Asian Journal of Physics Vol 30, No 12 (2021) 1707-1726

Quantum mechanics? It’s all fun and games until some one loses an i*

Christopher A Fuchs, Maxim Olchanyi, and Matthew B Weiss
Department of Physics, University of Massachusetts Boston, 100 Morrissey Blvd, Boston, MA 02125, USA
Dedicated to Prof D V G L N Rao

QBism regards quantum mechanics as an addition to probability theory. The addition provides an extra normative rule for decision-making agents concerned with gambling across experimental contexts, somewhat in analogy to the doubleslit experiment. This establishes the meaning of the Born Rule from a QBist perspective. Moreover, it suggests that the best way to formulate the Born Rule for foundational discussions is with respect to an informationally complete reference device. Recent work has demonstrated that reference devices employing symmetric informationally complete POVMs (or SICs) achieve a minimal quantumness. They witness the irreducible difference between classical and quantum. In this paper, we attempt to answer the analogous question for real-vector-space quantum theory. While standard quantum mechanics seems to allow SICs to exist in all finite dimensions, in the case of quantum theory over the real numbers it is known that SICs do not exist in most dimensions. We therefore attempt to identify the optimal reference device in the first real dimension without a SIC (i.e., d = 4) in hopes of better understanding the essential role of complex numbers in quantum mechanics. In contrast to their complex counterparts, the expressions that result in a QBist understanding of real-vector-space quantum theory are surprisingly complex. © Anita Publications. All rights reserved.
Keywords: QBist, Born Rule, Real-vector-space.

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