SIC-POVMs and Compatibility among Quantum States

Abstract

An unexpected connection exists between compatibility criteria for quantum states and symmetric informationally complete POVMs. Beginning with Caves, Fuchs and Schack's "Conditions for compatibility of quantum state assignments" [Phys. Rev. A 66 (2002), 062111], I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics.

15 pages, 4 MUBs, 2 errata for CFS (2002), 1 graph with chromatic number 4. v4: journal version

References in corpus (10)

• Hidden Variables and the Two Theorems of John Bell
• QBism, the Perimeter of Quantum Bayesianism
• Necessary and sufficient condition for quantum state-independent contextuality
• Experimental scheme for qubit and qutrit symmetric informationally complete positive operator-valued measurements using multiport devices
• Quasiprobability representations of quantum mechanics with minimal negativity
• Super-symmetric informationally complete measurements
• Quantum Conical Designs
• Informational power of the Hoggar SIC-POVM
• A toy model for quantum mechanics
• Multiscale Structure in Eco-Evolutionary Dynamics

Cited by in corpus (15)

• The SIC Question: History and State of Play
• Introducing the Qplex: A Novel Arena for Quantum Theory
• Symmetric Informationally Complete Measurements Identify the Irreducible Difference between Classical and Quantum Systems
• Informational power of the Hoggar SIC-POVM
• Von Neumann Was Not a Quantum Bayesian
• QBism: Quantum Theory as a Hero's Handbook
• Sporadic SICs and the Normed Division Algebras
• Discrete Wigner Functions from Informationally Complete Quantum Measurements
• Geometric and Information-Theoretic Properties of the Hoggar Lines
• Quantum Theory as Symmetry Broken by Vitality
• Ideas Abandoned en Route to QBism
• Is the SIC Outcome There When Nobody Looks?
• MUBs and SIC-POVMs of a spin-1 system from the Majorana approach
• Invariant Off-Diagonality: SICs as Equicoherent Quantum States
• Multiscale Structure of More-than-Binary Variables