SIC-POVMs and MUBs: Geometrical Relationships in Prime Dimension

Abstract

The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d^2-1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d+1 mutually orthogonal d-1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d+1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs.

Contribution to the Conference "Foundations of Probability and Physics-5", Vaxjo, 2008

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• Structure of Two-qubit Symmetric Informationally Complete POVMs
• The Number Behind the Simplest SIC-POVM
• Order 3 Symmetry in the Clifford Hierarchy
• Connections of geometric measure of entanglement of pure symmetric states to quantum state estimation
• Structure of the sets of mutually unbiased bases with cyclic symmetry
• Characterization of fiducial states in prime dimensions via mutually unbiased bases