Weak Values in Quantum Measurement Theory - Concepts and Applications - Yutaka Shikano 07M01099 Department of Physics, Tokyo Institute of Technology “Master Thesis’ Presentation”
Weak Values in Quantum Measurement Theory - Concepts and Applications - Yutaka Shikano 07M01099 Department of Physics, Tokyo Institute of Technology “Master.
Dec 28, 2015
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Weak Values in Quantum Measurement
Theory- Concepts and Applications
-Yutaka Shikano07M01099Department of Physics,Tokyo Institute of Technology
“Master Thesis’ Presentation”
2/19/2009 Master Thesis' Prsentation at Tokyo Tech
2
Outline
1. Aim
2. Conventional Quantum Measurement
3. Concepts of Weak Values
4. Quantum Operations for Weak Operators
5. Conclusions and Discussions
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.
2. Conventional Quantum Measurement
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time
Target system Probe system
Interaction between the target and probe systems.
1
We obtain the measurement outcome on the probe system.
2
We can evaluate the “measurement” outcome t = 0 on the measured system from the measurement outcome t = t.⊿
3
t = 0
t = t⊿
Quantum Measurement Theory
(M. Ozawa, J. Math. Phys. 25, 79 (1984))
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Representation of Quantum Measurement
Target state to obtain the measurement outcome “m” is
Kraus operator
Positive operator valued measure (POVM)
Probe observable associated with the measured observable is
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What information is obtained?
xx
Projective measurement (more generally speaking, POVM measurement) only gives information of the probability distribution.
eigenvalues
histogram
Experimentalist’s task
3. Concepts of Weak ValuesCould we construct another representation of the measurement outcome?
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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 and a measurement back action is also very small.
(Y. Aharonov, D. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988))
In order to measure the weak value…
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In order to Measure Weak Values
Target system
Observable A
Probe system
the pointer operator (position of the pointer) is q and its conjugate operator is p.
Probe state after measurement Probe state before measurement
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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))
: Initial probe variance for the momentum
Weak values are experimentally accessible by the shifts of expectation values for the probe observables.
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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.
Measuring the polarization.
Weak Measurement
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Weak Measurement by Slide Glass Use transverse position of each photon as pointer Weak measurement can be performed by tilting a
glass optical flat, where effective
gtFlat
Mode C
(N. M. W. Ritchie, J. G. Story, and R. G. Hulet, Phys. Rev. Lett. 66, 1107 (2003))
CCD camera
Probe
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Perform weak measurement on rail C.
Post-selection: rail A and B (No shift)
Post-selection: rail C (positive shift)
Post-selection: rail A+B-C (negative shift)
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Experimental RealizationPrepare the initial state
Post-selected state
0
0
1
-1
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
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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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
In order to define the quantum operations associated with the weak values,
Weak Operator
(YS and A. Hosoya, arXiv:0812.4507)
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Properties of Weak OperatorRelationship to Weak Value
Analogous to the expectation value
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Quantum Operations for Weak 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 weak operator
Key points of Proof:
1. Polar decomposition for the weak operator
2. Complete positivity of the quantum operation
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system
Pre-selected state
environment
environment
Possible history Post-selected state
Weak operator describes the entire history of the state evolution.
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Weak Measurement with Decoherence
Target system
Observable AEnvironment
No noisy operations with impulsive weak measurement
The shifts of the expectation values of the probe are
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.
Phase InformationProbability Distribution
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Discussions To construct the (differential) geometrical
structure for the weak operator. (<--> 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!
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