Escort Density Operators and Generalized Quantum Information Measures

Abstract

Parametrized families of density operators are studied. A generalization of the lower bound of Cramér and Rao is formulated. It involves escort density operators. The notion of φ-exponential family is introduced. This family, together with its escort, optimizes the generalized lower bound. It also satisfies a maximum entropy principle and exhibits a thermodynamic structure in which entropy and free energy are related by Legendre transform.

The q-exponential family in statistical physics

“…Indeed, assume that the density of states ω(U) is a strictly increasing function of U. This is for instance the case when V (q) ∼ |q| 2 , which implies ω(U) ∼ U 2 . Then the function ω(U) can be inverted and the knowledge of β uniquely determines the total energy U.…”

“…The transition to quantum models requires more attention but is feasible. An early step in this direction is found in [2]. In particular, the quantum analogue of (28) is I(ρ) = −Tr F (ρ), where ρ is the density matrix.…”

Quantum Statistical Manifold: the Linear Growth Case

“…The interest in deformed exponential families started with the q-statistics of Tsallis [11]. A further generalization was given by the author [12,13,14,15]. The latter formalism is used here and is explained below in Section 2 for the special case of linear growth.…”

Hermite–Gaussian model for quantum states

“…For instance, it has been applied in quantum mechanics leading to a quantum generalization of the Fisher-Rao metric [16], and recently, also in nuclear plasmas [17,18]. Moreover, generalized extensions of the information geometry approach to the non-extensive formulation of statistical mechanics [19] have been also considered [20][21][22][23]. Applications of information geometry to chaos can be also performed by considering complexity on curved manifolds [24][25][26][27][28], leading to a criterion for characterizing global chaos on statistical manifolds: the more negative is the curvature, the more chaotic is the dynamics; from which some consequences concerning dynamical systems have been explored [29].…”