Quantum measurements with prescribed symmetry

Abstract

We introduce a method to determine whether a given generalised quantum measurement is isolated or it belongs to a family of measurements having the same prescribed symmetry. The technique proposed reduces to solving a linear system of equations in some relevant cases. As consequence, we provide a simple derivation of the maximal family of Symmetric Informationally Complete measurements (SIC)-POVM in dimension 3. Furthermore, we show that the following remarkable geometrical structures are isolated, so that free parameters cannot be introduced: (a) maximal sets of mutually unbiased bases in prime power dimensions from 4 to 16, (b) SIC-POVM in dimensions from 4 to 16 and (c) contextuality Kochen-Specker sets in dimension 3, 4 and 6, composed of 13, 18 and 21 vectors, respectively.

10 pages, 2 figures

References in corpus (6)

• The SIC Question: History and State of Play
• Entropic uncertainty relation for mutually unbiased bases
• Entropic uncertainty relations and locking: tight bounds for mutually unbiased bases
• SICs: Extending the list of solutions
• Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension six
• Maximally informative ensembles for SIC-POVMs in dimension 3

Cited by in corpus (5)

• The SIC Question: History and State of Play
• Tight Frames, Hadamard Matrices and Zauner's Conjecture
• Detecting Entanglement can be More Effective with Inequivalent Mutually Unbiased Bases
• Two-Unitary Complex Hadamard Matrices of Order $36$
• Generalized Numerical Construction of MUBs: A Group Theoretical Investigation