1. Time in the Weak Value and the Discrete Time Quantum Walk Yutaka Shikano Theoretical Astrophysics Group, Department of Physics, Tokyo Institute of Technology Ph. D Final Defense

2. Ph.D defense on August 18th1 Background The concept of time is crucial to understand dynamics of the Nature. In quantum mechanics, When the Hamiltonian is bounded, the time operator is not self-adjoint (Pauli 1930).

3. Ph.D defense on August 18th2 How to characterize time in quantum mechanics? 1.Change the definition / interpretation of the observable –Extension to the symmetric operator YS and A. Hosoya, J. Math. Phys. 49, 052104 (2008). 2.Compare between the quantum and classical systems –Relationships between the quantum and classical random walks (Discrete Time Quantum Walk) –Weak Value YS and A. Hosoya, J. Phys. A 42, 025304 (2010). A. Hosoya and YS, J. Phys. A 43, 385307 (2010). 3.Construct an alternative framework. Aim: Construct a concrete method and a specific model to understand the properties of time

4. Ph.D defense on August 18th3 Organization of Thesis Chapter 1: Introduction Chapter 2: Preliminaries Chapter 3: Counter-factual Properties of Weak Value Chapter 4: Asymptotic Behavior of Discrete Time Quantum Walks Chapter 5: Decoherence Properties Chapter 6: Concluding Remarks

5. Ph.D defense on August 18th4 Appendixes A)Hamiltonian Estimation by Weak Measurement YS and S. Tanaka, arXiv:1007.5370. B)Inhomogeneous Quantum Walk with Self-Dual YS and H. Katsura, Phys. Rev. E 82, 031122 (2010). YS and H. Katsura, to appear in AIP Conf. Proc., arXiv:1104.2010. C)Weak Measurement with Environment YS and A. Hosoya, J. Phys. A 43, 0215304 (2010). D)Geometric Phase for Mixed States YS and A. Hosoya, J. Phys. A 43, 0215304 (2010).

6. Ph.D defense on August 18th5 Organization of Thesis Chapter 1: Introduction Chapter 2: Preliminaries Chapter 3: Counter-factual Properties of Weak Value Chapter 4: Asymptotic Behavior of Discrete Time Quantum Walks Chapter 5: Decoherence Properties Chapter 6: Concluding Remarks

7. Ph.D defense on August 18th6 Rest of Today’s talk 1.What is the discrete time quantum walk? 2.Asymptotic behaviors of the discrete time quantum walks 3.Discrete time quantum walk under the simple decoherence model 4.Conclusion Summary of the discrete time quantum walks Summary of the weak value Summary of this thesis

8. Ph.D defense on August 18th7 Discrete Time Random Walk (DTRW) Coin Flip Shift Repeat

9. Ph.D defense on August 18th8 Discrete Time Quantum Walk (DTQW) Quantum Coin Flip Shift Repeat (A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, in STOC’01 (ACM Press, New York, 2001), pp. 37 – 49.)

10. Ph.D defense on August 18th9 Example of DTQW Initial Condition –Position: n = 0 (localized) –Coin: Coin Operator: Hadamard Coin Let’s see the dynamics of quantum walk by 3 rd step! Probability distribution of the n-th cite at t step:

11. Ph.D defense on August 18th10 Example of DTQW 0123-2-3 step 0 1 2 3 1/129/121/12 prob. Quantum Coherence and Interference

12. Ph.D defense on August 18th11 Probability Distribution at the 1000-th step Initial Coin State Coin Operator DTQWDTRW Unbiased Coin (Left and Right with probability ½ )

13. Ph.D defense on August 18th12 Weak Limit Theorem (Limit Distribution) DTRW DTQW Central Limit Theorem (N. Konno, Quantum Information Processing 1, 345 (2002).) Probability density Coin operatorInitial state Prob. 1/2

14. Ph.D defense on August 18th13 Probability Distribution at the 1000-th step Initial Coin State Coin Operator DTQWDTRW Unbiased Coin (Left and Right with probability ½ )

15. Ph.D defense on August 18th14 Weak Limit Theorem (Limit Distribution) DTRW DTQW Central Limit Theorem (N. Konno, Quantum Information Processing 1, 345 (2002).) Probability density Coin operatorInitial state Prob. 1/2

