Weak Values with Decoherence (Typo in Program)

1. Weak Values with Decoherence(Typo in Program) Yutaka Shikano and Akio Hosoya Department of Physics, Tokyo Institute of Technology Aim Brief Review on Weak Values Quantum Operations for Weak Operators Conclusions Based on arXiv:0812.4502. (Typo in Abstract)

2. 1. Aim

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. ICSSUR'09 and the Feynman Festival

4. 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.

5. 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)) We have demonstrated some experiments to obtain weak values in optical systems. ICSSUR'09 and the Feynman Festival

6. 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 State of the probe after measurement Taylor expansion ICSSUR'09 and the Feynman Festival

7. 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, We assume the probe wave function for the position be real-valued. : 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)) ICSSUR'09 and the Feynman Festival

8. Strong Measurement Quantum State Projection “in vitro” experiment ICSSUR'09 and the Feynman Festival

9. Weak Measurement Cover Slightly Seeing “in vivo” experiment 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 -1 ICSSUR'09 and the Feynman Festival

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

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

13. q Mode C Flat gt 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 ICSSUR'09 and the Feynman Festival

14. Perform weak measurement on rail C. Post-selection: rail A+B-C (negative shift) Post-selection: rail C (positive shift) Post-selection: rail A and B (No shift) ICSSUR'09 and the Feynman Festival

15. Experimental Realization Prepare the initial state Post-selected state 0 0 1 -1 ICSSUR'09 and the Feynman Festival

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

17. Kraus Representation (Conventional) : Completely positive map (CP map) 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??? ICSSUR'09 and the Feynman Festival

18. Weak Operator (YS and A. Hosoya, arXiv:0812.4502) • To define the quantum operations associated with the weak values, Weak Operator ICSSUR'09 and the Feynman Festival

19. Properties of Weak Operator (1) Relationship to Weak Value Analogous to the expectation value 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 (ti) to the future (tf) and measurement performed at the time t. ICSSUR'09 and the Feynman Festival

21. Quantum Operations for Weak Operators • Key points of Proof: • Polar decomposition for the weak operator • 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 ICSSUR'09 and the Feynman Festival

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

23. 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 where ICSSUR'09 and the Feynman Festival

24. 5. Conclusions

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. ICSSUR'09 and the Feynman Festival

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