Environment-invariant measure of distance between evolutions of an open quantum system

TL;DR

This paper introduces an environment-invariant distance measure for comparing the evolution of open quantum systems to ideal closed-system operations, aiding quantum control and fidelity assessment.

Contribution

It develops a novel environment-invariant measure for quantifying differences between open and closed quantum system evolutions, useful for quantum control.

Findings

The measure is invariant to environmental effects and initial states.

It provides fidelity bounds for quantum channels relative to target unitaries.

Numerical simulations demonstrate its utility in quantum gate fidelity evaluation.

Abstract

The problem of quantifying the difference between evolutions of an open quantum system (in particular, between the actual evolution of an open system and the ideal target operation on the corresponding closed system) is important in quantum control, especially in control of quantum information processing. Motivated by this problem, we develop a measure for evaluating the distance between unitary evolution operators of a composite quantum system that consists of a sub-system of interest (e.g., a quantum information processor) and environment. The main characteristic of this measure is the invariance with respect to the effect of the evolution operator on the environment, which follows from an equivalence relation that exists between unitary operators acting on the composite system, when the effect on only the sub-system of interest is considered. The invariance to the environment's transformation makes it possible to quantitatively compare the evolution of an open quantum system and its closed counterpart. The distance measure also determines the fidelity bounds of a general quantum channel (a completely positive and trace-preserving map acting on the sub-system of interest) with respect to a unitary target transformation. This measure is also independent of the initial state of the system and straightforward to numerically calculate. As an example, the measure is used in numerical simulations to evaluate fidelities of optimally controlled quantum gate operations (for one- and two-qubit systems), in the presence of a decohering environment. This example illustrates the utility of this measure for optimal control of quantum operations in the realistic case of open-system dynamics.