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arXiv:1210.1298 · Progress of Theoretical and Experimental Physics

Complex probability measure and Aharonov's weak value

Tatsuya Morita, Toshihiko Sasaki, Izumi Tsutsui

Published May 1, 2013• 12 citations
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Abstract

We present a complex probability measure relevant for double (pairs of) states in quantum mechanics, as an extension of the standard probability measure for single states that underlies Born's statistical rule. When the double states are treated as the initial and final states of a quantum process, we find that Aharonov's weak value, which has attracted renewed interest as a novel observable quantity inherent in the process, arises as an expectation value associated with the probability measure. Despite being complex, our measure admits a physical interpretation of mixed processes, i.e., an ensemble of processes superposed with classical probabilities.

DOI
10.1093/ptep/ptt017

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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.

AIP conference proceedings•Jan 1, 2012•0 citations
We present a complex probability measure relevant for double (pairs of) states in quantum mechanics, as an extension of the standard probability measure for single states that underlies Born's statistical rule. When the double states are treated as the initial and final states of a quantum process, we find that Aharonov's weak value, which has attracted renewed interest as a novel observable quantity inherent in the process, arises as an expectation value associated with the probability measure. Despite being complex, our measure admits a physical interpretation of mixed processes, i.e., an ensemble of processes superposed with classical probabilities.
We show that the joint behavior of an arbitrary pair of (generally noncommuting) quantum observables can be described by quasi-probabilities, which are an extended version of the standard probabilities used for describing the outcome of measurement for a single observable. The physical situations that require these quasi-probabilities arise when one considers quantum measurement of an observable conditioned by some other variable, with the notable example being the weak measurement employed to obtain Aharonov’s weak value. Specifically, we present a general prescription for the construction of quasi-joint probability (QJP) distributions associated with a given combination of observables. These QJP distributions are introduced in two complementary approaches: one from a bottom-up, strictly operational construction realized by examining the mathematical framework of the conditioned measurement scheme, and the other from a top-down viewpoint realized by applying the results of the spectral theorem for normal operators and their Fourier transforms. It is then revealed that, for a pair of simultaneously measurable observables, the QJP distribution reduces to the unique standard joint probability distribution of the pair, whereas for a noncommuting pair there exists an inherent indefiniteness in the choice of such QJP distributions, admitting a multitude of candidates that may equally be used for describing the joint behavior of the pair. In the course of our argument, we find that the QJP distributions furnish the space of operators in the underlying Hilbert space with their characteristic geometric structures such that the orthogonal projections and inner products of observables can be given statistical interpretations as, respectively, “conditionings” and “correlations”. The weak value AwA_{w}Aw​ for an observable AAA is then given a geometric/statistical interpretation as either the orthogonal projection of AAA onto the subspace generated by another observable BBB, or equivalently, as the conditioning of AAA given BBB with respect to the QJP distribution under consideration.
Weak measurements of photon position can be used to obtain direct experimental evidence of the wavefunction of a photon between generation and ultimate detection. Significantly, these measurement results can also be understood as complex valued conditional probabilities of intermediate photon positions. It is therefore possible to interpret the quantum state as a complex valued probability distribution from which measurement probabilities can be derived according to Bayesian rules. The conventional measurement probabilities derived from squares of the wavefunction then describes the effects of measurement back-action, which originate from a non-classical relation between dynamics and statistics that is characteristic of quantum mechanics. It is pointed out that this relation can be used to derive the complete Hilbert space formalism directly from complex probabilities, without the axiomatic introduction of quantum states or operators.
In quantum mechanics, it is well known that counterfactual arguments sometimes lead to paradoxes, which implies that our classical intuition is not always reliable.However, it has been pointed out by several authors that the weak value, which was introduced by Aharonov et al. in 1988, may be useful to comprehend quantum mechanics intuitively.Another extraordinary aspect of the weak value is its amplification, which has been exploited for precision measurement.Nevertheless, the physical meaning of the weak value is still obscure due to its complexity of the value and the very fact of the amplification.In this thesis, by considering various situations of quantum interference, we attempt to provide a possible interpretation of the weak value.We find, on one hand, that the imaginary part of the weak value is relevant to the wave nature.On the other hand, in the double-slit experiment the particle nature can be seen in the real part of the position weak value.This property for the real part does not hold for more general cases such as the multiple (more than two) slit experiment.The position weak value, nevertheless, can be interpreted as the average of the classical trajectories based on a complex probability.We also investigate the possibility of knowing which path the particle takes in the multiple-slit experiment without destroying the interference fringes.We show that, because of the statistical nature of the weak value, obtaining the which-path information and the interference pattern is compatible with complementarity.Our study suggests that the weak value is a reasonable physical quantity.
Weak measurements carried out between quantum state preparation and post-selection result in complex values for self-adjoint operators, corresponding to complex conditional probabilities for the projections on specific eigenstates. In this paper it is shown that the complex phases of these weak conditional probabilities describe the dynamic response of the system to unitary transformations. Quantum mechanics thus unifies the statistical overlap of different states with the dynamical structure of transformations between these states. Specifically, it is possible to identify the phase of weak conditional probabilities directly with the action of a unitary transform that maximizes the overlap of initial and final states. This action provides a quantitative measure of how much quantum correlations can diverge from the deterministic relations between physical properties expected from classical physics or hidden variable theories. In terms of quantum information, the phases of weak conditional probabilities thus represent the logical tension between sets of three quantum states that is at the heart of quantum paradoxes.
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.