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

Abstract

In this article, we discuss the formal structure of a generalized information theory based on the extension of the probability calculus of Kolmogorov to a (possibly) non-commutative setting. By studying this framework, we argue that quantum information can be considered as a particular case of a huge family of non-commutative extensions of its classical counterpart. In any conceivable information theory, the possibility of dealing with different kinds of information measures plays a key role. Here, we generalize a notion of state spectrum, allowing us to introduce a majorization relation and a new family of generalized entropic measures.

Solutions for the MaxEnt problem with symmetry constraints

“…, r, we define q l = p j , with j an arbitrary index in J l . Now, we can express the Shannon entropy and the constraint equations (24), (25), (26) in terms of q l ,…”

Lattices and Their Consistent Quantification

“…There are efforts underway by Holik and colleagues to apply this approach to the non-distributive lattice of subspaces of the Hilbert space in quantum mechanics. [50,67,68] With this methodology firmly in place, it will be interesting to discover what additional applications await.…”

Generalized entropies in quantum and classical statistical theories

“…The approach to quantum theories based in convex sets dates back to the works of B. Mielnik [48] and G. Ludwig [51] (at least). Recently, the operational approach based in convex sets has attracted much attention, related to the search of operational and informational axioms characterizing quantum theory (see for example [34,36] and references therein).…”