1. 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!) Aephraim M. Steinberg Centre for Q. Info. & Q. Control Institute for Optical Sciences Dept. of Physics, U. of Toronto CANADA Manipulating and Measuring the Quantuum State of Photons and Atoms QUEST 05, Santa Fe

2. DRAMATIS PERSONAE Toronto quantum optics & cold atoms group: Postdocs: Morgan Mitchell (  Barcelona) Matt Partlow Optics: Jeff LundeenKevin Resch(  Zeilinger   ) Lynden(Krister) ShalmMasoud Mohseni (  Lidar) Rob Adamson Atoms: Jalani FoxStefan Myrskog (  Thywissen) Ana Jofre(  NIST)Mirco Siercke Samansa ManeshiChris Ellenor Some theory collaborators:...Daniel Lidar, Pete Turner, János Bergou, Mark Hillery, Paul Brumer, Howard Wiseman,...

3. There are many types of measurement! Projective measurement (or von Neumann); postselection Quantum state “tomography” (reconstruction of , W, etc) standard, adaptive,... incomplete? in the presence of inaccessible information Quantum process “tomography” (CP map from    ) standard, ancilla-assisted, “direct”,... POVMs “Direct” measurement of functions of  “Interaction-free” measurements “Weak measurements” (various senses) Aharonov/Vaidman application to postselection A few to keep in mind:

4. OUTLINE Tomography – characterizing quantum states & processes... brief review Motional states of atoms in optical lattices Process tomography Pulse echo Inverted states, negative Wigner functions,... Entangled photon pairs 2-photon process tomography Direct measurement of purity Generating entanglement by postselection Characterizing states with “inaccessible” info Bonus topic if you don’t interrupt me enough: Weak measurements and “paradoxes” (which-path debate; Hardy’s paradox)

5. Quantum tomography: what & why? 0 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

6. 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.

7. Wigner function of an ion in the excited state Liebfried, Meekhof, King, Monroe, Itano, Wineland, PRL 77, 4281 (96)

8. 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...

9. QPT of QFT Weinstein et al., J. Chem. Phys. 121, 6117 (2004) To the trained eye, this is a Fourier transform... From those superoperators, one can extract Kraus operator amplitudes, and their structure helps diagnose the process.

10. Ancilla-assisted process tomography Altepeter et al, PRL 90, 193601 (03) Proposed in 2000-01 (Leung; D’Ariano & Lo Presti; Dür & Cirac): one member of an maximally-entangled pair could be collapsed to any given state by a measurement on the other; replace multiple state preparations with coincidence measurement. 2-qbit state tomography with the entangled input is equivalent to 1-qbit process tomography using 4 different inputs (and both require 16 measurements). 1-qbit processes represented as deformations of Bloch sphere...some unphysical results with engineered decoherence.

11. Quantum tomography experiments on photons, and how to avoid them 1

12. HWP QWP PBS Argon Ion Laser Beamsplitter "Black Box" 50/50 Detector B Detector A Two waveplates per photon for state preparation Two waveplates per photon for state analysis SPDC source Two-photon Process Tomography [Mitchell et al., PRL 91, 120402 (2003)]

13. Hong-Ou-Mandel Interference How often will both detectors fire together? r r t t + r 2 +t 2 = 0; total destructive interf. (if photons indistinguishable). If the photons begin in a symmetric state, no coincidences. {Exchange effect; cf. behaviour of fermions in analogous setup!} The only antisymmetric state is the singlet state |HV> – |VH>, in which each photon is unpolarized but the two are orthogonal. This interferometer is a "Bell-state filter," needed for quantum teleportation and other applications. Our Goal: use process tomography to test this filter.

14. “Measuring” the superoperator of a Bell-state filter } Output DM Input HH HV VV VH } } } etc. 16 analyzer settings 16 input states Coincidencences [Mitchell et al., PRL 91, 120402 (2003)]

15. “Measuring” the superoperator Input Output DM HH HV VV VH etc. Superoperator Input Output

17. Superoperator after transformation to correct polarisation rotations: Dominated by a single peak; 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: The ideal filter would have a single peak. Leading Kraus operator allows us to determine unitary error. (Experimental demonstration delayed for technical reasons; now, after improved rebuild of system, first addressing some other questions...)

