Time in the Weak Value and the Discrete Time Quantum Walk

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

Ph.D defense on August 18th 2

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).


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


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


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).


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


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


Discrete Time Random Walk (DTRW)

Coin Flip

Shift

Repeat


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.)


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:


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


Probability Distribution at the 1000-th step

Initial Coin State

Coin Operator

DTQWDTRW

Unbiased Coin

(Left and Right with probability ½)


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


Probability Distribution at the 1000-th step

Initial Coin State

Coin Operator

DTQWDTRW

Unbiased Coin

(Left and Right with probability ½)


15

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


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)


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))


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


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).)


Dirac Equation from DTQW

Position of Dirac Particle : Walker Space

Spinor : Coin Space


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).])


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.


DTQW with decoherence

Simple Decoherence Model:

Position measurement for each step w/ probability “p”.


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


100th step of Walks


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.


Summary of DTQW


• 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).


Summary of Weak Value


• 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).


Quid est ergo tempus? Si nemo ex me quaerat, scio; si quaerenti explicare velim, nescio.

by St. Augustine

What is time?


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


DTRW v.s. DTQW

coin

position

Rolling the coin Shift of the position due to the coin

Classical Walk Quantum Walk

Unitary operator


DTRW v.s. DTQW

Classical Walk

Quantum Walk

coin

position

Rolling the coin Shift of the position due to the coin

Unitary operator


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


--

Probability Distribution at the 1000-th Step

Initial Coin state


Limit Distribution (Appendix B)

Theorem (YS and H. Katsura, Phys. Rev. E 82, 031122 (2010))


When is the probability space defined?

Hilbert space H

Observable A

Probability space

Case 1 Case 2

Hilbert space H

Observable A

Probability space


Definition of (Discrete) Probability Space

Event Space Ω

Probability Measure dP

Random Variable　 X: Ω -> K

The expectation value is


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


Example

Position Operator

Momentum OperatorNot Correspondence!!

Observable-dependent Probability Space


When is the probability space defined?

Hilbert space H

Observable A

Probability space

Case 1 Case 2

Hilbert space H

Observable A

Probability space


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))


Expectation Value?

is defined as the probability measure.

Born Formula Random Variable⇒ ＝Weak Value

(A. Hosoya and YS, J. Phys. A 43, 385307 (2010))


Definition of Probability Space

Event Space Ω

Probability Measure dP

Random Variable　 X: Ω -> K

The expectation value is


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


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…


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


• 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))



1. What is the Weak Value?• Observable-independent probability space

2. Counter-factual phenomenon: Hardy’s Paradox

3. Weak Value with Decoherence

4. Conclusion


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))


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.


Why does the paradox be occurred?

Before the annihilation point:

Annihilation must occur.

2nd Beam SplitterProb. 1/12

How to experimentally confirm this state?


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


Counter-factual argument

• For the pre-selected state, the following operators are equivalent:

Analogously,

(A. Hosoya and YS, J. Phys. A 43, 385307 (2010))


What is the state-dependent equivalence?

State-dependent equivalence


Counter-factual arguments

• For the pre-selected state, the following operators are equivalent:

Analogously,


Pre-Selected State and Weak Value

Experimentally realizable!!



1. What is the Weak Value?• Observable-independent probability space

2. Counter-factual phenomenon: Hardy’s Paradox

3. Weak Value with Decoherence

4. Conclusion


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))


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???


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))


Properties of W Operator

Relationship to Weak Value

Analogous to the expectation value


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


system

Pre-selected state

environment

environment

Post-selected state


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.

Time in the Weak Value and the Discrete Time Quantum Walk

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