Generalizing Entanglement via Informational Invariance for Arbitrary Statistical Theories

Abstract

Given an arbitrary statistical theory, different from quantum mechanics, how to decide which are the nonclassical correlations? We present a formal framework which allows for a definition of nonclassical correlations in such theories, alternative to the current one. This enables one to formulate extrapolations of some important quantum mechanical features via adequate extensions of "reciprocal" maps relating states of a system with states of its subsystems. These extended maps permit one to generalize i) separability measures to any arbitrary statistical model as well as ii) previous entanglement criteria. The standard definition of entanglement becomes just a particular case of the ensuing, more general notion.

Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory

“…This will be the analog of the convex set of separable states in quantum mechanics (see [53] for more discussion on this). We will write a product state as ν A ⊗ ν B satisfying:…”

“…This approach has been developed by many authors, and it is fair to say that it is based on the study of convex sets of states, which define measures on certain algebras of observables. These are usually called events or, more generally, effects (see, for example, [43][44][45][46][47][48][49][50][51][52][53]). The origins of this approach could be traced to the works of G. Ludwig [12,13] and G. Mackey [54], but also to von Neumann.…”

“…It is possible to study many important notions of information theory, such as entanglement, discord and many information protocols in generalized probabilistic models (see, for example, [45][46][47][48][49][50][51][52][53]58]). We will argue in favor of the existence of a generalized information theory, continuing the lines of previous works [49,50,59] (see also [60,61], where non-commutative versions of many statistical techniques are studied).…”

Generalized Probabilities in Statistical Theories

“…We remark that generalized probabilities can also be studied in what has been called the Convex Operational Models (COM) approach [54][55][56][57][58][59][60][61][62][63][64]. These are called "generalized probabilistic theories" (GPTs) or simply, "operational models" (see also [17]).…”

“…In the COM approach, the properties of the systems studied and their associated probabilities are encoded in a geometrical way in the form of a convex set and its space of observable magnitudes. The quantum formalism and many aspects of quantum information theory (such as entanglement, discord, and many information protocols) can be suitably described using this approach [55][56][57][58][59][60][61][62][63][64]. Non-linear generalizations of QM were studied using the convex approach in [65][66][67].…”