18. Some vague thoughts... (1)QPT is incredibly expensive (16 n msmts for n qbits) (2)Both density matrices and superoperators we measure typically are very sparse... a lot of time is wasted measuring coherences between populations which are zero. (a) If aiming for constant errors, can save time by making a rough msmt of a given rate first and then deciding how long to acquire data on that point. (b) Could also measure populations first, and then avoid wasting time on coherences which would close to 0. (c) Even if r has only a few significant eigenvalues, is there a way to quickly figure out in which basis to measure? (3)If one wants to know some derived quantity, are there short-cuts? (a) Direct (joint) measurements of polynomial functions (b) Optimize counting procedure based on a given cost function (c) Adaptive search E.g.: suppose you would like to find a DFS within a larger Hilbert space, but need not characterize the rest.

19. A sample error model: the "Sometimes-Swap" gate Consider an optical system with stray reflections – occasionally a photon-swap occurs accidentally: Two subspaces are decoherence-free: 1D: 3D: Experimental implementation: a slightly misaligned beam-splitter (coupling to transverse modes which act as environment) TQEC goal: let the machine identify an optimal subspace in which to compute, with no prior knowledge of the error model.

20. Some strategies for a DFS search (simulation; experiment underway) random tomography adaptive tomography Best known 2-D DFS (average purity). # of input states used averages Our adaptive algorithm always identifies a DFS after testing 9 input states, while standard tomography routinely requires 16 (complete QPT). standard tomography # of inputs tested purity of best 2D DFS found Surprise: in the absence of noise, simulations show that essentially any 2 input states suffice to identify the DFS (required max-lik to work). Project to revisit: add noise, do experiment, study scaling,...

21. Often, only want to look at a single figure of merit of a state (i.e. tangle, purity, etc…) Would be nice to have a method to measure these properties without needing to carry out full QST. Todd Brun showed that m th degree polynomial functions of a density matrix f m (  ) can be determined by measuring a single joint observable involving m identical copies of the state. (T. A. Brun, e-print: quant-ph/0401067) Polynomial Functions of a Density Matrix

22. For a pure state, P=1 For a maximally mixed state, P=(1/n) Quadratic  2-particle msmt needed Measuring the purity of a qubit Need two identical copies of the state Make a joint measurement on the two copies. In Bell basis, projection onto the singlet state HOM as Singlet State Filter + Pure State on either side = 100% visibility H H H H H H H HOM Visibility = Purity Mixed State = 50% visibility + V H H H H VV V Linear Purity of a Quantum State P = 1 – 2   –    –   Singlet-state probability can be measured by a singlet-state filter (HOM)

23. Use Type 1 spontaneous parametric downconversion to prepare two identical copies of a quantum state Vary the purity of the state Use a HOM to project onto the singlet Compare results to QST Coincidence Circuit Single Photon Detector Single Photon Detector Type 1 SPDC Crystal Singlet Filter Quartz Slab Quartz Slab Experimentally Measuring the Purity of a Qubit

24. Prepared the state |+45> Measured Purity from Singlet State Measurement P=0.92±0.02 Measured Purity from QST P=0.99±0.01 Results For a Pure State

25. Can a birefringent delay decohere polarization (when we trace over timing info) ? [cf. J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, and P. G. Kwiat, Phys. Rev. Lett., 90, 193601 ] Case 1: Same birefringence in each arm V V H H 100% interference Visibility = (90±2) % Case 2: Opposite birefringence in each arm V V H H 25% interference Visibility = (21±2) % Preparing a Mixed State The HOM isn’t actually insensitive to timing information.

26. The HOM is not merely a polarisation singlet-state filter Problem: Used a degree of freedom of the photon as our bath instead of some external environment The HOM is sensitive to all degrees of freedom of the photons The HOM acts as an antisymmetry filter on the entire photon state Y Kim and W. P. Grice, Phys. Rev. A 68, 062305 (2003) S. P. Kulik, M. V. Chekhova, W. P. Grice and Y. Shih, Phys. Rev. A 67,01030(R) (2003) Not a singlet filter, but an “Antisymmetry Filter”

27. Randomly rotate the half-waveplates to produce |45> and |-45> |45> |45> or |-45> Preliminary results Currently setting up LCD waveplates which will allow us to introduce a random phase shift between orthogonal polarizations to produce a variable degree of coherence Could produce a “better” maximally mixed state by using four photons. Similar to Paul Kwiat’s work on Remote State Preparation. Coincidence Circuit Visibility = (45±2) % Preparing a Mixed State

28. When the distinguishable isn’t… 2

29. Highly number-entangled states ("low- noon " experiment). The single-photon superposition state |1,0> + |0,1>, which may be regarded as an entangled state of two fields, is the workhorse of classical interferometry. The output of a Hong-Ou-Mandel interferometer is |2,0> + |0,2>. States such as |n,0> + |0,n> ("high-noon" states, for n large) have been proposed for high-resolution interferometry – related to "spin-squeezed" states. Multi-photon entangled states are the resource required for KLM-like efficient-linear-optical-quantum-computation schemes. A number of proposals for producing these states have been made, but so far none has been observed for n>2.... until now! M.W. Mitchell et al., Nature 429, 161 (2004); and cf. P. Walther et al., Nature 429, 158 (2004).

