Quantum mechanics? It's all fun and games until someone loses an $i$

Abstract

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 [21] 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. * This paper is dedicated to Prof. Gopal Rao upon his promotion to Distinguished Professor Emeritus status. Prof. Rao once remarked to one of us (CAF) that being at UMass Boston-rather than say at Harvard or Yale-allowed his career to excel. UMass Boston brought him academic freedoms he could not find elsewhere: He worked on whatever he pleased, without pressure for high-dollar funding or worries over journals' impact factors. We offer this paper to Prof. Rao in his own spirit. True science is founded upon the freedom to become fascinated by a simple pebble on a beach, whether it be polished or not, or whether it have any value at the local market.

The classification of rebit quantum channels

“…At this moment, the question remains a much debated and interesting open problem, see e.g. [1,6,[8][9][10]. If one wished to analyze the dynamical change of rebit states in terms of channels, it seems reasonable to assume that having available the complete classification of these maps would be of great benefit.…”

n-Lüders channels: a novel class of rebit channels, their properties and applications

“…This temporal gap is probably due to the lack of concrete examples of physical rebit systems, which, nevertheless, continue to inspire very insightful theoretical studies, see e.g. [3][4][5][6][7][8][9][10][11][12][13][14] just to have a few examples.…”