Weak Values with Decoherence(Typo in Program)

Yutaka Shikano and Akio HosoyaDepartment of Physics, Tokyo Institute of Technology

Based on arXiv:0812.4502. (Typo in Abstract)

1. Aim

2. Brief Review on Weak Values

3. Quantum Operations for Weak Operators

4. Conclusions

1. Aim

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Aim of This Talk To construct a mathematical 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 promote the weak value analysis.

I hope that you will consider the new and strange fashion “weak values” by the way to going your home.

2. Brief Review on Weak Values

Review of Weak Values•Y. Aharonov and D. Rohrlich, “Quantum Paradoxes” (Wiley-VCH, Weibheim, 2005).•Introduction Part of YS and A. Hosoya, arXiv:0812.4502.

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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.

(Y. Aharonov, D. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988))

To measure the weak value…

We have demonstrated some experiments to obtain weak values in optical systems.

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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.

State of the probe after measurement

Taylor expansion

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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.

We assume the probe wave function for the position be real-valued.

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Strong Measurement

Projection

“in vitro” experiment

Quantum State

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Weak MeasurementCover Slightly Seeing

“in vivo” experiment

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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

1st step: Check the post-selected state !!

Shifting the phase for each path.

Changeable

From the interference pattern, we can construct the post-selected state.

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Weak Measurement

2nd step: See the image of CCD camera.

Fixed

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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

3. Quantum Operations for Weak Operators

Could we construct the general framework analogous to the conventional quantum operations?

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Kraus Representation (Conventional)

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???

: Completely positive map (CP map)

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Weak Operator

To define the quantum operations associated with the weak values,

Weak Operator

(YS and A. Hosoya, arXiv:0812.4502)

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Properties of Weak Operator (1)Relationship to Weak Value

Analogous to the expectation value

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Properties of Weak Operator (2)Forward time evolution for the density operator

Backward time evolution for the density operator

The weak operator describes the entire history of the state from the past (ti) to the future (tf ) and measurement performed at the time t.

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Quantum Operations for Weak Operators

Key points of Proof:

1. Polar decomposition for the weak operator

2. Complete positivity of the quantum operation

Roughly speaking,

Kraus operator for the density operator on forward time

Kraus operator for the density operator on backward time

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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.

Impulsive Weak Measurement

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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

where

5. Conclusions

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Conclusions In analogous to the quantum operation for density

operator, we construct the quantum operation for the weak operator associated with the weak values.

We show that the probe shifts in weak measurement is given by the weak value defined by the quantum operation due to the environment.

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Thank you very much for your attention.

Please be careful getting home after the final talk.