Super-symmetric informationally complete measurements

Abstract

Symmetric informationally complete measurements (SICs in short) are highly symmetric structures in the Hilbert space. They possess many nice properties which render them an ideal candidate for fiducial measurements. The symmetry of SICs is intimately connected with the geometry of the quantum state space and also has profound implications for foundational studies. Here we explore those SICs that are most symmetric according to a natural criterion and show that all of them are covariant with respect to the Heisenberg-Weyl groups, which are characterized by the discrete analogy of the canonical commutation relation. Moreover, their symmetry groups are subgroups of the Clifford groups. In particular, we prove that the SIC in dimension~2, the Hesse SIC in dimension~3, and the set of Hoggar lines in dimension~8 are the only three SICs up to unitary equivalence whose symmetry groups act transitively on pairs of SIC projectors. Our work not only provides valuable insight about SICs, Heisenberg-Weyl groups, and Clifford groups, but also offers a new approach and perspective for studying many other discrete symmetric structures behind finite state quantum mechanics, such as mutually unbiased bases and discrete Wigner functions.

29 pages, to appear in Annals of Physics

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• Tight informationally complete quantum measurements
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Cited by in corpus (22)

• The SIC Question: History and State of Play
• Multiqubit Clifford groups are unitary 3-designs
• Constructing exact symmetric informationally complete measurements from numerical solutions
• The Clifford group fails gracefully to be a unitary 4-design
• Quasiprobability representations of quantum mechanics with minimal negativity
• Permutation Symmetry Determines the Discrete Wigner Function
• SICs and Algebraic Number Theory
• Informational power of the Hoggar SIC-POVM
• Iso-entangled mutually unbiased bases, symmetric quantum measurements and mixed-state designs
• SIC-POVMs and Compatibility among Quantum States
• Sporadic SICs and the Normed Division Algebras
• Mutually unbiased bases as minimal Clifford covariant 2-designs
• Dimension towers of SICs. I. Aligned SICs and embedded tight frames
• Magic informationally complete POVMs with permutations
• Geometric and Information-Theoretic Properties of the Hoggar Lines
• Quantum Measurements in the Light of Quantum State Estimation
• Quantum Theory is a Quasi-stochastic Process Theory
• Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories
• Two fundamental solutions to the rigid Kochen-Specker set problem and the solution to the minimal Kochen-Specker set problem under one assumption
• Invariant Off-Diagonality: SICs as Equicoherent Quantum States
• Maximal Sets of Equiangular Lines
• Informationally Overcomplete POVMs for Quantum State Estimation and Binary Detection