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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 1 How to characterize time in quantum mechanics? Aim: Construct a concrete method and a specific model to understand the properties of time 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. Ph.D defense on August 18th 2 Organization of Thesis Chapter 1: Introduction Chapter 2: Preliminaries Chapter 3: Chapter 4: Counter-factual Properties of Weak Value Asymptotic Behavior of Discrete Time Quantum Walks Chapter 5: Decoherence Properties Chapter 6: Concluding Remarks Ph.D defense on August 18th 3 Appendixes A) Hamiltonian Estimation by Weak Measurement • B) 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. 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). Ph.D defense on August 18th 4 Organization of Thesis Chapter 1: Introduction Chapter 2: Preliminaries Chapter 3: Chapter 4: Counter-factual Properties of Weak Value Asymptotic Behavior of Discrete Time Quantum Walks Chapter 5: Decoherence Properties Chapter 6: Concluding Remarks Ph.D defense on August 18th 5 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 Ph.D defense on August 18th 6 Discrete Time Random Walk (DTRW) Coin Flip Shift Repeat Ph.D defense on August 18th 7 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 8 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 9 Example of DTQW -3 -2 -1 0 1 2 3 0 1 2 3 1/12 step 9/12 1/12 1/12 prob. Quantum Coherence and Interference Ph.D defense on August 18th 10 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 11 Weak Limit Theorem (Limit Distribution) DTRW Central Limit Theorem Prob. 1/2 DTQW Prob. 1/2 (N. Konno, Quantum Information Processing 1, 345 (2002).) Coin operator Initial state Probability density 12 Ph.D defense on August 18th 12 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 13 Weak Limit Theorem (Limit Distribution) DTRW Central Limit Theorem Prob. 1/2 DTQW Prob. 1/2 (N. Konno, Quantum Information Processing 1, 345 (2002).) Coin operator Initial state Probability density 14 Ph.D defense on August 18th 14 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 15 Continuous Time Quantum Walk (CTQW) Dynamics of discretized Schroedinger Equation. (E. Farhi and S. Gutmann, Phys. Rev. A 58, 915 (1998)) Limit Distribution (Arcsin Law <- Quantum probability theory) p.d. • Experimental Realization • A. Peruzzo et al., Science 329, 1500 (2010). (Photon, Waveguide) Ph.D defense on August 18th 16 Connections in asymptotic behaviors From the viewpoint of the limit distribution, DTQW Lattice-size-dependent coin Time-dependent coin & Re-scale Increasing the dimension Dirac eq. CTQW (A. Childs and J. Goldstone, Phys. Rev. A 70, 042312 (2004)) Continuum Limit Schroedinger eq. Ph.D defense on August 18th 17 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 18 Dirac Equation from DTQW Position of Dirac Particle : Walker Space Spinor : Coin Space Ph.D defense on August 18th 19 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 20 Connections in asymptotic behaviors DTQW Lattice-size-dependent coin Time-dependent coin & Re-scale Increasing the dimension Dirac eq. CTQW (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 21 DTQW with decoherence Simple Decoherence Model: Position measurement for each step w/ probability “p”. Ph.D defense on August 18th 22 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).) Symmetric DTQW with position measurement with time-dependent probability 1 0 Ph.D defense on August 18th 1 23 100th step of Walks Ph.D defense on August 18th 24 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 Ph.D defense on August 18th 1 25 Summary of DTQW Ph.D defense on August 18th 26 • 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 27 Summary of Weak Value Ph.D defense on August 18th 28 • 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 29 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 30 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 31 Ph.D defense on August 18th 32 DTRW v.s. DTQW position coin Unitary operator Rolling the coin Classical Walk Shift of the position due to the coin Quantum Walk Ph.D defense on August 18th 33 DTRW v.s. DTQW position coin Unitary operator Rolling the coin Shift of the position due to the coin Classical Walk Quantum Walk Ph.D defense on August 18th 34 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 35 Probability Distribution at the 1000-th Step Initial Coin state - Ph.D defense on August 18th 36 Limit Distribution (Appendix B) Theorem (YS and H. Katsura, Phys. Rev. E 82, 031122 (2010)) Ph.D defense on August 18th 37 When is the probability space defined? Hilbert space H Hilbert space H Observable A Probability space Probability space Observable A Case 1 Case 2 Ph.D defense on August 18th 38 Definition of (Discrete) Probability Space Event Space Ω Probability Measure dP Random Variable X: Ω -> K The expectation value is Ph.D defense on August 18th 39 Event Space Expectation Value Number (Prob. Dis.) Even/Odd (Prob. Dis.) 1 1/6 1 1/6 2 1/6 0 1/6 3 1/6 1 1/6 6 1/6 0 1/6 21/6 = 7/2 Ph.D defense on August 18th 3/6 = 1/2 40 Example Position Operator Momentum Operator Not Correspondence!! Observable-dependent Probability Space Ph.D defense on August 18th 41 When is the probability space defined? Hilbert space H Hilbert space H Observable A Probability space Probability space Observable A Case 1 Case 2 Ph.D defense on August 18th 42 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 43 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 44 Definition of Probability Space Event Space Ω Probability Measure dP Random Variable X: Ω -> K The expectation value is Ph.D defense on August 18th 45 Event Space Expectation Value Number (Prob. Dis.) Even/Odd (Prob. Dis.) 1 1/6 1 1/6 2 1/6 0 1/6 3 1/6 1 1/6 6 1/6 0 1/6 21/6 = 7/2 Ph.D defense on August 18th 3/6 = 1/2 46 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 47 One example to measure the weak value 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, 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 48 • 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 49 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 Ph.D defense on August 18th 50 Hardy’s Paradox (L. Hardy, Phys. Rev. Lett. 68, 2981 (1992)) B 50/50 beam splitter Path O Mirror D Path I D BB annihilation Path I Positron Electron Path O Ph.D defense on August 18th DB B BD DD 51 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 August 18th 52 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 August 18th 53 Hardy’s Paradox B 50/50 beam splitter Path O Mirror D Path I D BB DB B Path I Positron Electron Path O Ph.D defense on August 18th BD DD 54 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 August 18th 55 What is the state-dependent equivalence? State-dependent equivalence Ph.D defense on August 18th 56 Counter-factual arguments • For the pre-selected state, the following operators are equivalent: Analogously, Ph.D defense on August 18th 57 Pre-Selected State and Weak Value Experimentally realizable!! Ph.D defense on August 18th 58 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 Ph.D defense on August 18th 59 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 August 18th 60 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 August 18th 61 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 August 18th 62 Properties of W Operator Relationship to Weak Value Analogous to the expectation value Ph.D defense on August 18th 63 Quantum Operations for W Operators Key points of Proof: 1. Polar decomposition for the W operator 2. Complete positivity of the quantum operation S-matrix for the combined system 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 Ph.D defense on August 18th 64 environment system Post-selected state Pre-selected state environment Ph.D defense on August 18th 65 Conclusion • We obtain the properties of the weak value; – To be naturally defined as the observableindependent 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 August 18th 66