Towards a state minimizing the output entropy of a tensor product of random quantum channels

TL;DR

This paper investigates how bipartite quantum states' outputs behave under tensor products of random quantum channels, revealing properties related to eigenvalues and entropy that inform quantum information theory questions.

Contribution

It generalizes the analysis of output eigenvalues for various input states under random channels, extending previous results to mixed states and new multi-scale random matrix models.

Findings

Bell states minimize output entropy asymptotically.

Eigenvalue properties depend on input state entanglement.

New models quantify eigenvalue differences between channels and their complements.

Abstract

We consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels. Depending on natural assumptions that we make on the states, the eigenvalues of their outputs have new properties which we describe. Our motivation is provided by the additivity questions in quantum information theory, and we build on the idea that a Bell state sent through a product of conjugated random channels has at least one large eigenvalue. We generalize this setting in two directions. First, we investigate general entangled pure inputs and show that that Bell states give the least entropy among those inputs in the asymptotic limit. We then study mixed input states, and obtain new multi-scale random matrix models that allow to quantify the difference of the outputs' eigenvalues between a quantum channel and its complementary version in the case of a non-pure input.