A thermofield-double model of Uhlmann anholonomy

TL;DR

This paper models a thermofield double quantum system to study Uhlmann anholonomy, revealing geometric phases, gauge structures, and their implications for quantum computation and state discrimination.

Contribution

It introduces a thermofield double model of Uhlmann anholonomy, analyzing geometric phases, gauge symmetries, and their applications to quantum computation and state discrimination.

Findings

Uhlmann connection related to higher-dimensional instantons.

Explicit calculation of anholonomy for geodesic triangles.

Sequence of geodesic triangles implementing the iSWAP gate.

Abstract

A simple parametrized family of quantum systems consisting of two entangled subsystems, dubbed left and right ones, both of them featuring N qubits is considered in the thermofield double formalism. We assume that the system evolves in a purely geometric manner based on the parallel transport condition due to Uhlmann. We explore the different interpretations of this evolution relative to observers either coupled to the left or to the right subsystems. The Uhlmann condition breaks the symmetry between left and right by regarding one of the two possible sets of local unitary operations as gauge degrees of freedom. Then gauging the right side we show that the geometric evolution on the left manifests itself via certain local operations reminiscent of non-unitary filtering measurements. On the other hand on the right the basic evolutionary steps are organized into a sequence of unitary operations of a holonomic quantum computation. We calculate the Uhlmann connection governing the transport for our model which turns out to be related to higher dimensional instantons. Then we evaluate the anholonomy of the connection for geodesic triangles with geodesic segments defined with respect to the Bures metric. By analysing the explicit form of the local filtering measurements showing up on the left side we realize that they are also optimal measurements for distinguishing two given mixed states in the statistical sense. We also point out that by conducting an interference experiment on the right side one can observe the physical effects of the anholonomic quantum computation. We demonstrate this by calculating explicit examples for phase shifts and visibility patterns arising in such interference experiments. Finally a sequence of geodesic triangles producing the iSWAP gate via anholonomy needed for computational universality is presented.