Dividing Quantum Channels

TL;DR

This paper explores the structure of quantum channels, identifying indivisible and infinitesimal divisible channels, providing complete characterizations for qubits, and establishing a representation theorem for continuous evolutions in finite dimensions.

Contribution

It introduces the concepts of indivisible and infinitesimal divisible quantum channels, offering a complete characterization for qubits and a new representation theorem for finite-dimensional channels.

Findings

Existence of indivisible quantum channels that cannot be decomposed into simpler channels.

Complete characterization of indivisible and infinitesimal divisible channels for qubits.

A new representation theorem for continuous quantum evolutions in finite dimensions.

Abstract

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.