Unitary Dilations of Discrete-Time Quantum-Dynamical Semigroups

TL;DR

This paper demonstrates how the evolution of open quantum systems via a single quantum channel can be represented as a unitary evolution of a larger closed system, with extensions to multiple channels and finite-dimensional cases.

Contribution

It constructs explicit unitary dilations for discrete-time quantum dynamical semigroups, including cases with multiple commuting channels and finite-dimensional auxiliary spaces.

Findings

Unitary dilation constructed for single quantum channels.

Extension of dilation results to multiple commuting channels.

Finite-dimensional auxiliary spaces possible for cyclic channels.

Abstract

We show that the discrete-time evolution of an open quantum system generated by a single quantum channel $T$ can be embedded in the discrete-time evolution of an enlarged closed quantum system, i.e. we construct a unitary dilation of the discrete-time quantum-dynamical semigroup $(T^n)_{n \in \mathbb N_0}$. In the case of a cyclic channel $T$, the auxiliary space may be chosen (partially) finite-dimensional. We further investigate discrete-time quantum control systems generated by finitely many commuting quantum channels and prove a similar unitary dilation result as in the case of a single channel.