Post on 05-Feb-2016
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Time in the Weak Value and the Discrete Time Quantum Walk
Yutaka ShikanoTheoretical Astrophysics Group,
Department of Physics,
Tokyo Institute of Technology
Ph. D Final Defense
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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).
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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) – 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.
Aim: Construct a concrete method and a specific model to understand the properties of time
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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
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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).
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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
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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
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Discrete Time Random Walk (DTRW)
Coin Flip
Shift
Repeat
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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.)
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Example of DTQW
• Initial Condition– Position: n = 0 (localized)– Coin:
• Coin Operator: Hadamard Coin
Let’s see the dynamics of quantum walk by 3rd step!
Probability distribution of the n-th cite at t step:
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Example of DTQW
0 1 2 3-1-2-3
step
0
1
2
3
1/12 9/12 1/12 1/12 prob.
Quantum Coherence and Interference
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Probability Distribution at the 1000-th step
Initial Coin State
Coin Operator
DTQWDTRW
Unbiased Coin
(Left and Right with probability ½)
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13
Weak Limit Theorem (Limit Distribution)
DTRW
DTQW
Central Limit Theorem
(N. Konno, Quantum Information Processing 1, 345 (2002).)
Probability density
Coin operator Initial state
Prob. 1/2 Prob. 1/2
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Probability Distribution at the 1000-th step
Initial Coin State
Coin Operator
DTQWDTRW
Unbiased Coin
(Left and Right with probability ½)
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Weak Limit Theorem (Limit Distribution)
DTRW
DTQW
Central Limit Theorem
(N. Konno, Quantum Information Processing 1, 345 (2002).)
Probability density
Coin operator Initial state
Prob. 1/2 Prob. 1/2
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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)
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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))
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Connections in asymptotic behaviors
From the viewpoint of the limit distribution,
DTQW
CTQWDirac eq.
Schroedinger eq.
(A. Childs and J. Goldstone, Phys. Rev. A 70, 042312 (2004))
Increasing the dimension
Continuum Limit
Time-dependent coin & Re-scaleLattice-size-dependent
coin
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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).)
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Dirac Equation from DTQW
Position of Dirac Particle : Walker Space
Spinor : Coin Space
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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).])
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Connections in asymptotic behaviors
DTQW
CTQWDirac eq.
Schroedinger eq.
(A. Childs and J. Goldstone, Phys. Rev. A 70, 042312 (2004).)
Increasing the dimension
Continuum Limit
Time-dependent coin & Re-scaleLattice-size-dependent
coin
DTQW can simulate some dynamical features in some quantum systems.
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DTQW with decoherence
Simple Decoherence Model:
Position measurement for each step w/ probability “p”.
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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, Quant. Inf. Comp. 11, 0741 (2011).)
Time Scaled Limit Distribution (Crossover!!)
Symmetric DTQW with position measurement with time-dependent probability
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100th step of Walks
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What do we know from this analytical results?
0
1
1
Almost all discrete time quantum walks with decoherence has the normal distribution.
This is the reason why the large steps of the DTQW have not experimentally realized yet.
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Summary of DTQW
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• 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).
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Summary of Weak Value
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• 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).
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Quid est ergo tempus? Si nemo ex me quaerat, scio; si quaerenti explicare velim, nescio.
by St. Augustine
What is time?
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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.
Quantum Classical
Measurement / Decoherence
Quantization
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DTRW v.s. DTQW
coin
position
Rolling the coin Shift of the position due to the coin
Classical Walk Quantum Walk
Unitary operator
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DTRW v.s. DTQW
Classical Walk
Quantum Walk
coin
position
Rolling the coin Shift of the position due to the coin
Unitary operator
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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
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--
Probability Distribution at the 1000-th Step
Initial Coin state
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Limit Distribution (Appendix B)
Theorem (YS and H. Katsura, Phys. Rev. E 82, 031122 (2010))
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When is the probability space defined?
Hilbert space H
Observable A
Probability space
Case 1 Case 2
Hilbert space H
Observable A
Probability space
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Definition of (Discrete) Probability Space
Event Space Ω
Probability Measure dP
Random Variable X: Ω -> K
The expectation value is
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Number (Prob. Dis.) Even/Odd (Prob. Dis.)
1
2
3
6
1/6
1/6
1/6
1/6
1/6
1/6
1/6
1/6
1
0
1
0
Expectation Value
Event Space
21/6 = 7/2 3/6 = 1/2
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Example
Position Operator
Momentum OperatorNot Correspondence!!
Observable-dependent Probability Space
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When is the probability space defined?
Hilbert space H
Observable A
Probability space
Case 1 Case 2
Hilbert space H
Observable A
Probability space
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Observable-independent Probability Space??
• We can construct the probability space independently on the observable by the weak values.
pre-selected state post-selected state
Def: Weak values of observable A
(Y. Aharonov, D. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988))
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Expectation Value?
is defined as the probability measure.
Born Formula Random Variable⇒ =Weak Value
(A. Hosoya and YS, J. Phys. A 43, 385307 (2010))
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Definition of Probability Space
Event Space Ω
Probability Measure dP
Random Variable X: Ω -> K
The expectation value is
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Number (Prob. Dis.) Even/Odd (Prob. Dis.)
1
2
3
6
1/6
1/6
1/6
1/6
1/6
1/6
1/6
1/6
1
0
1
0
Expectation Value
Event Space
21/6 = 7/2 3/6 = 1/2
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Definition of Weak Values
pre-selected state post-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…
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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
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• 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))
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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
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Hardy’s Paradox
B
D
B
D
50/50 beam splitter
Mirror
Path O
Path I
Path I
Path O
Positron
Electron
annihilation
BB
DB
BD
DD
(L. Hardy, Phys. Rev. Lett. 68, 2981 (1992))
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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.
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Why does the paradox be occurred?
Before the annihilation point:
Annihilation must occur.
2nd Beam SplitterProb. 1/12
How to experimentally confirm this state?
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Hardy’s Paradox
B
D
B
D
50/50 beam splitter
Mirror
Path O
Path I
Path I
Path O
Positron
Electron
BB
DB
BD
DD
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Counter-factual argument
• For the pre-selected state, the following operators are equivalent:
Analogously,
(A. Hosoya and YS, J. Phys. A 43, 385307 (2010))
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What is the state-dependent equivalence?
State-dependent equivalence
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Counter-factual arguments
• For the pre-selected state, the following operators are equivalent:
Analogously,
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Pre-Selected State and Weak Value
Experimentally realizable!!
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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
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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))
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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???
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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))
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Properties of W Operator
Relationship to Weak Value
Analogous to the expectation value
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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 operationS-matrix for the combined system
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system
Pre-selected state
environment
environment
Post-selected state
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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.