30. [See for example H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002)] ˘ Practical schemes? 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!

31. |1> a|0> + b|1> + c|2>a'|0> + b'|1> + c'|2> The germ of the KLM idea INPUT STATE ANCILLA TRIGGER (postselection) OUTPUT STATE |1> In particular: with a similar but somewhat more complicated setup, one can engineer a |0> + b |1> + c |2>  a |0> + b |1> – c |2> ; effectively a huge self-phase modulation (  per photon). More surprisingly, one can efficiently use this for scalable QC. KLM Nature 409, 46, (2001); Cf. experiments by Franson et al., White et al.,...

32. 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

33. Trick #2 How to combine three non-orthogonal photons into one spatial mode? Yes, it's that easy! If you see three photons out one port, then they all went out that port. "mode-mashing"

34. 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)

35. The basic optical scheme

36. It works! Singles: Coincidences: Triple coincidences: Triples (bg subtracted):

37. Generating / measuring other states With perfect detectors and perfect single-photon sources, such schemes can easily be generalized. With one or the other (and typically some feedback), many states may be synthesized by iteratively adding or subtracting photons, and in some cases implementing appropriate unitaries. Postselection has also been used to generate GHZ, W, and cluster states (to various degrees of fidelity). Photon subtraction can be used to generate non-gaussian states. Postselection is also the heart of KLM and competing schemes, and can be used to implement arbitrary unitaries, and hence to entangle anything. “Continuous” photon subtraction (& counting) can be used, even with inefficient detectors, to reconstruct the entire photon-number distribution.

38. 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?  Left  Arnold  Right  Danny OR  –  Arnold&Danny ?

39. 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 ?

40. 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:

41. Experimental Apparatus

42. 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 

43. 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

44. Tomography in optical lattices, and steps towards control... 3

45. Rb atom trapped in one of the quantum levels of a periodic potential formed by standing light field (30GHz detuning, 10s of  K depth) Tomography in Optical Lattices [Myrkog et al., quant-ph/0312210 Kanem et al., quant-ph/0506140] Complete characterisation of process on arbitrary inputs?

46. Towards QPT: Some definitions / remarks "Qbit" = two vibrational states of atom in a well of a 1D lattice Control parameter = spatial shifts of lattice (coherently couple states), achieved by phase-shifting optical beams (via AO) Initialisation: prepare |0> by letting all higher states escape Ensemble: 1D lattice contains 1000 "pancakes", each with thousands of (essentially) non-interacting atoms. No coherence between wells; tunneling is a decoherence mech. Measurement in logical basis: direct, by preferential tunneling under gravity Measurement of coherence/oscillations: shift and then measure. Typical experiment: Initialise |0> Prepare some other superposition or mixture (use shifts, shakes, and delays) Allow atoms to oscillate in well Let something happen on its own, or try to do something Reconstruct state by probing oscillations (delay + shift +measure)

47. First task: measuring state populations

48. Time-resolved quantum states

49. Recapturing atoms after setting them into oscillation...

50. ...or failing to recapture them if you're too impatient

51. Oscillations in lattice wells (Direct probe of centre-of-mass oscillations in 1  m wells; can be thought of as Ramsey fringes or Raman pump-probe exp’t.)

52. Wait… Quantum state reconstruction Shift…  x Cf. Poyatos,Walser,Cirac,Zoller,Blatt, PRA 53, 1966 ('96) & Liebfried,Meekhof,King,Monroe,Itano,Wineland, PRL77, 4281 ('96) Measure ground state population (former for HO only; latter requires only symmetry) Q(0,0) = P g 1  W(0,0) =  (-1) n P n 1 

53. Husimi distribution of coherent state

54. Atomic state measurement (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

55. Extracting a superoperator: prepare a complete set of input states and measure each output Likely sources of decoherence/dephasing: Real photon scattering (100 ms; shouldn't be relevant in 150  s period) Inter-well tunneling (10s of ms; would love to see it) Beam inhomogeneities (expected several ms, but are probably wrong) Parametric heating (unlikely; no change in diagonals) Other

56. 0 500  s 1000  s 1500  s 2000  s Towards bang-bang error-correction: pulse echo indicates T2 ≈ 1 ms... Free-induction-decay signal for comparison echo after “bang” at 800 ms echo after “bang” at 1200 ms echo after “bang” at 1600 ms decay of coherence introduced by echo pulses themselves (since they are not perfect  -pulses) (bang!)

