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Ph. D Defense
Time in the Weak Value and the
Discrete Time Quantum Walk
Yutaka Shikano
Theoretical Astrophysics Group,
Department of Physics,
Tokyo Institute of Technology
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?v=yENvgXYzGxw
Reduction
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Time in Physics
Induction
Ref: http://www.youtube.com/watch?v=yENvgXYzGxw
Reduction
Absolute time (Parameter)
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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
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)
•
•
•
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.
arXiv:1009.2131.
M. Gönülol, E. Aydiner, YS, and Ö. E. Mustecaplıo˜glu, New J. Phys.
13, 033037 (2011).
– Weak Value
3. Construct an alternative framework.
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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
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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).
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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
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In Chaps. 4 and 5, on Discrete Time Quantum Walks
Classical random walk
How to relate??
Simple decoherence
model
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Discrete Time
Quantum Walk
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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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When is the probability space defined?
Hilbert space H
Hilbert space H
Observable A
Probability space
Probability space
Observable A
Case 1
Case 2
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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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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
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3/6 = 1/2
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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
Observable A
Probability space
Probability space
Observable A
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 Variable＝Weak Value
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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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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
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3/6 = 1/2
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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
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))
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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
(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
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DB
B
BD
DD
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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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Hardy’s Paradox
B
50/50 beam splitter
Path O
Mirror
D
Path I
D
BB
DB
B
Path I
Positron
Electron
Path O
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BD
DD
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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 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
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:
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
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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 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.
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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
STOC’01 (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:
Let’s see the dynamics of quantum walk by 3rd step!
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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
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Probability Distribution at the 1000-th step
DTRW
DTQW
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
DTQW
Prob. 1/2
N. Konno, Quantum Information Processing 1, 345 (2002)
Coin operator
Initial state
Probability density
47
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Probability Distribution at the 1000-th step
DTRW
DTQW
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 <- 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
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.
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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
Coin operator
(K. Chisaki, N. Konno, E. Segawa, and YS, arXiv:1009.2131.)
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, arXiv:1009.2131.)
Symmetric DTQW with
position measurement with
time-dependent probability
1
0
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1
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