Bipartite depolarizing channels

TL;DR

This paper introduces a versatile class of bipartite depolarizing channels, characterizes their properties, and applies these findings to understand entanglement dynamics, including a proven conjecture and new insights into local operations.

Contribution

It defines a new 3-parameter family of bipartite maps, analyzes their positivity and entanglement properties, and applies these results to entanglement annihilation and local operations.

Findings

Identified parameter regions for positivity, complete positivity, and entanglement-breaking.

Discovered PPT but not entanglement-breaking maps within the model.

Proved a recent conjecture on entanglement annihilation by local depolarizing channels.

Abstract

We introduce a 3-parameter class of maps acting on a bipartite system that are a natural generalisation of the depolarizing channel (and include it as a special case). Then, we find the exact regions of the parameter space that alternatively determine a positive, completely positive, entanglement-breaking or entanglement-annihilating map. This model displays a much richer behaviour than the one shown by a simple depolarizing channel, yet it stays exactly solvable. As an example of this richness, PPT but not entanglement-breaking maps are found. A simple example of a positive yet indecomposable map is provided. Then, the study of the entanglement-annihilating property is fully addressed. Finally, we apply our results to solve the problem of the entanglement annihilation caused in a bipartite system by a tensor product of local depolarizing channels. In this context, a conjecture recently posed in the literature is proven, and many related open questions are answered. To arrive at this result we furthermore show how the Hadamard product between quantum states can be implemented via local operations.