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“Master Thesis’ Presentation” Weak Values in Quantum Measurement Theory - Concepts and Applications Yutaka Shikano 07M01099 Department of Physics, Tokyo Institute of Technology Outline 1. 2. 3. 4. 5. Aim Conventional Quantum Measurement Concepts of Weak Values Quantum Operations for Weak Operators Conclusions and Discussions 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 2 1. Aim 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 4 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. 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 5 2. Conventional Quantum Measurement Quantum Measurement Theory (M. Ozawa, J. Math. Phys. 25, 79 (1984)) Target system Probe system t=0 1 3 t = ⊿t We can evaluate the “measurement” outcome t = 0 on the measured system from the measurement outcome t = ⊿t. Interaction between the target and probe systems. 2 We obtain the measurement outcome on the probe system. time 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 7 Representation of Quantum Measurement Probe observable associated with the measured observable is Target state to obtain the measurement outcome “m” is Kraus operator Positive operator valued measure (POVM) 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 8 What information is obtained? Experimentalist’s task histogram x x eigenvalues Projective measurement (more generally speaking, POVM measurement) only gives information of the probability distribution. 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 9 3. Concepts of Weak Values Could we construct another representation of the measurement outcome? Definition of Weak Values Def: Weak values of observable A pre-selected state post-selected state In order to measure the weak value… 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)) 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 11 In order to Measure Weak Values Probe system Target system the pointer operator (position of the pointer) is q and its conjugate operator is p. Observable A Probe state after measurement 2/19/2009 Probe state before measurement Master Thesis' Prsentation at Tokyo Tech 12 Probe system Target system Observable A 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, : 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)) 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 13 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 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 14 Creating superposition of initial state Creating the postselected state. Weak Measurement 2/19/2009 Master Thesis' Prsentation at Tokyo Tech Measuring the polarization. 15 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 Mode C q Flat gt CCD camera 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 16 Perform weak measurement on rail C. Post-selection: rail A+B-C (negative shift) Post-selection: rail C (positive shift) 2/19/2009 Post-selection: rail A and B 17 Master Thesis' Prsentation at Tokyo Tech(No shift) Experimental Realization Prepare the initial state Post-selected state 0 0 1 -1 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 18 4. Quantum Operations for Weak Operators Could we construct the general framework analogous to the conventional quantum measurement? CP map for Quantum Operations Positive map Arbitrary extension of Hilbert space When is positive map, is called a completely positive map (CP map). (M. Ozawa, J. Math. Phys. 25, 79 (1984)) 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 20 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 21 Weak Operator (YS and A. Hosoya, arXiv:0812.4507) In order to define the quantum operations associated with the weak values, Weak Operator 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 22 Properties of Weak Operator Relationship to Weak Value Analogous to the expectation value 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 23 Quantum Operations for Weak Operators Key points of Proof: 1. Polar decomposition for the weak operator 2. Complete positivity of the quantum operation 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 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 24 Weak operator describes the entire history of the state evolution. environment Post-selected state Possible history Pre-selected state 2/19/2009 system environment Master Thesis' Prsentation at Tokyo Tech 25 Weak Measurement with Decoherence Environment Target system Observable A No noisy operations with impulsive weak measurement The shifts of the expectation values of the probe are 2/19/2009 Master Thesis' Prsentation at Tokyo Tech 26 5. Conclusions and Discussions 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. Probability Distribution 2/19/2009 Phase Information Master Thesis' Prsentation at Tokyo Tech 28 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-selfadjoint operators (e.g., phase operator and time operator.). 2/19/2009 Thank you for your attention! Master Thesis' Prsentation at Tokyo Tech 29