Beyond morphophoricity: $s$-tight IC measurements in geometric generalised probabilistic theories

TL;DR

This paper introduces a new class of $s$-tight informationally complete measurements in generalized probabilistic theories, unifying morphophoric and tight IC measurements and revealing their mathematical structure and relation to the generalized Urgleichung.

Contribution

It defines and analyzes the broader class of $s$-tight IC measurements, connecting them to the generalized Urgleichung and extending previous quantum-specific concepts.

Findings

$s$-tight IC measurements unify morphophoric and tight IC classes.

Tight IC measurements correspond to a simplified form of the generalized Urgleichung.

The mathematical structure reveals deeper connections between measurement classes and probabilistic frameworks.

Abstract

The analysed in this paper new class of $s$-tight IC measurements contains both morphophoric measurements, preserving the geometry of the states space, and tight IC measurements, introduced nearly 20 years ago by Scott in the quantum case as optimal for the task of linear quantum tomography. By looking at the mathematical side of these classes we discover their common feature, which is also preserved in the broader class of $s$-tight IC measurements: a particularly elegant form of the formula that can be seen as the generalised form of the Urgleichung known from the QBist approach to quantum theory. In particular, the tight IC measurements are identified as the ones for which this generalised Urgleichung takes an exceptionally simple form.