Time in the Weak Value and the Discrete Time Quantum Walk

1
Time in the Weak Value and the Discrete Time
Quantum Walk
Ph. D Defense

• Yutaka Shikano
• Theoretical Astrophysics Group,
• Department of Physics,
• Tokyo Institute of Technology

2
Quid est ergo tempus? Si nemo ex me quaerat,
scio si quaerenti explicare velim, nescio.
by St. Augustine, (in Book 11, Chapter 14,
Confessions (Latin Confessiones))
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Time in Physics
Induction
Ref http//www.youtube.com/watch?vyENvgXYzGxw
Reduction
4
Time in Physics
Induction
Ref http//www.youtube.com/watch?vyENvgXYzGxw
Reduction
Absolute time (Parameter)
5
Time in quantum mechanics
CM
Canonical Quantization
QM
Absolute time (Parameter)
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Time in quantum mechanics
CM
Canonical Quantization
QM
The time operator is not self-adjoint in the case
that the Hamiltonian is bounded proven by Pauli.
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How to characterize time in quantum mechanics?
Aim Construct a concrete method and a specific
model to understand the properties of time

• 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)
• YS, K. Chisaki, E. Segawa, N. Konno, Phys. Rev. A
  81, 062129 (2010).
• K. Chisaki, N. Konno, E. Segawa, YS, to appear in
  Quant. Inf. Comp. arXiv1009.2131.
• M. Gönülol, E. Aydiner, YS, and Ö. E.
  Mustecaplioglu, New J. Phys. 13, 033037 (2011).
• Weak Value
• Construct an alternative framework.

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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

• Hamiltonian Estimation by Weak Measurement
• YS and S. Tanaka, arXiv1007.5370.
• 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.,
  arXiv1104.2010.
• 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).

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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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In Chaps. 4 and 5, on Discrete Time Quantum Walks
Classical random walk
How to relate??
Discrete Time Quantum Walk
Simple decoherence model
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Rest of Todays talk

• What is the Weak Value?
• Observable-independent probability space
• Counter-factual phenomenon Hardys Paradox
• Weak Value with Decoherence
• Conclusion

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When is the probability space defined?
Hilbert space H
Hilbert space H
Probability space
Observable A
Observable A
Probability space
Case 1
Case 2
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Definition of (Discrete) Probability Space
Event Space O Probability Measure dP Random
Variable X O -gt K
The expectation value is
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Number (Prob. Dis.)
Even/Odd (Prob. Dis.)
Event Space
1
1/6
1/6
1
0
1/6
1/6
2
1/6
1
1/6
3
6
1/6
1/6
0
3/6 1/2
21/6 7/2
Expectation Value
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Example
Position Operator
Momentum Operator
Not Correspondence!!
Observable-dependent Probability Space
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When is the probability space defined?
Hilbert space H
Hilbert space H
Probability space
Observable A
Observable A
Probability space
Case 1
Case 2
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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))
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Expectation Value?
(A. Hosoya and YS, J. Phys. A 43, 385307 (2010))
is defined as the probability measure.
Born Formula ? Random VariableWeak Value
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Definition of Probability Space
Event Space O Probability Measure dP Random
Variable X O -gt K
The expectation value is
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Number (Prob. Dis.)
Even/Odd (Prob. Dis.)
Event Space
1
1/6
1/6
1
0
1/6
1/6
2
1/6
1
1/6
3
6
1/6
1/6
0
3/6 1/2
21/6 7/2
Expectation Value
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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))
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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))
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• Fundamental Test of Quantum Theory
• Direct detection of Wavefunction
• (J. Lundeen et al., Nature 474, 188 (2011))
• Trajectories in Youngs double slit experiment
• (S. Kocsis et al., Science 332, 1198 (2011))
• Violation of Leggett-Gargs 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 Todays talk

• What is the Weak Value?
• Observable-independent probability space
• Counter-factual phenomenon Hardys Paradox
• Weak Value with Decoherence
• Conclusion

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Hardys Paradox
(L. Hardy, Phys. Rev. Lett. 68, 2981 (1992))
B
50/50 beam splitter
Path O
Mirror
D
Path I
D
BB DB BD DD
B
Path I
Positron
Path O
Electron
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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.
How to experimentally confirm this state?
2nd Beam Splitter
Prob. 1/12
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Hardys Paradox
B
50/50 beam splitter
Path O
Mirror
D
Path I
D
BB DB BD DD
B
Path I
Positron
Path O
Electron
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Counter-factual argument
(A. Hosoya and YS, J. Phys. A 43, 385307 (2010))

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

Analogously,
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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 Todays talk

• What is the Weak Value?
• Observable-independent probability space
• Counter-factual phenomenon Hardys Paradox
• Weak Value with Decoherence
• Conclusion

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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))
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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
(YS and A. Hosoya, J. Phys. A 43, 0215304 (2010))

• In order to define the quantum operations
  associated with the weak values,

W Operator
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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

• Key points of Proof
• Polar decomposition for the W operator
• Complete positivity of the quantum operation

S-matrix for the combined system

• The properties of the quantum operation are
• Two Kraus operators
• Partial trace for the auxiliary Hilbert space
• Mixed states for the W operator

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environment
system
Post-selected state
Pre-selected state
environment
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Conclusion

• We obtain the properties of the weak value
• To be naturally defined as the observable-independ
  ent 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.
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Discrete Time Random Walk (DTRW)
Coin Flip
Shift
Repeat
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Discrete Time Quantum Walk (DTQW)
(A. Ambainis, E. Bach, A. Nayak, A. Vishwanath,
and J. Watrous, in STOC01 (ACM Press, New York,
2001), pp. 37 49.)
Quantum Coin Flip
Shift
Repeat
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Example of DTQW

• Initial Condition
• Position n 0 (localized)
• Coin
• Coin Operator Hadamard Coin

Probability distribution of the n-th cite at t
step
Lets see the dynamics of quantum walk by 3rd
step!
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Example of DTQW
0
1
2
3
prob.
1/12
9/12
1/12
1/12
step
Quantum Coherence and Interference
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Probability Distribution at the 1000-th step
DTQW
DTRW
Initial Coin State
Unbiased Coin (Left and Right with probability ½)
Coin Operator
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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
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Probability Distribution at the 1000-th step
DTQW
DTRW
Initial Coin State
Unbiased Coin (Left and Right with probability ½)
Coin Operator
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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)
Dynamics of discretized Schroedinger Equation.
(E. Farhi and S. Gutmann, Phys. Rev. A 58, 915
(1998))
Limit Distribution (Arcsin Law lt- Quantum
probability theory)
p.d.

• Experimental Realization
• A. Peruzzo et al., Science 329, 1500 (2010).
  (Photon, Waveguide)

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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.
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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
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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,
arXiv1009.2131.)
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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DTQW with decoherence
Simple Decoherence Model
Position measurement for each step w/ probability
p.
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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, arXiv1009.2131.)
1
Symmetric DTQW with position measurement with
time-dependent probability
0
1