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arXiv:1112.2347 · Springer eBooks

Geometry of the Set of Mixed Quantum States: An Apophatic Approach

Ingemar Bengtsson, Stephan Weis, Karol Życzkowski

Published Sep 28, 2012• 35 citations
arXivHTMLPDF
DOI
10.1007/978-3-0348-0448-6_15

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Pure coherent states are known as the most classical states in quantum mechanics. Similarly, pure spin coherent states based on the rotation group also exist in finite dimensions. We call these states pure classical states, and the mixed classical states are the statistical mixtures of these states. The set of all classical states is a subset of the set of all pure & mixed states. We present an essentially complete description of the geometry of this set for 3-dimensional Hilbert space. While studying its geometry we find a remarkable similarity between the geometry of this set and the geometry of the set of all classical probability distributions. This study also improves a recent work on this set done by Giraud at el [4] and relates it to an earlier work by Wootters [14].
Density matrices for mixed state qubits, parametrized by the Bloch vector in the open unit ball of the Euclidean 3-space, are well known in quantum information and computation theory. By presenting new identities for the qubit density matrix we indicate its intimate relationship with M\"obius addition and scalar multiplication. The latter, in turn, form the algebraic setting for the Poincar\'e ball model of hyperbolic geometry so that, as a result, the qubit density matrix is linked to hyperbolic geometry.