Morphophoric POVMs, generalised qplexes, and 2-designs

TL;DR

This paper introduces morphophoric POVMs and generalised qplexes, expanding the framework of quantum measurements beyond SIC-POVMs to include 2-designs, and explores their geometric and algebraic properties.

Contribution

It generalizes the concept of qplexes using morphophoric POVMs, including 2-designs, and extends QBism's primal equation to this broader class.

Findings

Generalised qplexes share the same intrinsic geometry as quantum states.

The external geometry involves algebraic and geometric analysis of polytopes.

Extension of QBism's primal equation to morphophoric POVMs.

Abstract

We study the class of quantum measurements with the property that the image of the set of quantum states under the measurement map transforming states into probability distributions is similar to this set and call such measurements morphophoric. This leads to the generalisation of the notion of a qplex, where SIC-POVMs are replaced by the elements of the much larger class of morphophoric POVMs, containing in particular 2-design (rank-1 and equal-trace) POVMs. The intrinsic geometry of a generalised qplex is the same as that of the set of quantum states, so we explore its external geometry, investigating, inter alia, the algebraic and geometric form of the inner (basis) and the outer (primal) polytopes between which the generalised qplex is sandwiched. In particular, we examine generalised qplexes generated by MUB-like 2-design POVMs utilising their graph-theoretical properties. Moreover, we show how to extend the primal equation of QBism designed for SIC-POVMs to the morphophoric case.