Composition of quantum states and dynamical subadditivity

TL;DR

This paper introduces a novel composition method for bipartite quantum states based on reshuffling density matrices, proving subadditivity of von Neumann entropy and establishing new bounds for quantum and classical stochastic maps.

Contribution

It presents a new non-Abelian product for quantum states, proves its entropy subadditivity, and derives bounds for quantum and classical map entropies, advancing understanding of quantum state composition.

Findings

Proved subadditivity of von Neumann entropy under the new composition.

Established strong dynamical subadditivity for concatenated bistochastic maps.

Derived new entropy bounds for classical stochastic matrix products.

Abstract

We introduce a composition of quantum states of a bipartite system which is based on the reshuffling of density matrices. This non-Abelian product is associative and stems from the composition of quantum maps acting on a simple quantum system. It induces a semi-group in the subset of states with maximally mixed partial traces. Subadditivity of the von Neumann entropy with respect to this product is proved. It is equivalent to subadditivity of the entropy of bistochastic maps with respect to their composition, where the entropy of a map is the entropy of the corresponding state under the Jamiolkowski isomorphism. Strong dynamical subadditivity of a concatenation of three bistochastic maps is established. Analogous bounds for the entropy of a composition are derived for general stochastic maps. In the classical case they lead to new bounds for the entropy of a product of two stochastic matrices.