Geometric uncertainty relation for mixed quantum states

TL;DR

This paper develops a geometric framework for mixed quantum states using symplectic reduction, deriving a new uncertainty relation and comparing it with classical results, highlighting differences and strengths.

Contribution

It introduces a geometric uncertainty relation for mixed states based on a novel principal fiber bundle construction, extending pure state geometry.

Findings

Derived a geometric uncertainty relation for mixed states.

Provided a geometric proof of the Robertson-Schrödinger relation.

Showed the geometric relation can be a stronger estimate in some cases.

Abstract

In this paper we use symplectic reduction in an Uhlmann bundle to construct a principal fiber bundle over a general space of unitarily equivalent mixed quantum states. The bundle, which generalizes the Hopf bundle for pure states, gives in a canonical way rise to a Riemannian metric and a symplectic structure on the base space. With these we derive a geometric uncertainty relation for observables acting on quantum systems in mixed states. We also give a geometric proof of the classical Robertson-Schr\"{o}dinger uncertainty relation, and we compare the two. They turn out not to be equivalent, because of the multiple dimensions of the gauge group for general mixed states. We give examples of observables for which the geometric relation provides a stronger estimate than that of Robertson and Schr\"{o}dinger, and vice versa.