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.
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.
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 Aw 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.
Statistical functions such as the moment-generating, characteristic, cumulant-generating, and second characteristic functions are standard tools in classical statistics and probability theory. They provide a systematic means to analyze the statistical properties of a system and find applications in diverse fields. While these functions are ubiquitous in classical theory, a quantum counterpart has remained underdeveloped because of the noncommutativity of operators. The absence of such a framework has obscured the connections between statistical quantities and the nonclassical features of quantum mechanics. Here, we construct a framework for quantum statistical functions that addresses these limitations and unifies the languages of quantum statistics. We show that the functions reproduce standard statistical quantities such as expectation values, variance, and covariance upon differentiation. By extending the framework to include pre- and post-selection, we define conditional functions that generate conditional statistical quantities, including the weak value and the weak variance. We further show that multivariable functions, defined with specific operator orderings, correspond to the Kirkwood--Dirac, Margenau--Hill, and Wigner distributions. By generalizing Bochner's theorem within the theory of compactly supported distributions, we obtain a criterion that separates classical statistics from quantum statistics, linking the failure of positive definiteness of the multivariable function to the emergence of quasiprobability. As an application, we import the classical method of moments and generalized method of moments into quantum estimation, introducing quantum estimators that exploit the proposed functions. Our framework reproduces quantum statistical quantities and incorporates the nonclassical features of quasiprobability, providing a basis for further study of quantum statistics.
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.
Methods are presented to generate conditional moments for all quasi-probability distributions, and explicit expressions for all conditional moments are given. In analogy to the generalized characteristic function operator, which generates all joint quasi-distribution functions, the conditional characteristic function operator which generates all conditional quasi-distributions is defined. This allows one to introduce the concept of conditional correspondence rules; formulations are given for all such rules.
Recently, weak measurements have attracted a lot of interest as an experimental method for the investigation of non-classical correlations between observables that cannot be measured jointly. Here, I explain how the complex valued statistics observed in weak measurements relate to the operator algebra of the conventional Hilbert space formalism and show that the algebra of operators originates from more fundamental relations between the physical properties of a quantum system that can be expressed in terms of complex conditional probabilities. In particular, commutation relations can be identified with fundamental imaginary correlations that characterize the relations between physical properties in terms of their transformation dynamics. Non-commutativity thus originates from a definition of relations between physical properties that replaces the assumption of joint reality with a complex-valued probability reflecting the dynamical response of the system to external forces, e.g. in measurement interactions.
In contrast to classical physics, the language of quantum mechanics involves operators and wave functions (or, more generally, density operators). However, in 1932, Wigner formulated quantum mechanics in terms of a distribution function W(q,p), the marginals of which yield the correct quantum probabilities for q and p separately \cite{wigner}. Its usefulness stems from the fact that it provides a re-expression of quantum mechanics in terms of classical concepts so that quantum mechanical expectation values are now expressed as averages over phase-space distribution functions. In other words, statistical information is transferred from the density operator to a quasi-classical (distribution) function.