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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

The post-selection probability current and its implications

John Edward Gray, Allen Parks

Published Apr 1, 2009• 1 citation
Publisher

Abstract

We introduce a notation for probability current and operator current in a normal quantum mechanics setting. We then extend these concepts to the concept of post-selection that was introduced by Schordinger and has found wide application by Aharonov and his colleagues. We introduce the concept of a post-selection operator current and then use it as an alternative means of examining concepts of weak values. The concept of weak energy introduced by Parks is derived in this setting. Also, the configuration space based on geometric phase and the Fubini-Study metric that was introduced by Parks defines an operator current as well. We provide a connection between this current and post-selection operator current.

DOI
10.1117/12.818097

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We introduce a notation for probability current and operator current in a normal quantum mechanics setting. We then extend these concepts to the concept of post-selection that was introduced by Schordinger and has found wide application by Aharonov and his colleagues. We introduce the concept of a post-selection operator current and then use it as an alternative means of examining concepts of weak values. The concept of weak energy introduced by Parks is derived in this setting. Also, the configuration space based on geometric phase and the Fubini-Study metric that was introduced by Parks defines an operator current as well. We provide a connection between this current and post-selection operator current.
In the previous meeting I demonstrated that the contextual value of an observable coincides with the weak value of Aharonov and his colleagues on the basis of consistency of quantum mechanics with Kolmogorov's measure theory of probability and that this consistency also leads to Born's rule as the unique probability measure of events to be discovered by the post-selection. I would like to point out in the meeting of this year that the error and disturbance defined by Ozawa in his work on the Heisenberg uncertainty relation can be interpreted in terms of the weak value. This observation further strengthens the viewpoint that the weak value is the "true value" of an observable before measurement.
The physical meaning of weak values and measurements can be completely understood with Born rule and the general probability theory. It is known that the weak value of an observable A^\hat {A}A^ with post-selection 〈F| may be out of the eigenvalue range of A^\hat {A}A^ . This is because the weak value of A^\hat {A}A^ with the post-selection is, in general, not the expectation value of A^\hat {A}A^ , but the expectation value of A^∣F⟩⟨F∣\hat {A}| F\rangle \langle F|A^∣F⟩⟨F∣ boosted by the post-selection.
The weak energy for a quantum state evolving relative to its fixed initial state (pointed weak energy) in a Hilbert space of arbitrary dimension is defined and aspects of the classical mechanical, transformational and geometric relationships between this quantity and the motion of the associated quantum state are studied in a two-dimensional configuration space with Pancharatnam (P) phase and Fubini-Study (FS) metric distance as generalized coordinates (PFS configuration space). It is shown that when expressed in these coordinates: (1) the evolution of the quantum state is described by an equation of motion for an associated correlation amplitude (pointed correlation amplitude) in which the dynamics are induced by the interaction of the pointed weak energy with this correlation amplitude; (2) the pointed weak energy defines a 1-form with the transformational properties of a U (1) gauge potential (pointed weak energy gauge potential) which produces infinitesimal changes in the pointed correlation amplitude via its interaction with the amplitude; (3) the pointed correlation amplitude is defined by an exponential function of a line integral of the pointed weak energy gauge potential; (4) pointed weak energy gauge transformations exist that are equivalent to local U (1) gauge transformations of quantum states and—when examined within the context of the pointed probability current and pointed weak energy stationary action principles—offer an interpretation of gauge invariance analogous to that found in classical mechanics; and (5) integral invariants exist which relate line integrals of the weak energy gauge potential to geometric phase and the Aharonov–Anandan connection 1-form in the associated Hilbert space. States which irreversibly decay and Grover's quantum search algorithm serve as simple illustrations of the theory.