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Time in the Weak Value and the Discrete Time Quantum Walk

Ph. D Defense. Time in the Weak Value and the Discrete Time Quantum Walk. Yutaka Shikano Theoretical Astrophysics Group, Department of Physics, Tokyo Institute of Technology. Quid est ergo tempus? Si nemo ex me quaerat, scio; si quaerenti explicare velim, nescio. by St. Augustine ,

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Time in the Weak Value and the Discrete Time Quantum Walk

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

  2. 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)) Ph.D defense on July 1st, 2011

  3. Time in Physics Induction Ref: http://www.youtube.com/watch?v=yENvgXYzGxw Reduction Ph.D defense on July 1st, 2011

  4. Time in Physics Induction Ref: http://www.youtube.com/watch?v=yENvgXYzGxw Reduction Absolute time (Parameter) Ph.D defense on July 1st, 2011

  5. Time in quantum mechanics CM Canonical Quantization QM Absolute time (Parameter) Ph.D defense on July 1st, 2011

  6. Time in quantum mechanics CM Canonical Quantization QM The time operator is not self-adjoint in the case that the Hamiltonian is bounded proven by Pauli. Ph.D defense on July 1st, 2011

  7. Change the definition / interpretation of the observable Extension to the symmetric operator YS and A. Hosoya, J. Math. Phys. 49, 052104 (2008). 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 Construct an alternative framework. How to characterize time in quantum mechanics? Aim: Construct a concrete method and a specific model to understand the properties of time Ph.D defense on July 1st, 2011

  8. 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 Ph.D defense on July 1st, 2011

  9. Appendixes • Hamiltonian Estimation by Weak Measurement • YS and S. Tanaka, arXiv:1007.5370. • 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. • Weak Measurement with Environment • YS and A. Hosoya, J. Phys. A 43, 0215304 (2010). • Geometric Phase for Mixed States • YS and A. Hosoya, J. Phys. A 43, 0215304 (2010). Ph.D defense on July 1st, 2011

  10. 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 Ph.D defense on July 1st, 2011

  11. In Chaps. 4 and 5, on Discrete Time Quantum Walks Classical random walk How to relate?? Discrete Time Quantum Walk Simple decoherence model Ph.D defense on July 1st, 2011

  12. Rest of Today’s talk • What is the Weak Value? • Observable-independent probability space • Counter-factual phenomenon: Hardy’s Paradox • Weak Value with Decoherence • Conclusion Ph.D defense on July 1st, 2011

  13. When is the probability space defined? Hilbert space H Hilbert space H Probability space Observable A Observable A Probability space Case 1 Case 2 Ph.D defense on July 1st, 2011

  14. Definition of (Discrete) Probability Space Event Space Ω Probability Measure dP Random VariableX: Ω -> K The expectation value is Ph.D defense on July 1st, 2011

  15. Event Space Number (Prob. Dis.) Even/Odd (Prob. Dis.) 1 1/6 1 1/6 0 1/6 1/6 2 1 1/6 1/6 3 6 1/6 0 1/6 3/6 = 1/2 21/6 = 7/2 Expectation Value Ph.D defense on July 1st, 2011

  16. Example Position Operator Momentum Operator Not Correspondence!! Observable-dependent Probability Space Ph.D defense on July 1st, 2011

  17. When is the probability space defined? Hilbert space H Hilbert space H Probability space Observable A Observable A Probability space Case 1 Case 2 Ph.D defense on July 1st, 2011

  18. Observable-independent Probability Space?? • We can construct the probability space independently on the observable by the weak values. Def: Weak values of observable A pre-selected state post-selected state (Y. Aharonov, D. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)) Ph.D defense on July 1st, 2011

  19. Expectation Value? (A. Hosoya and YS, J. Phys. A 43, 385307 (2010)) is defined as the probability measure. Born Formula ⇒ Random Variable＝Weak Value Ph.D defense on July 1st, 2011

  20. Definition of Probability Space Event Space Ω Probability Measure dP Random VariableX: Ω -> K The expectation value is Ph.D defense on July 1st, 2011

  21. Event Space Number (Prob. Dis.) Even/Odd (Prob. Dis.) 1 1/6 1 1/6 0 1/6 1/6 2 1 1/6 1/6 3 6 1/6 0 1/6 3/6 = 1/2 21/6 = 7/2 Expectation Value Ph.D defense on July 1st, 2011

  22. Definition of Weak Values Def: Weak values of observable A pre-selected state post-selected state To measure the weak value… 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)) Ph.D defense on July 1st, 2011

