Quantum dynamics is not strictly bidivisible

TL;DR

This paper investigates the divisibility properties of quantum channels, showing that for qubits such channels do not exist, and introducing a new decomposition method applicable to all finite dimensions.

Contribution

It introduces a novel decomposition of quantum channels into boundary and Markovian parts, linking divisibility classes with implementation types.

Findings

No channels divisible in two but not three parts for qubits.

The decomposition applies to any finite dimension.

Connects divisibility classes with quantum map implementations.

Abstract

We address the question of the existence of quantum channels that are divisible in two quantum channels but not in three or, more generally, channels divisible in $n$ but not in $n+1$ parts. We show that for the qubit those channels \textit{do not} exist, whereas for general finite-dimensional quantum channels the same holds at least for full Kraus rank channels. To prove these results, we introduce a novel decomposition of quantum channels which separates them into a boundary and Markovian part, and it holds for any finite dimension. Additionally, the introduced decomposition amounts to the well-known connection between divisibility classes and implementation types of quantum dynamical maps, and can be used to implement quantum channels using smaller quantum registers.