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 Defense

2. Ph.D defense on July 1st, 20111 Quid est ergo tempus? Si nemo ex me quaerat, scio; si quaerenti explicare velim, nescio. by St. Augustine, (in Book 11, Chapter 14, Confessions (Latin: Confessiones))

3. Ph.D defense on July 1st, 20112 Time in Physics Ref: http://www.youtube.com/watch?v=yENvgXYzGxw Induction Reduction

4. Ph.D defense on July 1st, 20113 Time in Physics Ref: http://www.youtube.com/watch?v=yENvgXYzGxw Induction Reduction Absolute time (Parameter)

5. Ph.D defense on July 1st, 20114 Time in quantum mechanics CM QM Canonical Quantization Absolute time (Parameter)

6. Ph.D defense on July 1st, 20115 Time in quantum mechanics CM QM Canonical Quantization The time operator is not self-adjoint in the case that the Hamiltonian is bounded proven by Pauli.

7. Ph.D defense on July 1st, 20116 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) YS, K. Chisaki, E. Segawa, N. Konno, Phys. Rev. A 81, 062129 (2010). K. Chisaki, N. Konno, E. Segawa, YS, to appear in Quant. Inf. Comp. arXiv:1009.2131. M. Gönülol, E. Aydiner, YS, and Ö. E. Mustecaplıo˜glu, New J. Phys. 13, 033037 (2011). –Weak Value 3.Construct an alternative framework. Aim: Construct a concrete method and a specific model to understand the properties of time

8. Ph.D defense on July 1st, 20117 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

9. Ph.D defense on July 1st, 20118 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).

10. Ph.D defense on July 1st, 20119 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

11. Ph.D defense on July 1st, 201110 In Chaps. 4 and 5, on Discrete Time Quantum Walks Classical random walk Discrete Time Quantum Walk How to relate?? Simple decoherence model

12. Ph.D defense on July 1st, 201111 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

13. Ph.D defense on July 1st, 201112 When is the probability space defined? Hilbert space H Observable A Probability space Case 1Case 2 Hilbert space H Observable A Probability space

14. Ph.D defense on July 1st, 201113 Definition of (Discrete) Probability Space Event Space Ω Probability Measure dP Random Variable X: Ω -> K The expectation value is

15. Ph.D defense on July 1st, 201114 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

16. Ph.D defense on July 1st, 201115 Example Position Operator Momentum Operator Not Correspondence!! Observable-dependent Probability Space

17. Ph.D defense on July 1st, 201116 When is the probability space defined? Hilbert space H Observable A Probability space Case 1Case 2 Hilbert space H Observable A Probability space

18. Ph.D defense on July 1st, 201117 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))

19. Ph.D defense on July 1st, 201118 Expectation Value? is defined as the probability measure. Born Formula ⇒ Random Variable ＝ Weak Value (A. Hosoya and YS, J. Phys. A 43, 385307 (2010))

20. Ph.D defense on July 1st, 201119 Definition of Probability Space Event Space Ω Probability Measure dP Random Variable X: Ω -> K The expectation value is

21. Ph.D defense on July 1st, 201120 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

22. Ph.D defense on July 1st, 201121 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…

23. Ph.D defense on July 1st, 201122 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

24. Ph.D defense on July 1st, 201123 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))

25. Ph.D defense on July 1st, 201124 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

26. Ph.D defense on July 1st, 201125 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))

27. Ph.D defense on July 1st, 201126 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.

28. Ph.D defense on July 1st, 201127 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?

29. Ph.D defense on July 1st, 201128 Hardy’s Paradox B D B D 50/50 beam splitter Mirror Path O Path I Path O Positron Electron BB DB BD DD

30. Ph.D defense on July 1st, 201129 Counter-factual argument For the pre-selected state, the following operators are equivalent: Analogously, (A. Hosoya and YS, J. Phys. A 43, 385307 (2010))

31. Ph.D defense on July 1st, 201130 What is the state-dependent equivalence? State-dependent equivalence

32. Ph.D defense on July 1st, 201131 Counter-factual arguments For the pre-selected state, the following operators are equivalent: Analogously,

33. Ph.D defense on July 1st, 201132 Pre-Selected State and Weak Value Experimentally realizable!!

34. Ph.D defense on July 1st, 201133 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

35. Ph.D defense on July 1st, 201134 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))

36. Ph.D defense on July 1st, 201135 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???

37. Ph.D defense on July 1st, 201136 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))

38. Ph.D defense on July 1st, 201137 Properties of W Operator Relationship to Weak Value Analogous to the expectation value

39. Ph.D defense on July 1st, 201138 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

40. Ph.D defense on July 1st, 201139 system Pre-selected state environment Post-selected state

41. Ph.D defense on July 1st, 201140 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.

42. Ph.D defense on July 1st, 201141

43. Ph.D defense on July 1st, 201142 Discrete Time Random Walk (DTRW) Coin Flip Shift Repeat

44. Ph.D defense on July 1st, 201143 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.)

45. Ph.D defense on July 1st, 201144 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:

46. Ph.D defense on July 1st, 201145 Example of DTQW 0123-2-3 step 0 1 2 3 1/129/121/12 prob. Quantum Coherence and Interference

47. Ph.D defense on July 1st, 201146 Probability Distribution at the 1000-th step Initial Coin State Coin Operator DTQWDTRW Unbiased Coin (Left and Right with probability ½ )

48. Ph.D defense on July 1st, 201147 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

49. Ph.D defense on July 1st, 201148 Probability Distribution at the 1000-th step Initial Coin State Coin Operator DTQWDTRW Unbiased Coin (Left and Right with probability ½ )

50. Ph.D defense on July 1st, 201149 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)

51. Ph.D defense on July 1st, 201150 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))

52. Ph.D defense on July 1st, 201151 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

53. Ph.D defense on July 1st, 201152 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))

54. Ph.D defense on July 1st, 201153 Dirac Equation from DTQW Position of Dirac Particle : Walker Space Spinor : Coin Space

55. Ph.D defense on July 1st, 201154 From DTQW to CTQW (K. Chisaki, N. Konno, E. Segawa, and YS, arXiv:1009.2131.) 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)])

56. Ph.D defense on July 1st, 201155 DTQW with decoherence Simple Decoherence Model: Position measurement for each step w/ probability “ p ”.

57. Ph.D defense on July 1st, 201156 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, arXiv:1009.2131. ) Time Scaled Limit Distribution (Crossover!!) Symmetric DTQW with position measurement with time-dependent probability