Geometric Aspects of Mixed Quantum States Inside the Bloch Sphere
Alsing P.M. Cafaro C. Felice D. Luongo O.
March 2024Multidisciplinary Digital Publishing Institute (MDPI)
Quantum Reports
2024#6Issue 190 - 109 pp.
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 the complexity and volume of quantum states. In this paper, we present an insightful discussion on the differences between the Bures and the Sjöqvist 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 explicitly illustrate the distinct behaviors of the geodesic paths on each one of the two metric manifolds. Third, we compare the finite distances between an initial state and the final mixed state when calculated with the two metrics. Interestingly, in analogy with what happens when studying the 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 a finite distance between mixed quantum states is not preserved when comparing distances determined with the Bures and the Sjöqvist metrics. Finally, we conclude with a brief discussion on the consequences of this violation of a metric-based relative ranking on the concept of the complexity and volume of mixed quantum states.
differential geometry , quantum computation , quantum information
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Air Force Research Laboratory, Information Directorate, Rome, 13441, NY, United States
Department of Nanoscale Science and Engineering, College of Nanotechnology, Science, and Engineering, University at Albany-SUNY, Albany, 12222, NY, United States
Department of Mathematics and Physics, SUNY Polytechnic Institute, Utica, 13502, NY, United States
Scuola Militare Nunziatella, Napoli, 80132, Italy
Scuola di Scienze e Tecnologie, Università di Camerino, Via Madonna delle Carceri 9, Camerino, 62032, Italy
INAF—Osservatorio Astronomico di Brera, Milano, 20121, Italy
INFN, Sezione di Perugia, Perugia, 06123, Italy
NNLOT, Al-Farabi Kazakh National University, Al-Farabi av. 71, Almaty, 050040, Kazakhstan
Air Force Research Laboratory
Department of Nanoscale Science and Engineering
Department of Mathematics and Physics
Scuola Militare Nunziatella
Scuola di Scienze e Tecnologie
INAF—Osservatorio Astronomico di Brera
INFN
NNLOT
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