16. Ph.D defense on August 18th15 Experimental and Theoretical Progresses –Trapped Atoms with Optical Lattice and Ion Trap M. Karski et al., Science 325, 174 (2009). 23 step F. Zahringer et al., Phys. Rev. Lett. 104, 100503 (2010). 15 step –Photon in Linear Optics and Quantum Optics A. Schreiber et al., Phys. Rev. Lett. 104, 050502 (2010). 5 step M. A. Broome et al., Phys. Rev. Lett. 104, 153602. 6 step –Molecule by NMR C. A. Ryan, M. Laforest, J. C. Boileau, and R. Laflamme, Phys. Rev. A 72, 062317 (2005). 8 step Applications –Universal Quantum Computation N. B. Lovett et al., Phys. Rev. A 81, 042330 (2010). –Quantum Simulator T. Oka, N. Konno, R. Arita, and H. Aoki, Phys. Rev. Lett. 94, 100602 (2005). (Landau-Zener Transition) C. M. Chandrashekar and R. Laflamme, Phys. Rev. A 78, 022314 (2008). (Mott Insulator-Superfluid Phase Transition) T. Kitagawa, M. Rudner, E. Berg, and E. Demler, Phys. Rev. A 82, 033429 (2010). (Topological Phase)

17. Ph.D defense on August 18th16 Continuous Time Quantum Walk (CTQW) Experimental Realization A. Peruzzo et al., Science 329, 1500 (2010). (Photon, Waveguide) p.d. Limit Distribution (Arcsin Law <- Quantum probability theory) Dynamics of discretized Schroedinger Equation. (E. Farhi and S. Gutmann, Phys. Rev. A 58, 915 (1998))

18. Ph.D defense on August 18th17 Connections in asymptotic behaviors From the viewpoint of the limit distribution, DTQW CTQW Dirac eq. Schroedinger eq. (A. Childs and J. Goldstone, Phys. Rev. A 70, 042312 (2004)) Increasing the dimension Continuum Limit Time-dependent coin & Re-scale Lattice-size-dependent coin

19. Ph.D defense on August 18th18 Dirac Equation from DTQW Coin Operator Time Evolution of Quantum Walk Note that this cannot represents arbitrary coin flip. (F. W. Strauch, J. Math. Phys. 48, 082102 (2007).)

20. Ph.D defense on August 18th19 Dirac Equation from DTQW Position of Dirac Particle : Walker Space Spinor : Coin Space

21. Ph.D defense on August 18th20 From DTQW to CTQW ( K. Chisaki, N. Konno, E. Segawa, and YS, Quant. Inf. Comp. 11, 0741 (2011). ) Coin operator Limit distribution By the re-scale, this model corresponds to the CTQW. (Related work in [A. Childs, Commun. Math. Phys. 294, 581 (2010).])

22. Ph.D defense on August 18th21 Connections in asymptotic behaviors DTQW CTQW Dirac eq. Schroedinger eq. (A. Childs and J. Goldstone, Phys. Rev. A 70, 042312 (2004).) Increasing the dimension Continuum Limit Time-dependent coin & Re-scale Lattice-size-dependent coin DTQW can simulate some dynamical features in some quantum systems.

23. Ph.D defense on August 18th22 DTQW with decoherence Simple Decoherence Model: Position measurement for each step w/ probability “ p ”.

24. Ph.D defense on August 18th23 0 1 1 ( YS, K. Chisaki, E. Segawa, and N. Konno, Phys. Rev. A 81, 062129 (2010).) (K. Chisaki, N. Konno, E. Segawa, and YS, Quant. Inf. Comp. 11, 0741 (2011). ) Time Scaled Limit Distribution (Crossover!!) Symmetric DTQW with position measurement with time-dependent probability

25. Ph.D defense on August 18th24 100th step of Walks

26. Ph.D defense on August 18th25 What do we know from this analytical results? 0 1 1 Almost all discrete time quantum walks with decoherence has the normal distribution. This is the reason why the large steps of the DTQW have not experimentally realized yet.