  23. One example to measure the weak value Probe system the pointer operator (position of the pointer) is Q and its conjugate operator is P. Target system Observable A Since the weak value of A is complex in general, Weak values are experimentally accessible by some experiments. (This is not unique!!) (R. Jozsa, Phys. Rev. A 76, 044103 (2007)) Ph.D defense on July 1st, 2011

  24. 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)) Ph.D defense on July 1st, 2011

  25. Rest of Today’s talk • What is the Weak Value? • Observable-independent probability space • Counter-factual phenomenon: Hardy’s Paradox • Weak Value with Decoherence • Conclusion Ph.D defense on July 1st, 2011

  26. annihilation Hardy’s Paradox (L. Hardy, Phys. Rev. Lett. 68, 2981 (1992)) B 50/50 beam splitter Path O Mirror D Path I D BB DB BD DD B Path I Positron Path O Electron Ph.D defense on July 1st, 2011

  27. 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. Ph.D defense on July 1st, 2011

  28. Why does the paradox be occurred? Before the annihilation point: Annihilation must occur. How to experimentally confirm this state? 2nd Beam Splitter Prob. 1/12 Ph.D defense on July 1st, 2011

  29. Hardy’s Paradox B 50/50 beam splitter Path O Mirror D Path I D BB DB BD DD B Path I Positron Path O Electron Ph.D defense on July 1st, 2011

  30. Counter-factual argument (A. Hosoya and YS, J. Phys. A 43, 385307 (2010)) • For the pre-selected state, the following operators are equivalent: Analogously, Ph.D defense on July 1st, 2011

  31. What is the state-dependent equivalence? State-dependent equivalence Ph.D defense on July 1st, 2011

  32. Counter-factual arguments • For the pre-selected state, the following operators are equivalent: Analogously, Ph.D defense on July 1st, 2011

  33. Pre-Selected State and Weak Value Experimentally realizable!! Ph.D defense on July 1st, 2011

  34. Rest of Today’s talk • What is the Weak Value? • Observable-independent probability space • Counter-factual phenomenon: Hardy’s Paradox • Weak Value with Decoherence • Conclusion Ph.D defense on July 1st, 2011

  35. Completely Positive map Positive map Arbitrary extension of Hilbert space When is positive map, is called a completely positive map (CP map). (M. Ozawa, J. Math. Phys. 25, 79 (1984)) Ph.D defense on July 1st, 2011

  36. 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??? Ph.D defense on July 1st, 2011

  37. W Operator (YS and A. Hosoya, J. Phys. A 43, 0215304 (2010)) • In order to define the quantum operations associated with the weak values, W Operator Ph.D defense on July 1st, 2011

  38. Properties of W Operator Relationship to Weak Value Analogous to the expectation value Ph.D defense on July 1st, 2011

  39. Quantum Operations for W Operators • Key points of Proof: • Polar decomposition for the W operator • Complete positivity of the quantum operation S-matrix for the combined system • The properties of the quantum operation are • Two Kraus operators • Partial trace for the auxiliary Hilbert space • Mixed states for the W operator Ph.D defense on July 1st, 2011

  40. environment system Post-selected state Pre-selected state environment Ph.D defense on July 1st, 2011

  41. 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. Ph.D defense on July 1st, 2011

  42. Ph.D defense on July 1st, 2011

  43. Discrete Time Random Walk (DTRW) Coin Flip Shift Repeat Ph.D defense on July 1st, 2011

  44. Discrete Time Quantum Walk (DTQW) (A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, in STOC’01 (ACM Press, New York, 2001), pp. 37 – 49.) Quantum Coin Flip Shift Repeat Ph.D defense on July 1st, 2011

  45. Example of DTQW Initial Condition Position: n = 0 (localized) Coin: Coin Operator: Hadamard Coin Probability distribution of the n-th cite at t step: Let’s see the dynamics of quantum walk by 3rd step! Ph.D defense on July 1st, 2011

  46. Example of DTQW -3 -2 -1 0 1 2 3 0 1 2 3 prob. 1/12 9/12 1/12 1/12 step Quantum Coherence and Interference Ph.D defense on July 1st, 2011

  47. Probability Distribution at the 1000-th step DTRW DTQW Initial Coin State Unbiased Coin (Left and Right with probability ½) Coin Operator Ph.D defense on July 1st, 2011

  48. Weak Limit Theorem (Limit Distribution) DTRW Central Limit Theorem Prob. 1/2 Prob. 1/2 DTQW N. Konno, Quantum Information Processing 1, 345 (2002) Coin operator Initial state Probability density 47 Ph.D defense on July 1st, 2011

  49. Probability Distribution at the 1000-th step DTRW DTQW Initial Coin State Unbiased Coin (Left and Right with probability ½) Coin Operator Ph.D defense on July 1st, 2011

  50. 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) Ph.D defense on July 1st, 2011

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