The cross states of a composite quantum system: separability and entanglement in any Hilbert space dimension

TL;DR

This paper introduces cross states in infinite-dimensional bipartite quantum systems, characterizes separability via a projective norm, and defines an entanglement measure that captures finite entanglement in such systems.

Contribution

It generalizes the classical cross norm criterion of separability to infinite-dimensional systems and introduces an entanglement function for cross states.

Findings

Cross states form a dense subset in infinite dimensions.

Separable states minimize the projective norm among cross states.

The entanglement function equals the projective norm on cross states.

Abstract

We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.