57. Cf. Hannover experiment Buchkremer, Dumke, Levsen, Birkl, and Ertmer, PRL 85, 3121 (2000). Far smaller echo, but far better signal-to-noise ("classical" measurement of ) Much shorter coherence time, but roughly same number of periods – dominated by anharmonicity, irrelevant in our case.

58. A better "bang" pulse for QEC? Under several (not quite valid) approximations, the double-shift is a momentum displacement. We expected a momentum shift to be at least as good as a position shift. In practice: we want to test the idea of letting learning algorithms search for the best pulse shape on their own, and this is a first step. A = –60 ° t t = 0 T = 900  s measurement time position shift (previous slides) initial state A = –60 ° t variable hold delay =  t = 0 T = 900  s pulse measurement double shift (similar to a momentum shift) initial state

59. time ( microseconds ) single-shift echo (≈10% of initial oscillations) double-shift echo (≈30% of initial oscillations) Echo from compound pulse Future: More parameters; find best pulse. Step 2 (optional): figure out why it works! Also: optimize # of pulses (given imper- fection of each) Pulse 900 us after state preparation, and track oscillations

60. 0.1090.003+0.007 i-0.006+0.028 i0.14+0.037 i 0.003-0.007 i0.2590.018+0.024 i0.011-0.034 i -0.006-0.028 i0.018-0.024 i0.414-0.019-0.037 i 0.14-0.037 i0.011+0.034 i-0.019+0.037 i 0.202 A pleasant surprise from tomography… To characterize processes such as our echo pulses, we extract the completely positive map or “superoperator,” shown here in the Choi-matrix representation: () Upper left-hand quadrant indicates output density matrix expected for a ground-state input Ironic fact: when performing tomography, none of our inputs was a very pure ground state, so in this extraction, we never saw P e > 55% or so, though this predicts 70% – upon observing this superoperator, we went back and confirmed that our echo can create 70% inversion!

61. Data:"W-like" [P g -P e ](x,p) for a mostly-excited incoherent mixture

62. Why does our echo decay? Finite bath memory time: So far, our atoms are free to move in the directions transverse to our lattice. In 1 ms, they move far enough to see the oscillation frequency change by about 10%... which is about 1 kHz, and hence enough to dephase them.

63. What if we try “bang-bang”? (Repeat pulses before the bath gets amnesia; trade-off since each pulse is imperfect.)

64. Some coherence out to > 3 ms now...

65. How to tell how much of the coherence is from the initial state?

66. Future: Tailor phase & amplitude of successive pulses to cancel out spurious coherence Study optimal number of pulses for given total time. (Slow gaussian decay down to exponential?)

67. Can we talk about what goes on behind closed doors? And now for something completely different (?)

70. The Rub

71. By using a pointer with a big uncertainty (relative to the strength of the measurement interaction), one can obtain information, without creating entanglement between system and apparatus (effective "collapse"). H int =gAp x System-pointer coupling What does that really mean?

72. What will that look like?

74. A Gedankenexperiment... e-e- e-e- e-e- e-e-

75. 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)]

76. 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.

77. What do you mean, interaction-free? Measurement, by definition, makes some quantity certain. This may change the state, and (as we know so well), disturb conjugate variables. How can we measure where the bomb is without disturbing its momentum (for example)? But if we disturbed its momentum, where did the momentum go? What exactly did the bomb interact with, if not our particle? It destroyed the relative phase between two parts of the particle's wave function.

78. 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 D- e+ was in D+D- ? But … if they were both in, they should have annihilated! D+ e- was in

79. What does this mean? Common conclusion: We've got to be careful about how we interpret these "interaction-free measurements." You're not always free to reason classically about what would have happened if you had measured something other than what you actually did. (Do we really have to buy this?)

80. How to make the experiment possible: The "Switch"  PUMP   Coinc. Counts 22  PUMP - 2 x  LO  LO   +  PUMP 2 x  LO = 2  LO -  PUMP =  K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Rev. Lett. 87, 123603 (2001).

