SIC-POVMs and Compatibility among Quantum States

TL;DR

This paper explores the deep connection between compatibility criteria for quantum states and symmetric informationally complete POVMs, revealing new properties of qutrit SIC-POVMs and their implications for quantum computation and hidden-variable theories.

Contribution

It demonstrates that compatibility criteria can elucidate the relationship between SIC-POVMs and mutually unbiased bases, offering new insights into quantum resources and foundational models.

Findings

Qutrit SIC-POVMs have additional interesting properties.

Compatibility criteria relate SIC-POVMs to mutually unbiased bases.

Insights into quantum resources for magic-state quantum computation.

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.