Morphophoric POVMs, generalised qplexes, and 2-designs

Wojciech Słomczyński and Anna Szymusiak

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

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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.

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[3] Matthew B. Weiss, "Depolarizing Reference Devices in Generalized Probabilistic Theories", arXiv:2312.12790, (2023).

[4] Michele Dall'Arno and Francesco Buscemi, "Tight conic approximation of testing regions for quantum statistical models and measurements", arXiv:2309.16153, (2023).

[5] Gerhard Zauner, "The finite Fourier Transform and projective 2-designs", arXiv:2207.09922, (2022).

[6] Blake C. Stacey, "Contradictions or Curiosities? On Kent's Critique of the Masanes--Galley--Müller Derivation of the Quantum Measurement Postulates", arXiv:2405.17733, (2024).

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