

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

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

33. What makes a computer quantum? across the Danube (...Another talk, or more!) 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. (One partial answer...)

34. The Serious Problem For QI The danger of errors grows exponentially with the size of the quantum system. Without error-correction techniques, quantum computation would be a pipe dream. A major goal is to learn to completely characterize the evolution (and decoherence) of physical quantum systems in order to design and adapt error-control systems. $The tools are "quantum state tomography" and "quantum process tomography": full characterisation of the density matrix or Wigner function, and of the " $ uperoperator" which describes its time-evolution.

35. 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.

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

37. 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)

38. Superoperator after transformation to correct polarisation rotations: Residuals allow us to estimate degree of decoherence and other errors. Superoperator provides information needed to correct & diagnose operation Measured superoperator, in Bell-state basis: 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)

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

40. The status of quantum cryptography

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

42. Theory: H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002) ˘ Highly number-entangled states ("low- noon " experiment). 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! States such as |n,0> + |0,n> ("noon" states) have been proposed for high-resolution interferometry – related to "spin-squeezed" states.

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

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

45. Trick #3 But how do you get the two down-converted photons to be at 120 o to each other? More post-selected (non-unitary) operations: if a 45 o photon gets through a polarizer, it's no longer at 45 o. If it gets through a partial polarizer, it could be anywhere... (or nothing) (or <2 photons) (or nothing)

46. The basic optical scheme

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

48. Complete characterisation when you have incomplete information 4b

49. 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?

50. 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 ?

51. The sections of the density matrix labelled “inaccessible” correspond to information about the ordering of photons with respect to inaccessible degrees of freedom. 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: (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

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

53. More Photons… So the total number of operators accessible to measurement is If you have a collection of spins, what are the permutation-blind observables that describe the system? They correspond to measurements of angular momentum operators J and m j... for N photons, J runs to N/2

54. 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: