# Quantum and classical resources for unitary design of open-system evolutions

## TL;DR

This paper develops a framework for implementing quantum channels through controlled unitary dynamics involving the environment, providing conditions for channel realization and connecting to existing protocols.

## Contribution

It formalizes a controllability framework for open-system quantum dynamics and extends Stinespring's dilation, including classical randomization, to realize various quantum channels.

## Key findings

- Conditions for realizing quantum channels based on environment state and dimension
- Disproof of Lloyd's conjecture on environment size, with classical randomization as a resource
- Recasting measurement-based protocols within a coherent control framework

## Abstract

A variety of tasks in quantum control, ranging from purification and cooling, to quantum stabilization and open-system simulation, rely on the ability to implement a target quantum channel over a specified time interval within prescribed accuracy. This can be achieved by engineering a suitable unitary dynamics of the system of interest along with its environment -- which, depending on the available level of control, is fully or partly exploited as a coherent quantum controller. After formalizing a controllability framework for completely positive trace-preserving quantum dynamics, we provide sufficient conditions on the environment state and dimension that allow for the realization of relevant classes of quantum channels -- including extreme channels, stochastic unitaries, or simply any channel. The results hinge on generalizations of Stinespring's dilation via a subsystem principle. In the process, we show that a conjecture by Lloyd on the minimal dimension of the environment required for arbitrary open-system simulation, albeit formally disproved, can in fact be salvaged -- provided that classical randomization is included among the available resources. Existing measurement-based feedback protocols for universal simulation, dynamical decoupling, and dissipative state preparation are recast within the proposed coherent framework as concrete applications, and the resources they employ discussed in the light of the general results.