81. GaN Diode Laser PBS Det. H (D-)Det. V (D+) DC BS 50-50 BS1 50-50 BS2 Switch H V CC PBS Experimental Setup (W)(W)

82. H Pol DC V Pol DC 407 nm Pump

83. What do we need to measure? Ideally If D + clicks  - photon is in I - IFM + If D - clicks  + photon is in I + IFM - There is never a photon in I - and I + Annihilation at W Sometimes both D + and D - click The paradox For a Real Apparatus Vis Hor Int Vis Vert Int Vis Switch Vis Switch + Vis Int 2  7/4

84. 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? Upcoming experiment: demonstrate that "weak measurements" (à la Aharonov + Vaidman) will bear out these predictions. [Y. Aharanov, A. Botero, S. Popescu, B. Reznik, J. Tollaksen, quant-ph/0104062]

85. PROBLEM SOLVED!(?)

86. Problem: For two-particle weak measurements we need a strong nonlinearity to implement a Von Neuman measurement interaction (H int =gPÂ 1 Â 2 ). Two-Particle Weak Measurements Pointer Polarization Correlations for  Â 1 Â 2  weak D + Polarizer Angle (rad.) D - Polarizer Angle (rad.) Weak Measurement for a Polarization Pointer (N particles): Resch & Steinberg, PRL 92,130402 (2004) Lundeen & Resch, Phys. Lett. A 334 (2005) 337–344 If Pointer1 and Pointer2 always move together, then the uncertainty in their difference never changes. If Pointer1 and Pointer 2 both move, but never together, then Δ( Pointer1-Pointer2) must increase. Spin Lowering Operator Solution: Do two single-particle weak measurements and study correlations →

87. N(I - )N(O  ) N(I + ) 011 N(O + ) 10 10 Weak Measurements in Hardy’s Paradox N(I - ) N(O  ) N(I + )0.243±0.0680.663±0.0830.882±0.015 N(O + )0.721±0.074-0.758±0.0830.087±0.021 0.925±0.024-0.039±0.023 Experimental Weak Values Ideal Weak Values

88. Which-path controversy (Scully, Englert, Walther vs the world?) Suppose we perform a which-path measurement using a microscopic pointer, e.g., a single photon deposited into a cavity. Is this really irreversible, as Bohr would have all measurements? Is it sufficient to destroy interference? Can the information be “erased,” restoring interference? Scully et al, Nature 351, 111(1991)

89. Which-path measurements destroy interference (modify p-distrib!)

90. How is complementarity enforced? The fringe pattern (momentum distribution) is clearly changed – yet every moment of the momentum distribution remains the same.

91. The debate since then... Storey, Tan, Collett, & Walls proved that all WWMs must disturb the momentum of any momentum eigenstate. But Scully, Englert, and Walther were right in that every moment of the momentum distribution of the two-slit wavefunction was unchanged by their proposed WWM. Wiseman and Harrison argued that aside from considering different initial conditions, the two sides had different definitions of momentum transfer (probability versus amplitude, roughly). Shouldn’t one be able to measure some momentum transfer kernel, regardless of the choice of initial state? Typically, only by starting in momentum eigenstates.

92. Weak measurements to the rescue! To find the probability of a given momentum transfer, measure the weak probability of each possible initial momentum, conditioned on the final momentum observed at the screen...

93. A practical weak measurement Use transverse position of each photon as the pointer. Weak measurements can be performed by tilting a glass optical flat. The position of each photon is uncertain to within the beam waist... a small shift does not provide any photon with distinguishing info. But after many photons arrive, the shift of the beam may be measured. gtFlat  cf. Ritchie et al., PRL 68, 1107 ('91).

94. Convoluted implementation... Glass plate in focal plane measures P(p i ) weakly (shifting photons along y) Half-half-waveplate in image plane measures path strongly CCD in Fourier plane measures for each position x; this determines wk for each final momentum p f.

95. Calibration of the weak measurement

96. A few distributions P(p i | p f ) Note: not delta-functions; i.e., momentum may have changed. Of course, these "probabilities" aren't always positive, etc etc... EXPERIMENT THEORY (finite width due to finite width of measuring plate)

97. The distribution of the integrated momentum-transfer EXPERIMENT THEORY Note: the distribution extends well beyond h/d. On the other hand, all its moments are (at least in theory, so far) 0.

98. Some concluding remarks/questions... 1.Quantum process tomography can be useful for characterizing and "correcting" quantum systems 2.It taught us how to “invert” the c-o-m oscillation of atoms 3.What other quantities can one extract from superop’s? 4.How much control is possible with a single knob (translating our lattice, e.g.), in the presence of strong dephasing? How to find optimal processes? 5.It’s really expensive! How much will feedback help us do more efficient tomography? In what circumstances can one simply avoid tomography altogether? 6.“Effective” decoherence is very subtle when the “environment” is a degree of freedom of the system itself 7.State-tomography ideas can be generalized to a situation where experimentally indistinguishable particles may or may not have some degree of distinguishing information 8.Weak measurements on subensembles are very strange... but perhaps less strange that the paradoxes they resolve?