Symmetric informationally complete positive operator valued measure and probability representation of quantum mechanics

TL;DR

This paper explores SIC-POVMs within quantum probability representations, establishing their properties, relations to other frameworks, and implications for quantum state measurement, especially focusing on qubits and algebraic structures.

Contribution

It formulates SIC-POVM existence in terms of star-product symbols and derives new relations, linking SIC-POVMs to probability representations and algebraic structures.

Findings

SIC-POVMs are a special case of probability representation

Derived new relations on SIC-projectors

Established connections with mutually unbiased bases

Abstract

Symmetric informationally complete positive operator valued measures (SIC-POVMs) are studied within the framework of the probability representation of quantum mechanics. A SIC-POVM is shown to be a special case of the probability representation. The problem of SIC-POVM existence is formulated in terms of symbols of operators associated with a star-product quantization scheme. We show that SIC-POVMs (if they do exist) must obey general rules of the star product, and, starting from this fact, we derive new relations on SIC-projectors. The case of qubits is considered in detail, in particular, the relation between the SIC probability representation and other probability representations is established, the connection with mutually unbiased bases is discussed, and comments to the Lie algebraic structure of SIC-POVMs are presented.