Symmetric informationally complete measurements identify the irreducible difference between classical and quantum systems

Abstract

We describe a general procedure for associating a minimal informationally complete quantum measurement (or MIC) with a purely probabilistic representation of the Born Rule. Such representations provide a way to understand the Born Rule as a consistency condition between probabilities assigned to the outcomes of one experiment in terms of the probabilities assigned to the outcomes of other experiments. In this setting, the difference between quantum and classical physics is the way their physical assumptions augment bare probability theory: Classical physics corresponds to a trivial augmentation-one just applies the Law of Total Probability (LTP) between the scenarioswhile quantum theory makes use of the Born Rule expressed in one or another of the forms of our general procedure. To mark the irreducible difference between quantum and classical, one should seek the representations that minimize the disparity between the expressions. We prove that the representation of the Born Rule obtained from a symmetric informationally complete measurement (or SIC) minimizes this distinction in at least two senses-the first to do with unitarily invariant distance measures between the rules, and the second to do with available volume in a reference probability simplex (roughly speaking a new kind of uncertainty principle). Both of these arise from a useful result in majorization theory. This work complements recent studies in quantum computation where the deviation of the Born Rule from the LTP is measured in terms of negativity of Wigner functions. arXiv:1805.08721v3 [quant-ph] 30 Apr 2020

SICs: Some Explanations

“…I think so. The QBist approach to the foundations of quantum mechanics certainly suggests it [1,14,15]. The number theoretical angle suggests additional arguments.…”

“…There is a school of thought maintaining that SICs will ultimately prove to be as important [1,14,15] for quantum foundations as are the orthonormal bases [16]. We do not pursue this argument here, but we do hope to convince the reader that SICs deserve to be spelt with capital letters.…”

Morphophoric POVMs, generalised qplexes, and 2-designs

“…ii. for every POVM Ξ = (Ξ k ) N k=1 , (23) holds with α given by (24); iii. for some IC-POVM Ξ = (Ξ k ) N k=1 and α > 0, (23) holds.…”

“…Over the last ten years in a series of papers [2, 3, 30-32, 49, 68] Fuchs, Schack, Appleby, Stacey, Zhu and others have introduced first the idea of QBism (formerly quantum Bayesianism) and then its probabilistic embodiment -the (Hilbert) qplex, both based on the notion of symmetric informationally complete positive operator-valued measure (SIC-POVM ), representing the standard (or reference) measurement of a quantum system. In spite of the fact that SIC-POVMs do indeed exhibit some unique properties that distinguish them among other IC-POVMs [24,25,52], we believe that similar (or, in a sense, even richer from a purely mathematical point of view) structures, say, generalised qplexes, can be successfully used to faithfully represent quantum states as probability distributions if we replace SIC-POVMs with the broader class of morphophoric POVMs, containing in particular rank-1 measurements generated by complex projective 2-designs. As in the QBism interpretation of quantum mechanics, the result is a new formalism for quantum theory, which involves only probabilities.…”

Is the Reality Criterion Analytic?

“…Alternatively (and more precisely), one could characterize QBism and related views as doxastic approaches to QM (e.g.,Boge 2018, Sect. 7.2;DeBrota et al 2018).…”