Complexity of pure and mixed qubit geodesic paths on curved manifolds

Carlo Cafaro and Paul M. Alsing
Phys. Rev. D 106, 096004 – Published 8 November 2022

Abstract

It is known that mixed quantum states are highly entropic states of imperfect knowledge (i.e., incomplete information) about a quantum system, while pure quantum states are states of perfect knowledge (i.e., complete information) with vanishing von Neumann entropy. In this paper, we propose an information geometric theoretical construct to describe and, to a certain extent, understand the complex behavior of evolutions of quantum systems in pure and mixed states. The comparative analysis is probabilistic in nature, it uses a complexity measure that relies on a temporal averaging procedure along with a long-time limit, and is limited to analyzing expected geodesic evolutions on the underlying manifolds. More specifically, we study the complexity of geodesic paths on the manifolds of single-qubit pure and mixed quantum states equipped with the Fubini-Study metric and the Sjöqvist metric, respectively. We analytically show that the evolution of mixed quantum states in the Bloch ball is more complex than the evolution of pure states on the Bloch sphere. We also verify that the ranking based on our proposed measure of complexity, a quantity that represents the asymptotic temporal behavior of an averaged volume of the region explored on the manifold during the evolution of the systems, agrees with the geodesic length-based ranking. Finally, focusing on geodesic lengths and curvature properties in manifolds of mixed quantum states, we observed a softening of the complexity on the Bures manifold compared to the Sjöqvist manifold.

  • Received 21 September 2022
  • Accepted 19 October 2022

DOI:https://doi.org/10.1103/PhysRevD.106.096004

© 2022 American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & TechnologyInterdisciplinary Physics

Authors & Affiliations

Carlo Cafaro1 and Paul M. Alsing2

  • 1SUNY Polytechnic Institute, 12203 Albany, New York, USA
  • 2Air Force Research Laboratory, Information Directorate, 13441 Rome, New York, USA

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Issue

Vol. 106, Iss. 9 — 1 November 2022

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