27. Ph.D defense on August 18th26 Summary of DTQW

28. Ph.D defense on August 18th27 I showed the limit distributions of the DTQWs on the one dimensional system. Under the simple decoherence model, I showed that the DTQW can be linearly mapped to the DTRW. –YS, K. Chisaki, E. Segawa, N. Konno, Phys. Rev. A 81, 062129 (2010). –K. Chisaki, N. Konno, E. Segawa, YS, Quant. Inf. Comp. 11, 0741 (2011).

29. Ph.D defense on August 18th28 Summary of Weak Value

30. Ph.D defense on August 18th29 I showed that the weak value was independently defined from the quantum measurement to characterize the observable-independent probability space. I showed that the counter-factual property could be characterized by the weak value. I naturally characterized the weak value with decoherence. –YS and A. Hosoya, J. Phys. A 42, 025304 (2010). –A. Hosoya and YS, J. Phys. A 43, 385307 (2010).

31. Ph.D defense on August 18th30 Quid est ergo tempus? Si nemo ex me quaerat, scio; si quaerenti explicare velim, nescio. by St. Augustine What is time?

32. Ph.D defense on August 18th31 Conclusion of this Thesis Toward understanding what time is, I compared the quantum and the classical worlds by two tools, the weak value and the discrete time quantum walk. Quantum Classical Measurement / Decoherence Quantization

33. Ph.D defense on August 18th32

34. Ph.D defense on August 18th33 DTRW v.s. DTQW coin position Rolling the coinShift of the position due to the coin Classical WalkQuantum Walk Unitary operator

35. Ph.D defense on August 18th34 DTRW v.s. DTQW Classical Walk Quantum Walk coin position Rolling the coinShift of the position due to the coin Unitary operator

36. Ph.D defense on August 18th35 Cf: Localization of DTQW (Appendix B) In the spatially inhomogeneous case, what behaviors should we see? Our Model Self-dual model inspired by the Aubry-Andre model In the dual basis, the roles of coin and shift are interchanged. Dual basis

37. Ph.D defense on August 18th36 - - Probability Distribution at the 1000-th Step Initial Coin state

38. Ph.D defense on August 18th37 Limit Distribution (Appendix B) Theorem (YS and H. Katsura, Phys. Rev. E 82, 031122 (2010))

39. Ph.D defense on August 18th38 When is the probability space defined? Hilbert space H Observable A Probability space Case 1Case 2 Hilbert space H Observable A Probability space

40. Ph.D defense on August 18th39 Definition of (Discrete) Probability Space Event Space Ω Probability Measure dP Random Variable X: Ω -> K The expectation value is

41. Ph.D defense on August 18th40 Number (Prob. Dis.)Even/Odd (Prob. Dis.) 1 2 3 6 1/6 1 0 1 0 Expectation Value Event Space 21/6 = 7/2 3/6 = 1/2

42. Ph.D defense on August 18th41 Example Position Operator Momentum Operator Not Correspondence!! Observable-dependent Probability Space

43. Ph.D defense on August 18th42 When is the probability space defined? Hilbert space H Observable A Probability space Case 1Case 2 Hilbert space H Observable A Probability space

44. Ph.D defense on August 18th43 Observable-independent Probability Space?? We can construct the probability space independently on the observable by the weak values. pre-selected statepost-selected state Def: Weak values of observable A (Y. Aharonov, D. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988))

45. Ph.D defense on August 18th44 Expectation Value? is defined as the probability measure. Born Formula ⇒ Random Variable ＝ Weak Value (A. Hosoya and YS, J. Phys. A 43, 385307 (2010))

46. Ph.D defense on August 18th45 Definition of Probability Space Event Space Ω Probability Measure dP Random Variable X: Ω -> K The expectation value is

47. Ph.D defense on August 18th46 Number (Prob. Dis.)Even/Odd (Prob. Dis.) 1 2 3 6 1/6 1 0 1 0 Expectation Value Event Space 21/6 = 7/2 3/6 = 1/2

48. Ph.D defense on August 18th47 Definition of Weak Values pre-selected statepost-selected state Def: Weak values of observable A Def: Weak measurement is called if a coupling constant with a probe interaction is very small. (Y. Aharonov, D. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)) To measure the weak value…

49. Ph.D defense on August 18th48 Target system Observable A Probe system the pointer operator (position of the pointer) is Q and its conjugate operator is P. Since the weak value of A is complex in general, (R. Jozsa, Phys. Rev. A 76, 044103 (2007)) Weak values are experimentally accessible by some experiments. (This is not unique!!) One example to measure the weak value

