The moment generating function of a set of stochastic variables defines the cumulants or the semi-invariants and the cumulant function. It is possible, simply by formal properties of exponential functions, to generaiize to a great extent the concepts of cumulants and cumulant function. The stochastic variables to be considered need not be ordinary c -numbers but they may be q -numbers such as used in quantum mechanics. The exponential function which defines a moment generating function may be any kind of generalized exponential, for example an ordered exponential with a certain prescription for ordering q -number variables. The definition of average may be greatly generalized as far as the condition is fulfilled that the average of unity is unity. After statements of a few basic theorems these generalizations are discussed here with certain examples of application. This generalized cumulant expansion provides us with a point of view from which many existent methods in quantum mechanics and statistical mechanics can be unified.
The moment generating function of a set of stochastic variables defines the cumulants or the semi-invariants and the cumulant function. It is possible, simply by formal properties of exponential functions, to generaiize to a great extent the concepts of cumulants and cumulant function. The stochastic variables to be considered need not be ordinary c -numbers but they may be q -numbers such as used in quantum mechanics. The exponential function which defines a moment generating function may be any kind of generalized exponential, for example an ordered exponential with a certain prescription for ordering q -number variables. The definition of average may be greatly generalized as far as the condition is fulfilled that the average of unity is unity. After statements of a few basic theorems these generalizations are discussed here with certain examples of application. This generalized cumulant expansion provides us with a point of view from which many existent methods in quantum mechanics and statistical mechanics can be unified.
When a dynamical system has a perturbation which is considered as a stochastic process, the Liouville equation for the system in the phase space or the space of quantum-mechanical density operators is a sort of stochastic equation. The ensemble average of its formal integral defines the relaxation operator Φ(t) of the system. By the definition Φ(t) = exp K(t), the cumulant function K(t) may be introduced. Some general properties are first discussed for a simple example of an oscillator with random frequency modulation and then, concepts of slow and fast modulation are considered. These concepts can be generalized to more general types of stochastic Liouville equations. It is shown that by various possibilities of defining generalized exponential functions, this approach may be useful to understand some essential features of the problem from an unified point of view.
Abstract More than fifty years ago, in a couple of seminal works (Kubo 1962 J. Phys. Soc. Japan 17 1100–20; Kubo 1963 J. Math. Phys . 4 174–83) Kubo introduced the important idea of generalized cumulants, extending to stochastic operators this concept implicitly introduced by Laplace in 1810. Kubo’s idea has been applied in several branches of physics, where the result of the average process is a Lioville operator or an effective time evolution operator for the density matrix of spin systems or the reduced density matrix for boson-fermions etc. Despite this success, the theoretical developments in these Kubo works pose problems that were highlighted many years ago by Fox and van Kampen and never solved. These weaknesses and errors, in particular concerning the factorization property of exponentials of cumulants and the explicit expressions that give generalized cumulants in terms of generalized moments and vice-versa, caused some perplexity (and confusion) about the possible application of this procedure, limiting its use, in practice. In the present paper, we give a sound ground to the approach to cumulant operators, working in a general framework that shows the potentiality of the old Kubo idea, today not yet fully exploited. It results that, for the same moment operators, different definitions of generalized cumulants can be adopted. A general Kubo–Meeron closed-form formula giving cumulant operators in terms of moment operators cannot be obtained, but the reverse one, cumulants in terms of operators, is given and, noticeably, formally it does not depend on the specific nature of the moments, but just on the definition of the generalized cumulants.
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.
Starting from the characteristic function of an operator, we investigate cumulant expansions in quantum optics and apply them to two-dimensional distributions for the canonical variables of the phase space in the case of one degree of freedom (Wigner quasiprobability and its Fourier transform, uncertainty matrix) and to one-dimensional distributions (phase operator, time evolution operator to Hamiltonian). In the relations between cumulants and moments, we make emphasis on the central moments of an operator. It is shown that the determinant of the uncertainty matrix (modified uncertainty product) is invariant with respect to rotation and squeezing of the state in the phase space, whereas the uncertainty sum is only invariant with respect to rotations. We examine some problems for exponentials of the phase operator and show how mean values and variances are connected with the cumulants. The Hilbert–Schmidt distance of a state during time evolution to an initial state is discussed by cumulants.
In the past two years, several points of view have been proposed to address the question of the generalization of the theory of free probability to random tensors with different invariances, and it is unclear at this point whether they lead to the same notions of tensorial free cumulants and freeness. One way to approach this problem, developed by Collins, Gurau and the second named author for local unitary invariant random tensors, relies on finite size quantities involving averages over the invariance group, and whose asymptotics naturally possess the properties expected for tensorial generalizations of free cumulants of arbitrary orders. At this point, this approach has only been carried out for certain distributions, and for a subset of the moments that define such theories, and a more systematic and exhaustive study is lacking. This is the program initiated in this paper: we link this approach to the one proposed by Nechita and Park; extend a number of their results as well as those of the aforementioned paper to arbitrary orders of fluctuations, thereby generalizing higher order free cumulants; push further the study of distributions with larger invariance groups; detail the link with the asymptotics of the free-energies of the tensor HCIZ and BGW integrals; and provide formulae for tensorial free cumulants of products of tensors. Another important question is that of the definition of concrete distributions whose tensorial free-cumulants take non-trivial values. We compute the tensorial free cumulants for Gaussian random tensors with non-trivial covariances, and show that they provide such examples.