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1. title From Quantum Tomography to Quantum Error Correction: playing games with the information in atoms and photons Aephraim Steinberg Dept. of Physics, University of Toronto

2. Acknowledgments Acknowledgments U of T quantum optics & laser cooling group: PDFs: Morgan Mitchell Marcelo Martinelli Optics: Kevin Resch ( Vienna) Jeff Lundeen Chris Ellenor ( Korea) Masoud Mohseni ( Lidar) Reza Mir Rob Adamson Atom Traps: Stefan Myrskog Jalani Fox Ana Jofre Mirco Siercke Samansa Maneshi Salvatore Maone ( real world) Theory friends: Daniel Lidar, Janos Bergou, Mark Hillery, John Sipe, Paul Brumer, Howard Wiseman

3. OUTLINE OUTLINE • Introduction: • Photons and atoms are promising for QI. • Need for real-world process characterisation • and tailored error correction. • Can there be nonlinear optics with <1 photon? • - Using our "photon switch" to test Hardy's Paradox. • Quantum process tomography on entangled photon pairs • - E.g., quality control for Bell-state filters. • - Input data for tailored Quantum Error Correction. • Quantum tomography (state and process) on center-of-mass states of atoms in optical lattices. • Summary / Coming attractions… • (Optimal discrimination of non-orthogonal states… • Tunneling-induced coherence between lattice sites… • Coherent control of quantum chaos… • Quantum computation in the presence of noise…)

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

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

6. The Rub

7. What makes a quantum computer?

8. What makes a computer quantum?

9. PART 1:Can we build a two-photon switch? Photons don't interact (good for transmission; bad for computation) Nonlinear optics: photon-photon interactions Generally exceedingly weak. Potential solutions: Cavity QED Better materials (1010 times better?) Measurement as nonlinearity (KLM) Novel effects (slow light, EIT, etc) Interferometrically-enhanced nonlinearity

10. Entangled photon pairs(spontaneous parametric down-conversion) The time-reverse of second-harmonic generation. A purely quantum process (cf. parametric amplification) Each energy is uncertain, yet their sum is precisely defined. Each emission time is uncertain, yet they are simultaneous.

11. Is SPDC really the time-reverse of SHG? (And if so, then why doesn't it exist in classical e&m?) The probability of 2 photons upconverting in a typical nonlinear crystal is roughly 10-10 (as is the probability of 1 photon spontaneously down-converting).

12. Quantum Interference

13. Suppression/Enhancementof Spontaneous Down-Conversion (57% visibility)

14. Switch cartoon

15. PART 1a:Applications of 2-photon switch N.B.: Does not work on Fock states! Have demonstrated controlled-phase operation. Have shown theoretically that a polarisation version could be used for Bell-state determination (and, e.g., dense coding)… but not for projective Bell measurements. Present "application," however, is to a novel test of QM.

16. "Interaction-Free Measurements" D C BS2 BS1 (AKA: The Elitzur-Vaidman bomb experiment) 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

17. Hardy Cartoon D+ D- C+ C- BS2+ BS2- I+ I- O- O+ W BS1+ BS1- e- e+ Hardy’s Paradox D+ e- was in D- e+ was in D+D- ? But …

18. Hardy's Paradox: Setup Det. A Det. B CC 50-50 BS2 PBS 50-50 BS1 GaN Diode Laser CC V H DC BS DC BS Switch (W) Cf. Torgerson et al., Phys. Lett. A. 204, 323 (1995)

19. Conclusions when both "dark" detectors fire 0 1 1 -1 Upcoming experiment: demonstrate that "weak measurements" (à la Aharonov + Vaidman) will bear out these predictions.

20. 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. The Real Problem

21. PART 2:State and process tomography

22. Density matrices and superoperators

23. Part 2a:Two-photon Process Tomography "Black Box" 50/50 Beamsplitter Two waveplates per photon for state preparation Detector A HWP HWP PBS QWP QWP SPDC source QWP QWP PBS HWP HWP Detector B Argon Ion Laser Two waveplates per photon for state analysis

24. Hong-Ou-Mandel Interference r t + t r How often will both detectors fire together? r2+t2 = 0; total destructive interference. If the photons begin in a symmetric state, no coincidences. 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.

25. “Measuring” the superoperator Coincidencences Output DM Input } HH } } 16 input states } HV etc. VV 16 analyzer settings VH

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

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

28. Testing the superoperator LL= input state Predicted Nphotons = 297 ± 14

29. Testing the superoperator LL= input state Predicted Nphotons = 297 ± 14 Observed Nphotons = 314

30. So, How's Our Singlet State Filter? 1/2 -1/2 -1/2 1/2 Bell singlet state:  = (HV-VH)/√2 Observed  , but a different maximally entangled state:

31. Model of real-world beamsplitter Singlet filter multi-layer dielectric AR coating 45° “unpolarized” 50/50 dielectric beamsplitter at 702 nm (CVI Laser) birefringent element + singlet-state filter + birefringent element

32. Comparison to ideal filter Measured superoperator, in Bell-state basis: Superoperator after transformation to correct polarisation rotations: A singlet-state filter would have a single peak, indicating the one transmitted state. Dominated by a single peak; residuals allow us to estimate degree of decoherence and other errors.

33. Part 2b:Tomography in Optical Lattices Part I: measuring state populations in a lattice…

34. Time-resolved quantum states

35. Lattice experimental setup Setup for lattice with adjustable position & velocity

36. Atoms oscillating

37. Also Atoms oscillating

38. Oscillations in lattice wells Ground-state population vs. time bet. translations

39. Quantum state reconstruction Wait… Shift… Initial phase- space distribution Measure ground state population

40. Q(x,p) for a coherent H.O. state?

41. Exp't:"W" or [Pg-Pe](x,p)

42. W(x,p) for 80% excitation

43. Atom superoperators sitting in lattice, quietly decohering… being shaken back and forth resonantly Initial Bloch sphere

44. Coming attractions… State Discrimination • Non-orthogonal quantum states cannot be distinguished • with certainty. • 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.

45. Theory: how to distinguish non-orthogonal states optimally Step 1: Repeat the letters "POVM" over and over. Step 2: Ask Janos, Mark, and Yuqing for help. 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.

46. 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

47. Experimental layout (ancilla)

48. 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.

49. SUMMARY • Quantum interference allows huge enhancements of optical nonlinearities. Useful for quantum computation? • Two-photon switch useful for studies of quantum weirdness • (Hardy's paradox, weak measurement,…) • Two-photon process tomography useful for characterizing • (e.g.) Bell-state filters. • Next round of experiments on tailored quantum error correction • (w/ D. Lidar et al.). • Wigner-function and Superoperator reconstruction also underway in optical lattices, a strong candidate system for quantum comp-utation. Characterisation and control of decoherence expected soon. • Other work: Implementation of a quantum algorithm in the presence of noise; Optimal discrimination of non-orthogonal states; • Tunneling-induced coherence; et cetera…