Uhlmann phase in composite systems with entanglement

TL;DR

This paper investigates the Uhlmann phase in entangled mixed states of a two-spin system, revealing topological transitions linked to entanglement and depolarization, and compares it with an interferometric phase.

Contribution

It provides an exact analytical expression for the Uhlmann phase in entangled states and connects it to entanglement measures, highlighting topological phase transitions.

Findings

Uhlmann phase exhibits a topological transition with phase singularity.

The phase transition depends on field direction, coupling, and depolarization.

Uhlmann and interferometric phases differ significantly in strong entanglement regimes.

Abstract

We study the geometric Uhlmann phase of entangled mixed states in a composite system made of two coupled spin-$\frac 1 2$ particles with a magnetic field acting on one of them. Within a depolarizing channel setup, an exact analytical expression for such a phase in each subsystem is derived. We find an explicit connection to the concurrence of the depolarizing channel density matrix, which allows to characterize the features of the Uhlmann phase in terms of the degree of entanglement in the system. In the space of field direction and coupling parameter, it exhibits a phase singularity revealing a topological transition between orders with different winding numbers. The transition occurs for fields lying in the equator of the sphere of directions and at critical values of the coupling which can be controlled by tuning the depolarization strength. Notably, under these conditions the concurrence of the composite system is bounded to the range $[0,1/2]$. We also compare the calculated Uhlmann phase to an interferometric phase, which has been formulated as an alternative for density matrices. The latter does not present a phase vortex, although they coincide in the weak entanglement regime, for vanishing depolarization (pure states). Otherwise they behave clearly different in the strong entanglement regime.