Holonomy for Quantum Channels

TL;DR

This paper extends the concept of holonomy to quantum channels, establishing a connection with the Uhlmann holonomy, and proposes a physical interferometry-based realization, linking geometric phase concepts to quantum information processing.

Contribution

It introduces a novel notion of holonomy for quantum channels using the Jamio{kowski} isomorphism and relates it to physical interferometry experiments.

Findings

Channel holonomy is related to Uhlmann holonomy.

Interferometry can realize channel holonomy physically.

Past neutron spin experiments are early examples of channel holonomy.

Abstract

A quantum holonomy reflects the curvature of some underlying structure of quantum mechanical systems, such as that associated with quantum states. Here, we extend the notion of holonomy to families of quantum channels, i.e., trace preserving completely positive maps. By the use of the Jamio{\l}kowski isomorphism, we show that the proposed channel holonomy is related to the Uhlmann holonomy. The general theory is illustrated for specific examples. We put forward a physical realization of the channel holonomy in terms of interferometry. This enables us to identify a gauge invariant physical object that directly relates to the channel holonomy. Parallel transport condition and concomitant gauge structure are delineated in the case of smoothly parametrized families of channels. Finally, we point out that interferometer tests that have been carried out in the past to confirm the $4\pi$ rotation symmetry of the neutron spin, can be viewed as early experimental realizations of the channel holonomy.