Representation of quantum states as points in a probability simplex associated to a SIC-POVM

TL;DR

This paper explores how quantum states can be represented as probability vectors in a simplex using SIC-POVMs and shows that this representation is equivalent to the Bloch vector representation with proper scaling.

Contribution

It proves the equivalence of the SIC-POVM probability simplex and the Bloch vector set for quantum states with appropriate scaling.

Findings

The sets of vectors $\\mathcal{Q}$ and $\mathcal{B}$ are equivalent after scaling.

Features of the shape of the set $\mathcal{Q}$ are discussed.

Representation of quantum states in a probability simplex is clarified.

Abstract

The quantum state of a $d$-dimensional system can be represented by the $d^2$ probabilities corresponding to a SIC-POVM, and then this distribution of probability can be represented by a vector of $\R^{d^2-1}$ in a simplex, we will call this set of vectors $\mathcal{Q}$. Other way of represent a $d$-dimensional system is by the corresponding Bloch vector also in $\R^{d^2-1}$, we will call this set of vectors $\mathcal{B}$. In this paper it is proved that with the adequate scaling $\mathcal{B}=\mathcal{Q}$. Also we indicate some features of the shape of $\mathcal{Q}$.