Geometric aspects of mixed quantum states inside the Bloch sphere

TL;DR

This paper compares the geometric properties of mixed quantum states within the Bloch sphere using Bures and Sjöqvist metrics, revealing differences in geodesic paths and distance rankings that impact the understanding of quantum complexity and volume.

Contribution

It provides a formal comparison of two quantum state metrics, analyzes their geodesic behaviors, and discusses implications for quantum state complexity and volume.

Findings

Distances differ significantly between the two metrics.

Geodesic paths exhibit distinct behaviors under each metric.

Relative rankings of states by distance are not preserved across metrics.

Abstract

When studying the geometry of quantum states, it is acknowledged that mixed states can be distinguished by infinitely many metrics. Unfortunately, this freedom causes metric-dependent interpretations of physically significant geometric quantities such as complexity and volume of quantum states. In this paper, we present an insightful discussion on the differences between the Bures and the Sj\"oqvist metrics inside a Bloch sphere. First, we begin with a formal comparative analysis between the two metrics by critically discussing three alternative interpretations for each metric. Second, we illustrate explicitly the distinct behaviors of the geodesic paths on each one of the two metric manifolds. Third, we compare the finite distances between an initial and final mixed state when calculated with the two metrics. Interestingly, in analogy to what happens when studying topological aspects of real Euclidean spaces equipped with distinct metric functions (for instance, the usual Euclidean metric and the taxicab metric), we observe that the relative ranking based on the concept of finite distance among mixed quantum states is not preserved when comparing distances determined with the Bures and the Sj\"oqvist metrics. Finally, we conclude with a brief discussion on the consequences of this violation of a metric-based relative ranking on the concept of complexity and volume of mixed quantum states.