Entanglements and compound states in quantum information theory

TL;DR

This paper explores quantum entanglements and compound states, introducing d-entanglements and analyzing their impact on quantum information measures like entropy and channel capacity.

Contribution

It introduces the concept of d-entanglements, linking classical-quantum correspondences to separable states and analyzing their role in quantum information theory.

Findings

von Neumann entropy as supremum over d-entanglements

Dimensional entropy achieved at standard entanglement

Quantum channel capacity can double classical capacity

Abstract

Quantum entanglements, describing truly quantum couplings, are stu died and classified from the point of view of quantum compound states. We show that c lassical-quantum correspondences such as quantum encodings can be treated as d-entanglements leading to a special class of the separable compound states. The mutual information of the d-compound and entangled states lead to two di fferent types of entropies for a given quantum state: the von Neumann entrop y, which is achieved as the supremum of the information over all d-entanglem ents, and the dimensional entropy, which is achieved at the standard entangl ement, the true quantum entanglement, coinciding with a d-entanglement only in the commutative case. The q-capacity of a quantum noiseless channel, defi ned as the supremum over all entanglements, is given as the logarithm of the dimensionality of the input von Neumann algebra. It can double the classical capacity, achieved as the supremum over all semi-quantum couplings (d-entang lements, or encodings), which is bounded by the logarithm of the dimensional ity of a maximal Abelian subalgebra.