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arXiv:2210.17513 · Journal of Physics A Mathematical and Theoretical

Asymptotics of quantum channels

Daniele Amato, Paolo Facchi A, Arturo Konderak

Published May 23, 2023• 17 citations
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Abstract

Abstract We discuss several aspects concerning the asymptotic dynamics of discrete-time semigroups associated with a quantum channel. By using an explicit expression of the asymptotic map, which describes the action of the quantum channel on its attractor manifold, we investigate the role of permutations in the asymptotic dynamics. We show that, in general, they make the asymptotic evolution non-unitary, and they are related to the divisibility of the quantum channel. Also, we derive several results about the asymptotics of faithful and non-faithful channels, and we establish a constructive unfolding theorem for the asymptotic dynamics.

DOI
10.1088/1751-8121/acd828

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We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘ indivisible ’ channels which can not be written as non-trivial products of other channels and study the set of ‘ infinitesimal divisible ’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.
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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 PPT2\hbox {PPT}^2PPT2 PPT2 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 Poincaré inequalities for non-primitive semigroups to the discrete setting to quantify the convergence of quantum semigroups in discrete time, which is of independent interest.