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## Time in the Weak Value and the Discrete Time Quantum Walk

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**Ph. D Final Defense**Time in the Weak Value and the Discrete Time Quantum Walk Yutaka Shikano Theoretical Astrophysics Group, Department of Physics, Tokyo Institute of Technology**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). Ph.D defense on August 18th**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) Weak Value YS and A. Hosoya, J. Phys. A 42, 025304 (2010). A. Hosoya and YS, J. Phys. A 43, 385307 (2010). 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 August 18th**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 August 18th**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 August 18th**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 August 18th**Rest of Today’s talk**• What is the discrete time quantum walk? • Asymptotic behaviors of the discrete time quantum walks • Discrete time quantum walk under the simple decoherence model • Conclusion • Summary of the discrete time quantum walks • Summary of the weak value • Summary of this thesis Ph.D defense on August 18th**Discrete Time Random Walk (DTRW)**Coin Flip Shift Repeat Ph.D defense on August 18th**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 August 18th**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 August 18th**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 August 18th**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 August 18th**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 12 Ph.D defense on August 18th**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 August 18th**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 14 Ph.D defense on August 18th**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 August 18th**Continuous Time Quantum Walk (CTQW)**Dynamics of discretized Schroedinger Equation. • Experimental Realization • A. Peruzzo et al., Science 329, 1500 (2010). (Photon, Waveguide) (E. Farhi and S. Gutmann, Phys. Rev. A 58, 915 (1998)) Limit Distribution (Arcsin Law <- Quantum probability theory) p.d. Ph.D defense on August 18th**Connections in asymptotic behaviors**From the viewpoint of the limit distribution, DTQW Time-dependent coin & Re-scale Lattice-size-dependent coin Increasing the dimension CTQW Dirac eq. (A. Childs and J. Goldstone, Phys. Rev. A 70, 042312 (2004)) Continuum Limit Schroedinger eq. Ph.D defense on August 18th**Dirac Equation from DTQW**(F. W. Strauch, J. Math. Phys.48, 082102 (2007).) Coin Operator Note that this cannot represents arbitrary coin flip. Time Evolution of Quantum Walk Ph.D defense on August 18th**Dirac Equation from DTQW**Position of Dirac Particle : Walker Space Spinor : Coin Space Ph.D defense on August 18th**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).]) Ph.D defense on August 18th**Connections in asymptotic behaviors**DTQW Time-dependent coin & Re-scale Lattice-size-dependent coin Increasing the dimension CTQW Dirac eq. (A. Childs and J. Goldstone, Phys. Rev. A 70, 042312 (2004).) Continuum Limit Schroedinger eq. DTQW can simulate some dynamical features in some quantum systems. Ph.D defense on August 18th**DTQW with decoherence**Simple Decoherence Model: Position measurement for each step w/ probability “p”. Ph.D defense on August 18th**Time Scaled Limit Distribution (Crossover!!)**(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).) 1 Symmetric DTQW with position measurement with time-dependent probability 0 1 Ph.D defense on August 18th**100th step of Walks**Ph.D defense on August 18th**What do we know from this analytical results?**Almost all discrete time quantum walks with decoherence has the normal distribution. 1 This is the reason why the large steps of the DTQW have not experimentally realized yet. 0 1 Ph.D defense on August 18th**Summary of DTQW**Ph.D defense on August 18th**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). Ph.D defense on August 18th**Summary of Weak Value**Ph.D defense on August 18th**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). Ph.D defense on August 18th**What is time?**Quid est ergo tempus? Si nemo ex me quaerat, scio; si quaerenti explicare velim, nescio. by St. Augustine Ph.D defense on August 18th**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. Quantization Quantum Classical Measurement / Decoherence Ph.D defense on August 18th**DTRW v.s. DTQW**position Unitary operator coin Rolling the coin Shift of the position due to the coin Classical Walk Quantum Walk Ph.D defense on August 18th**DTRW v.s. DTQW**position Unitary operator coin Rolling the coin Shift of the position due to the coin Classical Walk Quantum Walk Ph.D defense on August 18th**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 Ph.D defense on August 18th**-**- Probability Distribution at the 1000-th Step Initial Coin state Ph.D defense on August 18th**Limit Distribution (Appendix B)**(YS and H. Katsura, Phys. Rev. E 82, 031122 (2010)) Theorem Ph.D defense on August 18th**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 August 18th**Definition of (Discrete) Probability Space**Event Space Ω Probability Measure dP Random VariableX: Ω -> K The expectation value is Ph.D defense on August 18th**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 August 18th**Example**Position Operator Momentum Operator Not Correspondence!! Observable-dependent Probability Space Ph.D defense on August 18th**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 August 18th**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 August 18th**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 August 18th**Definition of Probability Space**Event Space Ω Probability Measure dP Random VariableX: Ω -> K The expectation value is Ph.D defense on August 18th**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 August 18th**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 August 18th**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 August 18th**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 August 18th