Geometry of the set of mixed quantum states: An apophatic approach

Ingemar Bengtsson, Stephan Weis, and Karol Życzkowski

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Submission date: 11. Dec. 2011
Pages: 14
published in: Geometric methods in physics : XXX Workshop, Bialowieza, Poland, June 26 to July 2, 2011 / P. Kielanowski (ed.)
Basel : Springer, 2013. - P. 175 - 197
(Trends in mathematics) 
DOI number (of the published article): 10.1007/978-3-0348-0448-6_15
Bibtex
MSC-Numbers: 81P16, 52A20
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Abstract:
The set of quantum states consists of density matrices of order N, which are hermitian, positive and normalized by the trace condition. We analyze the structure of this set in the framework of the Euclidean geometry naturally arising in the space of hermitian matrices. For N = 2 this set is the Bloch ball, embedded in ℝ³. For N ≥ 3 this set of dimensionality N² - 1 has a much richer structure. We study its properties and at first advocate an apophatic approach, which concentrates on characteristics not possessed by this set. We also apply more constructive techniques and analyze two dimensional cross-sections and projections of the set of quantum states. They are dual to each other. At the end we make some remarks on certain dimension dependent properties.