Geometric aspects of mixed quantum states inside the Bloch sphere
- URL: http://arxiv.org/abs/2312.02004v2
- Date: Tue, 6 Feb 2024 18:46:21 GMT
- Title: Geometric aspects of mixed quantum states inside the Bloch sphere
- Authors: Paul M. Alsing, Carlo Cafaro, Domenico Felice, Orlando Luongo
- Abstract summary: We discuss the differences between the Bures and the Sj"oqvist metrics inside a Bloch sphere.
We show 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.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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