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Marginal distributions in $(\\bf 2N)$-dimensional phase space and the quantum $(\\bf N+1)$ marginal theorem

Guy Auberson, Gilbert Mahoux, S. M. Roy, Singh, V.

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

We study the problem of constructing a probability density in $2N$-dimensional phase space which reproduces a given collection of $n$ joint probability distributions as marginals. Only distributions authorized by quantum mechanics, i.e. depending on a (complete) commuting set of $N$ variables, are considered. A diagrammatic or graph theoretic formulation of the problem is developed. We then exactly determine the set of ``admissible'' data, i.e. those types of data for which the problem always admits solutions. This is done in the case where the joint distributions originate from quantum mechanics as well as in the case where this constraint is not imposed. In particular, it is shown that a necessary (but not sufficient) condition for the existence of solutions is $n\\leq N+1$. When the data are admissible and the quantum constraint is not imposed, the general solution for the phase space density is determined explicitly. For admissible data of a quantum origin, the general solution is given in certain (but not all) cases. In the remaining cases, only a subset of solutions is obtained.

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Marginal distributions in (bf2N)(\\bf 2N)(bf2N)-dimensional phase space and the quantum (bfN+1)(\\bf N+1)(bfN+1) marginal theorem
We study the problem of constructing a probability density in 2N2N2N-dimensional phase space which reproduces a given collection of nnn joint probability distributions as marginals. Only distributions authorized by quantum mechanics, i.e. depending on a (complete) commuting set of NNN variables, are considered. A diagrammatic or graph theoretic formulation of the problem is developed. We then exactly determine the set of ``admissible'' data, i.e. those types of data for which the problem always admits solutions. This is done in the case where the joint distributions originate from quantum mechanics as well as in the case where this constraint is not imposed. In particular, it is shown that a necessary (but not sufficient) condition for the existence of solutions is nleqN+1n\\leq N+1nleqN+1. When the data are admissible and the quantum constraint is not imposed, the general solution for the phase space density is determined explicitly. For admissible data of a quantum origin, the general solution is given in certain (but not all) cases. In the remaining cases, only a subset of solutions is obtained.
We study the problem of constructing a probability density in 2N-dimensional phase space which reproduces a given collection of n joint probability distributions as marginals. Only distributions authorized by quantum mechanics, i.e., depending on a (complete) commuting set of N variables, are considered. A diagrammatic or graph theoretic formulation of the problem is developed. We then exactly determine the set of “admissible” data, i.e., those types of data for which the problem always admits solutions. This is done in the case where the joint distributions originate from quantum mechanics as well as in the case where this constraint is not imposed. In particular, it is shown that a necessary (but not sufficient) condition for the existence of solutions is n⩽N+1. When the data are admissible and the quantum constraint is not imposed, the general solution for the phase space density is determined explicitly. For admissible data of a quantum origin, the general solution is given in certain (but not all) cases. In the remaining cases, only a subset of solutions is obtained.
We address the classical and quantum marginal problems, namely the question of simultaneous realizability through a common probability density in phase space of a given set of compatible probability distributions. We consider only distributions authorized by quantum mechanics, i.e., those corresponding to complete commuting sets of observables. For four-dimensional phase space with position variables q⃗ and momentum variables p⃗, we establish the two following points: (i) given four compatible probabilities for (q1,q2), (q1,p2), (p1,q2), and (p1,p2), there does not always exist a positive phase space density ρ(q⃗,p⃗) reproducing them as marginals; this settles a long standing conjecture; it is achieved by first deriving Bell-type inequalities in phase space which have their own theoretical and experimental interest. (ii) Given instead at most three compatible probabilities, there always exist an associated phase space density ρ(q⃗,p⃗); the solution is not unique and its general form is worked out. These two points constitute our “three marginal theorem.”
This paper deals with the topics of modeling joint distributions on a generalized probability space. An algebraic structure known as quantum logic is taken as the basic model. There is a brief summary of some earlier published findings concerning a function s-map, which is a mathematical tool suitable for constructing virtual joint probabilities of even non-compatible propositions. The paper completes conclusions published in 2020 and extends the results for three or more random variables if the marginal distributions are known. The existence of a (n+1)-variate joint distribution is shown in special cases when the quantum logic consists of at most n blocks of Boolean algebras.
We examine the construction of joint probabilities for non-commuting observables. We show that there are indications in standard quantum mechanics that imply the existence of conditional expectation values, which in turn implies the existence of a joint distribution. We also argue that the uncertainty principle has no bearing on the existence of joint distributions but only constrains the marginal distributions. In addition, we show that within classical probability theory there are mathematical quantities that are similar to quantum mechanical wave functions. This is shown by generalising a theorem of Khinchin on the necessary and sufficient conditions for a function to be a characteristic function.
A marginal problem asks whether a given family of marginal distributions for some set of random variables arises from some joint distribution of these variables. Here, we point out that the existence of such a joint distribution imposes nontrivial conditions already on the level of Shannon entropies of the given marginals. These entropic inequalities are necessary (but not sufficient) criteria for the existence of a joint distribution. For every marginal problem, a list of such Shannon-type entropic inequalities can be calculated by Fourier-Motzkin elimination, and we offer a software interface to a Fourier-Motzkin solver for doing so. For the case that the hypergraph of given marginals is a cycle graph, we provide a complete analytic solution to the problem of classifying all relevant entropic inequalities, and use this result to bound the decay of correlations in stochastic processes. Furthermore, we show that Shannon-type inequalities for differential entropies are not relevant for continuous-variable marginal problems; non-Shannon-type inequalities are both in the discrete and in the continuous case. In contrast to other approaches, our general framework easily adapts to situations where one has additional (conditional) independence requirements on the joint distribution, as in the case of graphical models. We end with a list of open problems. A complementary article discusses applications to quantum nonlocality and contextuality.
It is shown that phase-space distribution functions that characterize a single-particle quantum system obey a certain "generalized non-negativity condition," which reflects the fact that the density operator is a positive operator. A corresponding criterion is obtained for the associated characteristic functions and is found to resemble, to some extent, Bochner's theorem of classical probability theory. Necessary and sufficient conditions on a phase-space representation of quantum mechanics are also derived, which ensure that all the possible distribution functions in that representation are non-negative; but it is also shown that such distribution functions are not joint probabilities for position and momentum. In fact, our results readily provide a new proof of theorems of Wigner, and of Cohen and Margenau, which imply that quantum mechanics cannot be formulated as a stochastic theory in phase space.