# Eventually entanglement breaking Markovian dynamics: structure and characteristic times

## TL;DR

This paper studies when Markovian quantum evolutions become entanglement breaking, providing characterizations, bounds, and extending Poincaré inequalities to discrete time to understand their convergence and entanglement destruction times.

## Contribution

It characterizes eventually entanglement breaking Markovian evolutions, proves finite entanglement breaking index for faithful PPT channels, and extends Poincaré inequalities to discrete quantum semigroups.

## Key findings

- Faithful PPT channels have finite entanglement breaking index.
- Provided bounds on the entanglement breaking index for quantum channels.
- Extended Poincaré inequalities to discrete quantum semigroups.

## Abstract

We investigate entanglement breaking times of Markovian evolutions in discrete and continuous time. In continuous time, we characterize which Markovian evolutions are eventually entanglement breaking, that is, evolutions for which there is a finite time after which any entanglement initially present has been destroyed by the noisy evolution. In the discrete time framework, we consider the entanglement breaking index, that is, the number of times a quantum channel has to be composed with itself before it becomes entanglement breaking. The PPT-square conjecture is that every PPT quantum channel has an entanglement breaking index of at most 2; we prove that every faithful PPT quantum channel has a finite entanglement breaking index, and more generally, any faithful PPT CP map whose Hilbert-Schmidt adjoint is also faithful is eventually entanglement breaking. We also provide a method to obtain concrete bounds on this index for any faithful quantum channel. To obtain these estimates, we use a notion of robustness of separability to obtain bounds on the radius of the largest separable ball around faithful product states. We also extend the framework of Poincar\'e inequalities for nonprimitive semigroups to the discrete setting to quantify the convergence of quantum semigroups in discrete time, which is of independent interest.

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Source: https://tomesphere.com/paper/1902.08173