Weak Values in Quantum Measurement Theory - Concepts and Applications -

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Weak Values in Quantum Measurement Theory-
Concepts and Applications -
Master Thesis Presentation

• Yutaka Shikano
• 07M01099
• Department of Physics,
• Tokyo Institute of Technology

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Outline

1. Aim
2. Conventional Quantum Measurement
3. Concepts of Weak Values
4. Quantum Operations for Weak Operators
5. Conclusions and Discussions

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1. Aim
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Motivations

• Measurement and state changes are highly
  non-trivial in quantum mechanics.
• In conventional quantum measurement theory, we
  have only obtained the probability distribution.
• Experimentalists obtain the probability
  distribution from the experimental data to show
  quantum phenomena.
• However, is the representation of the measurement
  outcome only the probability distribution?

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Aim

• To construct the general framework of the weak
  values advocated by Aharonov and his
  collaborators, which are experimentally
  accessible by the shift of the probe wave
  function in weak measurement.
• To show the efficiency of our proposed framework.

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2. Conventional Quantum Measurement
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Quantum Measurement Theory
(M. Ozawa, J. Math. Phys. 25, 79 (1984))
Target system
Probe system
t 0
t ?t
time
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Representation of Quantum Measurement
Probe observable associated with the measured
observable is
Target state to obtain the measurement outcome
m is
Kraus operator
Positive operator valued measure (POVM)
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What information is obtained?
Experimentalists task
histogram
x
x
eigenvalues
Projective measurement (more generally speaking,
POVM measurement) only gives information of the
probability distribution.
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3. Concepts of Weak Values

• Could we construct another representation of the
  measurement outcome?

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Definition of Weak Values
Def Weak values of observable A
pre-selected state
post-selected state
In order to measure the weak value
Def Weak measurement is called if a coupling
constant with a probe interaction is very small
and a measurement back action is also very small.
(Y. Aharonov, D. Albert, and L. Vaidman, Phys.
Rev. Lett. 60, 1351 (1988))
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In order to Measure Weak Values
Probe system the pointer operator (position of
the pointer) is q and its conjugate operator is p.
Target system
Observable A
Probe state after measurement
Probe state before measurement
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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,
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, 044103 (2007))
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Experimental Realization
(K. Resch, J. S. Lundeen and A. Steinberg, Phys.
Lett. A 324, 125 (2003))
Prepare the initial state
Post-selected state
0
0
1
-1
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Creating superposition of initial state
Creating the post-selected state.
Weak Measurement
Measuring the polarization.
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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
CCD camera
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Perform weak measurement on rail C.
Post-selection rail AB-C (negative shift)
Post-selection rail C (positive shift)
Post-selection rail A and B (No shift)
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Experimental Realization
Prepare the initial state
Post-selected state
0
0
1
-1
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4. Quantum Operations
for Weak Operators

• Could we construct the general framework
  analogous to the conventional quantum measurement?

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CP map for Quantum Operations
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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Kraus 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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Weak Operator
(YS and A. Hosoya, arXiv0812.4507)

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

Weak Operator
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Properties of Weak Operator
Relationship to Weak Value
Analogous to the expectation value
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Quantum Operations for Weak Operators

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

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

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Weak operator describes the entire history of the
state evolution.
environment
system
Post-selected state
Possible history
Pre-selected state
environment
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Weak Measurement with Decoherence
Target system
Environment
Observable A
No noisy operations with impulsive weak
measurement
The shifts of the expectation values of the probe
are
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5. Conclusions and Discussions
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Conclusions

• We have introduced the weak values and reviewed
  the experimental realization in the optical
  system.
• In analogous to the quantum operation for density
  operator, we construct the quantum operation for
  the weak operator associated with the weak values.

Probability Distribution
Phase Information
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Discussions

• To construct the (differential) geometrical
  structure for the weak operator. (lt--gt the Bloch
  sphere representation for the density operator.)
• To extend the concept of the observable. The weak
  values can be defined for non-self-adjoint
  operators (e.g., phase operator and time
  operator.).

Thank you for your attention!