Can QBism exist without Q? Morphophoric measurements in generalised probabilistic theories

TL;DR

This paper introduces morphophoric measurements within generalized probabilistic theories, extending QBism's foundational equation beyond quantum SIC-POVMs to a broader class of measurements with operational significance.

Contribution

It defines morphophoric measurements in GPTs, generalizes the QBism 'Urgleichung' to these measurements, and demonstrates the retention of QBism's core features in this broader setting.

Findings

Morphophoric measurements generalize 2-design POVMs and SIC-POVMs.

The QBism 'Urgleichung' is extended to morphophoric measurements in GPTs.

Quantities in the extended equation are interpretable operationally.

Abstract

In a Generalised Probabilistic Theory (GPT) equipped additionally with some extra geometric structure we define the morphophoric measurements as those for which the measurement map sending states to distributions of the measurement results is a similarity. In the quantum case, morphophoric measurements generalise the notion of a 2-design POVM, thus in particular that of a SIC-POVM. We show that the theory built on this class of measurements retains the chief features of the QBism approach to the basis of quantum mechanics. In particular, we demonstrate how to extend the primal equation ('Urgleichung') of QBism, designed for SIC-POVMs, to the morphophoric case of GPTs. In the latter setting, the equation takes a different, albeit more symmetric, form, but all the quantities that appear in it can be interpreted in probabilistic and operational terms, as in the original 'Urgleichung'.