Weak Values with Decoherence (Typo in Program) Yutaka Shikano and Akio Hosoya Department of Physics, Tokyo Institute of Technology Based on arXiv: (Typo in Abstract) 1. Aim 2. Brief Review on Weak Values 3. Quantum Operations for Weak Operators 4. Conclusions

1. Aim

6/26/2009 ICSSUR'09 and the Feynman Festival 3 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:

6/26/2009 ICSSUR'09 and the Feynman Festival 5 Definition of Weak Values pre-selected statepost-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.

6/26/2009 ICSSUR'09 and the Feynman Festival 6 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

6/26/2009 ICSSUR'09 and the Feynman Festival 7 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, (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.

6/26/2009 ICSSUR'09 and the Feynman Festival 8 Strong Measurement Projection “in vitro” experiment Quantum State

6/26/2009 ICSSUR'09 and the Feynman Festival 9 Weak Measurement Cover Slightly Seeing “in vivo” experiment

6/26/2009 ICSSUR'09 and the Feynman Festival 10 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

6/26/2009 ICSSUR'09 and the Feynman Festival 11 Creating superposition of initial state 1 st step: Check the post- selected state !! Shifting the phase for each path. Changeable From the interference pattern, we can construct the post- selected state.

6/26/2009 ICSSUR'09 and the Feynman Festival 12 Weak Measurement 2 nd step: See the image of CCD camera. Fixed

6/26/2009 ICSSUR'09 and the Feynman Festival 13 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

6/26/2009 ICSSUR'09 and the Feynman Festival 14 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)

6/26/2009 ICSSUR'09 and the Feynman Festival 15 Experimental Realization Prepare the initial state Post-selected state 0 0 1

3. Quantum Operations for Weak Operators Could we construct the general framework analogous to the conventional quantum operations?

6/26/2009 ICSSUR'09 and the Feynman Festival 17 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)

6/26/2009 ICSSUR'09 and the Feynman Festival 18 Weak Operator To define the quantum operations associated with the weak values, Weak Operator (YS and A. Hosoya, arXiv: )

6/26/2009 ICSSUR'09 and the Feynman Festival 19 Properties of Weak Operator (1) Relationship to Weak Value Analogous to the expectation value

6/26/2009 ICSSUR'09 and the Feynman Festival 20 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 (t i ) to the future (t f ) and measurement performed at the time t.

6/26/2009 ICSSUR'09 and the Feynman Festival 21 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

6/26/2009 ICSSUR'09 and the Feynman Festival 22 system Pre-selected state environment Possible history Post-selected state Weak operator describes the entire history of the state evolution. Impulsive Weak Measurement

6/26/2009 ICSSUR'09 and the Feynman Festival 23 Weak Measurement with Decoherence Target system Observable A Environment No noisy operations with impulsive weak measurement The shifts of the expectation values of the probe are where

5. Conclusions

6/26/2009 ICSSUR'09 and the Feynman Festival 25 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.

6/26/2009 ICSSUR'09 and the Feynman Festival 26 Thank you very much for your attention. Please be careful getting home after the final talk.