Weak Values in Quantum Measurement Theory - Concepts and Applications - Yutaka Shikano 07M01099 Department 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

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 4 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?

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 5 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

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 7 time Target systemProbe 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))

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 8 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

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 9 What information is obtained? x x Projective measurement (more generally speaking, POVM measurement) only gives information of the probability distribution. eigenvalues histogram Experimentalist’s task

3. Concepts of Weak Values Could we construct another representation of the measurement outcome?

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 11 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 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…

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 12 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 measurementProbe state before measurement

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 13 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.

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 14 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

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 15 Creating superposition of initial state Creating the post- selected state. Measuring the polarization. Weak Measurement

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 16 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

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 17 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)

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 18 Experimental Realization Prepare the initial state Post-selected state 0 0 1

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

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 20 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))

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 21 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???

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 22 Weak Operator In order to define the quantum operations associated with the weak values, Weak Operator (YS and A. Hosoya, arXiv: )

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 23 Properties of Weak Operator Relationship to Weak Value Analogous to the expectation value

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 24 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

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 25 system Pre-selected state environment Possible history Post-selected state Weak operator describes the entire history of the state evolution.

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 26 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

5. Conclusions and Discussions

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 28 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 Information Probability Distribution

2/19/2009 Master Thesis' Prsentation at Tokyo Tech 29 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!