Abstract We derive a general operator transformation between any two functions and we show that the operator transformation of quantum quasi-distributions is a special case. We also show that our approach generalizes the Edgeworth series of standard probability theory, and results in a unified approach for the transformation of probability densities. Our approach generalizes the classical Edgeworth series in two ways. It relates two arbitrary distributions rather than one of them being a Gaussian and secondly we generalize to an arbitrary Hermitian operator rather than the differentiation operator which appears in the Edgeworth formulation. We also consider applying the transformation to the same distribution but at different times.
Abstract We derive a general operator transformation between any two functions and we show that the operator transformation of quantum quasi-distributions is a special case. We also show that our approach generalizes the Edgeworth series of standard probability theory, and results in a unified approach for the transformation of probability densities. Our approach generalizes the classical Edgeworth series in two ways. It relates two arbitrary distributions rather than one of them being a Gaussian and secondly we generalize to an arbitrary Hermitian operator rather than the differentiation operator which appears in the Edgeworth formulation. We also consider applying the transformation to the same distribution but at different times.
Husimi distributions [11] and Wigner distributions [20] are well-known quasi-probability distributions which appear in several contexts. In this paper, we show some remarkable aspects of these distribution functions related to geometric structures of generalized coherent state systems [15,16] and operational quantum physics [7], and a scheme of formulating generalized version of quasi-probability distributions. Our scheme gives concrete formulae of quasi-probability distributions in more direct way from the theory of coherent state systems and clarify their operational meanings, especially of Husimi distributions and mutual relation between Husimi distributions and other classes of quasi-probability distributions.
The primary focus of this work is to investigate how the most emblematic classical probability density, namely a Gaussian, can be mapped to a valid quantum states. To explore this issue, we consider a Gaussian whose squared variance depends on a parameter λ. Specifically, depending on the value of λ, we study what happens in the classical-quantum correspondence as we change the indeterminacy of the classical particle. Furthermore, finding a correspondence between a classical state and a quantum state is not a trivial task. Quantum observables, described by Hermitian operators, do not generally commute, so a precise ordering must be introduced to resolve this ambiguity. In this work, we study two different arbitrary orderings: the first is an arbitrary ordering of the position and momentum observables; the second, which is the main focus of the present work, is an arbitrary ordering of the annihilation and creation operators. In this latter case, we find the interesting result that even a δ-function, which in general has no quantum correspondence, can be mapped into a valid quantum state for a particular ordering, specifically the antinormal one (all creation operators are to the right of all annihilation operators in the product). This means that the Gaussian probability density corresponds to a valid quantum state, regardless of how localized classical particles are in phase space.
We study a generalization of the Wigner function to arbitrary tuples of hermitian operators. We show that for any collection of hermitian operators A1...An , and any quantum state there is a unique joint distribution on R^n, with the property that the marginals of all linear combinations of the operators coincide with their quantum counterpart. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution, because for position and momentum this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.
We study a generalization of the Wigner function to arbitrary tuples of Hermitian operators. We show that for any collection of Hermitian operators A1, …, An and any quantum state, there is a unique joint distribution on Rn with the property that the marginals of all linear combinations of the Ak coincide with their quantum counterparts. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution because for position and momentum, this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties, and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.
The primary focus of this work is to investigate how the most emblematic classical probability density, namely a Gaussian, can be mapped to a valid quantum states. To explore this issue, we consider a Gaussian whose squared variance depends on a parameter [Formula: see text]. Specifically, depending on the value of [Formula: see text], we study what happens in the classical-quantum correspondence as we change the indeterminacy of the classical particle. Furthermore, finding a correspondence between a classical state and a quantum state is not a trivial task. Quantum observables, described by Hermitian operators, do not generally commute, so a precise ordering must be introduced to resolve this ambiguity. In this work, we study two different arbitrary orderings: the first is an arbitrary ordering of the position and momentum observables; the second, which is the main focus of the present work, is an arbitrary ordering of the annihilation and creation operators. In this latter case, we find the interesting result that even a [Formula: see text]-function, which in general has no quantum correspondence, can be mapped into a valid quantum state for a particular ordering, specifically the antinormal one (all creation operators are to the right of all annihilation operators in the product). This means that the Gaussian probability density corresponds to a valid quantum state, regardless of how localized classical particles are in phase space.
Derivation of two-time second-order correlation function by following approaches such as stochastic differential equation, coherent-state propagator, and quasi-statistical distribution function is presented. In the process, the time dependence of the operators is transferred to the density operator by making use of trace operation in which the coherent state propagator and Q-function that represent the quantum system under consideration are expressed in terms of different time parameters. Even though the number of resulting integrations are found to be large, the accompanying implementation turns out to be straightforward in view that the associated c-number functions are Gaussian by nature. In relation to the asserted possibility of rewriting the result of one of the approaches in terms of the other, the presented derivation is expected to lay a strong foundation for viable technique of calculating correlations of various moments at different times that can be deployed in revealing quantum correlations.