Time in Quantum Walk and Weak Value

Transcription

1 Time in Quantum Walk and Weak Value Yutaka Shikano Department of Physics, Tokyo Institute of Technology Department of Mechanical Engineering, Massachusetts Institute of Technology 1

2 Outline 1. Two Stories on Weak Value and Probability 2. What is Weak Value? Definition Experimental Realization, ex. 3-box paradox 3. Why is useful to consider the weak value? Non-Contextual Probability Space 4. What is Quantum Walk? Definition Properties 5. Meet Quantum Walk and Weak Value 2

3 On Feb 8th, Anaheim I This think book the book is a good Quantum Paradoxes introduction is for pedagogical. quantum What mechanics. do you Did think? you buy it? 3

4 On Jan. 22nd, Oxford What is probability in Yes we can using the weak quantum mechanics? value. Look!! How do we confirm this?? d/david.html 4

5 Change the viewpoint!! Weak values can change the viewpoint and give a new insight of probability and the stochastic process. Probability -> Non-Contextual Probability Space Stochastic Process -> Quantum Walk 5

6 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)) 6

7 To Measure Weak Values Target system Observable A Probe system the pointer operator (position of the pointer) is q and its conjugate operator is p. State of the probe after measurement 7

8 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, We assume the probe wave function for the position be real-valued. : Initial probe variance for the momentum Weak values are experimentally accessible by the shifts of expectation values for the probe observables. (R. Jozsa, Phys. Rev. A 76, (2007)) 8

9 Strong Measurement Quantum State Projection in vitro experiment 9

10 Weak Measurement Cover Slightly Seeing in vivo experiment 10

11 Experimental Realizations All experiments are in optical systems. Spin Hall Effect of Light (Magnification) O. Hosten and P. Kwiat, Science 319, 787 (2008). 3-Box Paradox : strange shift of the light axis (Negative Probability) K. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Lett. A 324, 125 (2003) Hardy s Paradox (Negative Probability) K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, New J. Phys. 11, (2009). J. S. Lundeen and A. M. Steinberg, Phys. Rev. Lett. 102, (2009). Economists, May 5 th Wall Street Journal, May 5 th Stability of the Sagnac Interferometer (Application) P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, (2009). Violation of the Leggett-Garg Inequality (Application) M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O Brien, A. G. White, and G. J. Pryde, arxiv:

12 Experimental Realization (K. Resch, J. S. Lundeen and A. Steinberg, Phys. Lett. A 324, 125 (2003)) Prepare the initial state Post-selected state

13 Creating superposition of initial state 1 st step: Check the postselected state!! Shifting the phase for each path. Changeable From the interference pattern, we can construct the postselected 13 state.

14 2 nd step: See the image of CCD camera. Fixed Weak Measurement 14

15 Weak Measurement by Slide Glass (N. M. W. Ritchie, J. G. Story, and R. G. Hulet, Phys. Rev. Lett. 66, 1107 (2003)) Use transverse position of each photon as pointer Weak measurement can be performed by tilting a glass optical flat, where effective Probe Mode C θ Flat gt CCD camera 15

16 Perform weak measurement on rail C. Post-selection: rail A+B-C (negative shift) Post-selection: rail C (positive shift) Post-selection: rail A and B 16 (No shift)

17 Experimental Realization Prepare the initial state Post-selected state

18 Geometric Phase and Weak Value (E. Sjoqvist, Phys. Lett. A 359, 187 (2006)) Geometric Phase To measure the weak value, we measure the geometric phase. To measure the geometric phase, we can measure the weak value somewhat. S. Tamate, H. Kobayashi, T. Nakanishi, K. Sugiyama, and M. Kitano, New J. Phys. 11, (2009). 18

19 Quantum Eraser Analogue Experimental Realization: H. Kobayashi, S. Tamate, T. Nakanishi, K. Sugiyama and M. Kitano, arxiv:

20 Theoretical Analysis 1. No Which Way Path Measurement 2. Which Way Path Measurement 3. Which Way Path Measurement with Post-Selection 20

21 On the Bloch Sphere 21

22 Quantum Eraser Analogue 22

23 Labeled Weak Measurement 1. No Post-Selection 2. Post-Selection 23

24 Properties of Weak Value Experimental Accessible Quantity To relate the Quantum Phases To relate the Entanglement To solve the Quantum Paradoxes Strange Weak Value, Cf: Hofmann s talk Consistency with Quantum Mechanics Cf: Hosoya s talk Vancouver Summer Olympic? Vancouver is not snowing this Main Medal Place 24

