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

Anna Szymusiak; Wojciech Słomczyński

doi:10.22331/q-2025-01-15-1598

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

Anna Szymusiak and Wojciech Słomczyński

Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

Published:	2025-01-15, volume 9, page 1598
Editor:	Alexander Wilce
Eprint:	arXiv:2302.04957v2
Doi:	https://doi.org/10.22331/q-2025-01-15-1598
Citation:	Quantum 9, 1598 (2025).

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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'.

Featured image: The intersection of the simplex $\Delta_4$ by the affine plane $\mathcal A$ (green) with the primal polytope $\Delta$ (square with yellow edges on the left and with green edges on the right), the set of possible probabilities $\mathcal P$ (the same as $\Delta$ on the left and yellow square on the right) and the dual polytope $D$ (purple inner square) in two different scenarios for square GGPT.

► BibTeX data

@article{Szymusiak2025canqbismexist,
  doi = {10.22331/q-2025-01-15-1598},
  url = {https://doi.org/10.22331/q-2025-01-15-1598},
  title = {Can {QB}ism exist without {Q}? {M}orphophoric measurements in generalised probabilistic theories},
  author = {Szymusiak, Anna and S{\l{}}omczy{\'{n}}ski, Wojciech},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1598},
  month = jan,
  year = {2025}
}

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