50. Ph.D defense on August 18th49 Fundamental Test of Quantum Theory –Direct detection of Wavefunction (J. Lundeen et al., Nature 474, 188 (2011)) –Trajectories in Young’s double slit experiment (S. Kocsis et al., Science 332, 1198 (2011)) –Violation of Leggett-Garg’s inequality (A. Palacios-Laloy et al. Nat. Phys. 6, 442 (2010)) Amplification (Magnify the tiny effect) –Spin Hall Effect of Light (O. Hosten and P. Kwiat, Science 319, 787 (2008)) –Stability of Sagnac Interferometer (P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009)) (D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 011802 (2010) (R)) –Negative shift of the optical axis (K. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Lett. A 324, 125 (2004)) Quantum Phase (Geometric Phase) (H. Kobayashi et al., J. Phys. Soc. Jpn. 81, 034401 (2011))

51. Ph.D defense on August 18th50 Rest of Today’s talk 1.What is the Weak Value? Observable-independent probability space 2.Counter-factual phenomenon: Hardy’s Paradox 3.Weak Value with Decoherence 4.Conclusion

52. Ph.D defense on August 18th51 Hardy’s Paradox B D B D 50/50 beam splitter Mirror Path O Path I Path O Positron Electron annihilation BB DB BD DD (L. Hardy, Phys. Rev. Lett. 68, 2981 (1992))

53. Ph.D defense on August 18th52 From Classical Arguments Assumptions: –There is NO non-local interaction. –Consider the intermediate state for the path based on the classical logic. The detectors DD cannot simultaneously click.

54. Ph.D defense on August 18th53 Why does the paradox be occurred? Before the annihilation point: Annihilation must occur. 2 nd Beam Splitter Prob. 1/12 How to experimentally confirm this state?

55. Ph.D defense on August 18th54 Hardy’s Paradox B D B D 50/50 beam splitter Mirror Path O Path I Path O Positron Electron BB DB BD DD

56. Ph.D defense on August 18th55 Counter-factual argument For the pre-selected state, the following operators are equivalent: Analogously, (A. Hosoya and YS, J. Phys. A 43, 385307 (2010))

57. Ph.D defense on August 18th56 What is the state-dependent equivalence? State-dependent equivalence

58. Ph.D defense on August 18th57 Counter-factual arguments For the pre-selected state, the following operators are equivalent: Analogously,

59. Ph.D defense on August 18th58 Pre-Selected State and Weak Value Experimentally realizable!!

60. Ph.D defense on August 18th59 Rest of Today’s talk 1.What is the Weak Value? Observable-independent probability space 2.Counter-factual phenomenon: Hardy’s Paradox 3.Weak Value with Decoherence 4.Conclusion

61. Ph.D defense on August 18th60 Completely Positive map Positive map When is positive map, is called a completely positive map (CP map). Arbitrary extension of Hilbert space (M. Ozawa, J. Math. Phys. 25, 79 (1984))

62. Ph.D defense on August 18th61 Operator-Sum Representation Any quantum state change can be described as the operation only on the target system via the Kraus operator. In the case of Weak Values???

63. Ph.D defense on August 18th62 W Operator In order to define the quantum operations associated with the weak values, W Operator (YS and A. Hosoya, J. Phys. A 43, 0215304 (2010))

64. Ph.D defense on August 18th63 Properties of W Operator Relationship to Weak Value Analogous to the expectation value

65. Ph.D defense on August 18th64 Quantum Operations for W Operators The properties of the quantum operation are 1.Two Kraus operators 2.Partial trace for the auxiliary Hilbert space 3.Mixed states for the W operator Key points of Proof: 1.Polar decomposition for the W operator 2.Complete positivity of the quantum operation S-matrix for the combined system

66. Ph.D defense on August 18th65 system Pre-selected state environment Post-selected state

67. Ph.D defense on August 18th66 Conclusion We obtain the properties of the weak value; –To be naturally defined as the observable- independent probability space. –To quantitatively characterize the counter-factual phenomenon. –To give the analytical expression with the decoherence. The weak value may be a fundamental quantity to understand the properties of time. For example, the delayed-choice experiment. Thank you so much for your attention.