Entanglement Breaking Rank and the existence of SIC POVMs

TL;DR

This paper explores the entanglement breaking rank of a specific quantum channel and establishes its equivalence to the existence of symmetric informationally-complete POVMs in quantum information theory.

Contribution

It introduces the entanglement breaking rank concept and proves its value characterizes the existence of SIC POVMs in a given dimension.

Findings

Entanglement breaking rank of the channel is $d^2$ if and only if SIC POVMs exist in dimension d.

Provides a new characterization linking quantum channel properties to measurement structures.

Advances understanding of the mathematical conditions for SIC POVMs in quantum information.

Abstract

We introduce and study the entanglement breaking rank of an entanglement breaking channel. We show that the entanglement breaking rank of the channel $\mathfrak Z: M_d \to M_d$ defined by \begin{align*} \mathfrak Z(X) = \frac{1}{d+1}(X+\text{Tr}(X)\mathbb I_d) \end{align*} is $d^2$ if and only if there exists a symmetric informationally-complete POVM in dimension $d$.