Mutually unbiased measurements with arbitrary purity

TL;DR

This paper introduces a new class of mutually unbiased measurements with arbitrary purity levels, exploring their properties and applications in entanglement detection in quantum systems.

Contribution

It provides a comprehensive class of MUMs covering all purity levels and demonstrates their use in constructing entanglement witnesses for bipartite states.

Findings

Class of MUMs spans full purity range from 1/d to 1.

Spectra of MUMs enable construction of special orthogonal matrices.

MUM-based entanglement witnesses effectively detect bipartite entanglement.

Abstract

Mutually unbiased measurements are a generalization of mutually unbiased bases in which the measurement operators need not to be rank one projectors. In a $d$-dimension space, the purity of measurement elements ranges from $1/d$ for the measurement operators corresponding to maximally mixed states to $1$ for the rank one projectors. In this contribution, we provide a class of MUM that encompasses the full range of purity. Similar to the MUB in which the operators corresponding to different outcomes of the same measurement commute mutually, our class of MUM possesses this sense of compatibility within each measurement. This makes the provided class more similar to the MUB, so that the main difference between them and MUB is due to the purity of the measurement operators. The spectra of these MUMs provides a way to construct a class of $d$-dimensional orthogonal matrices which leave the vector of equal components invariant. Based on this property, and by using the MUM-based entanglement witnesses, we investigate the role of purity to detect entanglement of bipartite states.