Notes on general SIC-POVMs

TL;DR

This paper explores the properties of general symmetric informationally complete measurements (SIC-POVMs) in quantum systems, deriving new relations, bounds, and uncertainty principles relevant for quantum information tasks.

Contribution

It formulates novel properties of general SIC-POVMs and derives state-dependent entropic bounds and uncertainty relations for these measurements.

Findings

Exact calculation of the index of coincidence for any density matrix and SIC-POVM.

Derived lower entropic bounds using Rényi and Tsallis entropies.

Established entropic uncertainty relations for pairs of SIC-POVMs.

Abstract

An unavoidable task in quantum information processing is how to obtain data about the state of an individual system by suitable measurements. Informationally complete measurements are relevant in quantum state tomography, quantum cryptography, quantum cloning, and other questions. Symmetric informationally complete measurements (SIC-POVMs) form an especially important class of such measurements. We formulate some novel properties and relations for general SIC-POVMs in a finite-dimensional Hilbert space. For a given density matrix and any general SIC-POVM, the so-called index of coincidence of generated probability distribution is exactly calculated. Using this result, we obtain state-dependent entropic bounds for a single general SIC-POVM. Lower entropic bounds are derived in terms of the R\'{e}nyi $\alpha$-entropies for $\alpha\in[2;\infty)$ and the Tsallis $\alpha$-entropies for $\alpha\in(0;2]$. A lower bound on the min-entropy of a SIC-POVM is separately examined. For a pair of general SIC-POVMs, entropic uncertainty relations of the Maassen-Uffink type are considered.