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arXiv:1607.06406 · Progress of Theoretical and Experimental Physics

Quasi-probabilities in conditioned quantum measurement and a geometric/statistical interpretation of Aharonov’s weak value

Jaeha Lee, Izumi Tsutsui

Published Feb 15, 2017• 6 citations
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Abstract

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 $A_{w}$ for an observable $A$ is then given a geometric/statistical interpretation as either the orthogonal projection of $A$ onto the subspace generated by another observable $B$, or equivalently, as the conditioning of $A$ given $B$ with respect to the QJP distribution under consideration.

DOI
10.1093/ptep/ptx024

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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.
A general prescription for the construction of quasi-joint-probability distributions intended to describe the "joint behaviour" of an arbitrary pair of (generally non-commuting) quantum observables, along with its application to the geometric/statistical interpretation of Aharonov's weak value and a proposal for new uncertainty relations, are presented.The introduction of such distributions is motivated by two complementary approaches: one from a bottom-up and strictly operational construction by carefully examining the mathematical framework of the conditioned measurement scheme, and the other from a top-down viewpoint by applying the results of spectral theorem for normal operators and its Fourier transforms.It is revealed that, for a pair of simultaneously measurable observables, the distribution reduces to the unique standard joint-probability distribution of the pair as expected, whereas for a non-commuting pair, there exists an inherent indefiniteness for the choice of such distributions, admitting a multitude of candidates that qualify to describe the "joint behaviour" of the pair.The L 2 structures induced by the quasi-joint-probability distributions are then found to provide statistical interpretation of the geometric structures that can be naturally introduced on the space of operators on the underlying Hilbert space.As possible applications of our findings, we focus on orthogonal projections and inner products of observables, and observe that such geometric concepts can respectively be given statistical interpretations as "conditionings" and "correlations".Incidentally, weak values A w are given a geometric/statistical interpretation as either the orthogonal projection of an observable A on the subspace generated by another observable B (or equivalently, as the conditioning of A given B with respect to the quasi-joint-probability distribution under consideration).Moreover, the Cauchy-Schwartz inequality applied on inner products of two observables (or equivalently, correlations between the pair) leads to novel inequalities interpreted as uncertainty relations for approximation/estimation, which entails the time-energy uncertainty relation as its special case.
We propose a general framework of the quantum/quasi-classical transformations by introducing the concept of quasi-joint-spectral distribution (QJSD). Specifically, we show that the QJSDs uniquely yield various pairs of quantum/quasi-classical transformations, including the Wigner-Weyl transform. We also discuss the statistical behaviour of combinations of generally non-commutin quantum observables by introducing the concept of quantum correlations and conditional expectations defined analogously to the classical counterpart. Based on these, Aharonov's weak value is given a statistical interpretation as one realisation of the quantum conditional expectations furnished in our formalism.
Bipartite quantum entangled systems can exhibit measurement correlations that violate Bell inequalities, revealing the profoundly counter-intuitive nature of the physical universe. These correlations reflect the impossibility of constructing a joint probability distribution for all values of all the different properties observed in Bell inequality tests. Physically, the impossibility of measuring such a distribution experimentally, as a set of relative frequencies, is due to the quantum back-action of projective measurements. Weakly coupling to a quantum probe, however, produces minimal back-action, and so enables a weak measurement of the projector of one observable, followed by a projective measurement of a noncommuting observable. By this technique it is possible to empirically measure weak-valued probabilities for all of the values of the observables relevant to a Bell test. The marginals of this joint distribution, which we experimentally determine, reproduces all of the observable quantum statistics including a violation of the Bell inequality, which we independently measure. This is possible because our distribution, like the weak values for projectors on which it is built, is not constrained to the interval [0,1][0,1][0,1]. It was first pointed out by Feynman that, for explaining singlet-state correlations within ``a [local] hidden variable view of nature \dots{} everything works fine if we permit negative probabilities.'' However, there are infinitely many such theories. Our method, involving ``weak-valued probabilities,'' singles out a unique set of probabilities, and moreover does so empirically.
Quantum mechanics does not permit joint measurements of non-commuting observables.However, it is possible to measure the weak value of a projection operator, followed by the precise measurement of a different property.The results can be interpreted as complex joint probabilities of the two non-commuting measurement outcomes.Significantly, it is possible to predict the outcome of completely different measurements by combining the joint probabilities of the initial state with complex conditional probabilities relating the new measurement to the possible combinations of measurement outcomes used in the characterization of the quantum state.We can therefore conclude that the complex conditional probabilities observed in weak measurements describe fundamental state-independent relations between non-commuting properties that represent the most fundamental form of universal laws in quantum physics.
We investigate the intrinsic ambiguity in the definition of correlation functions arising from the inevitable invasiveness of quantum measurements. While algebraic correlations defined as expectation values of products of observables are widely used, their relationship to operational ones defined through actual measurement procedures remain unclear. We demonstrate that the differences among various definitions of correlation functions and those among their underlying (quasi-)joint probability distributions are bounded above by a quantitative measure of measurement invasiveness. We further obtain a lower bound on the discrepancy among operational and algebraic (quasi-)joint probability distributions, providing a new form of the uncertainty relation. In addition, we identify an equivalence condition under which operational and algebraic correlations coincide. As an application, we analyze the quantum violation of the Leggett-Garg inequality and clarify the structural origin of the equivalence among different approaches to observing the violation, including sequential projective measurements and weak-measurement. Our results provide an operational foundation for the commonly used algebraic concepts of quantum theory.
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.