25 Definition of Probability Space Event Space Ω Probability Measure dp Random Variable X: Ω -> K The expectation value is 25

26 Example in Conventional Analysis Position Operator Momentum Operator Not Correspondence!! 26

27 Contextuality for Probability Space Fact Probability space is decided depending on the measured observable. Theorem (Kochen and Specker 1967) On dim(h) 3, for any observables, there does not exist the value, which is not related to measurement. In the case of C = A + B, Outcome c Outcome a + Outcome b 27

28 Demerit of Contextual Probability Space How do we understand the uncertainty relationship proven by Robertson in 1929? cf. Botero s talk The statistics on the different probability space 28

29 Property of Random Variable Experimental accessible quantity 29

30 Expectation Value? is defined as the probability measure. Born Formula Random Variable=Weak Value 30

31 Variance? Probability measure is corresponded. 31

32 Why is useful to consider the noncontextual probability space? To apply this non-contextual probability space to a typical example of the stochastic process in quantum mechanics, the quantum walk. 32

33 Definition of Quantum Walk Original Works Quantum Probability S. P. Gudder, Quantum Probability (Academic Press Inc., San Diego, 1988). Quantization of Random Walk and Time Symmetric Property Y. Aharonov, L. Davidovich, and N. Zagury, Phys. Rev. A 48, 1687 (1993). Quantization of Cellular Automaton D. Meyer, J. Stat. Phys. 85, 551 (1996). Reviews J. Kempe, Contemp. Phys. 44, 307 (2003). N. Konno, in Quantum Potential Theory, pp (2008). S. E. Venegas-Andraca, Quantum Walks for Computer Scientists (Morgan and Claypool, 2008). 33

34 Discrete Time Quantum Walk (A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, in STOC 01 (ACM Press, New York, 2001), pp ) Quantum Coin Flip Shift Repeat Aharonov s quantum random walk is a bit different!! 34

35 Example of Quantum Walk Initial Condition Position: x = 0 (localized) Coin: Coin Operator: Hadamard Coin Let s see the dynamics of quantum walk by 3 step!! 35

36 Example of Quantum Walk step 1/12 9/12 1/12 1/12 prob. Quantum Coherence and Interference 36

37 Experimental Realizations and Applications Experimental Realizations Trapped Atom with Optical Lattice and Ion Trap M. Karski et al., Science 325, 174 (2009). 23 step F. Zahringer et al., arxiv: step Photon in Linear Optics (restricted) A. Schreiber et al., arxiv: step Applications Universal Quantum Computation N. B. Lovett et al. arxiv: (On Graph) Quantum Simulator (Phase Transition) T. Oka, N. Konno, R. Arita, and H. Aoki, Phys. Rev. Lett. 94, (2005). Realization of Quantum Walk = Quantum Computation / Simulation Device 37

38 Dirac Equation and Quantum Walk Quantum Coin Flip (F. W. Strauch, J. Math. Phys. 48, (2007)) Note that this cannot represents arbitrary coin flip. Time Evolution of Quantum Walk 38

39 Dirac Equation from Quantum Walk (F. W. Strauch, J. Math. Phys. 48, (2007)) Motion of Dirac Particle : Walker Space Spinor : Coin Space 39

40 At 100th step for the Numerical Simulation Initial Coin State Random Walk Quantum Walk 40

41 Weak Limit Theorem (Convergence in terms of the distribution) Random Walk Central Limit Theorem in the case of right and left probability ½. Quantum Walk N. Konno, Quantum Information Processing 1, 345 (2002) with the initial coin state density 11/4/2009 Duality and Scale in Quantum Science at YKIS 41

42 Weak Value and Quantum Walk 1 0 Pre-Selected State Known m Position and Coin Unknown n step Post-Selected State Known 42

43 Weak Value and Quantum Walk 1 0 Pre-Selected State Known m Position and Coin Known n Post-Selected State Calculate step 43

44 Conclusion Weak Value may be useful for every quantum people. In the mathematical context. Probability = Non-Contextual Probability Space (A. Hosoya and YS, in preparation) Stochastic Process = Analysis for Quantum Walks (YS, in preparation) 44

45 Open Questions Relationship to Quantum Probability? Where is the Gudder motivation? Relationship to General Probabilistic Theory? Can we define the stochastic process in the general probability theory? Relationship to Thermodynamics? To characterize the non-equilibrium? Relationship to Quantum Computation? Relationship to Quantum Annealing and Quantum Adiabatic Computation? 45