Von Neumann was not a Quantum Bayesian Stacey, Blake C.
Philosophical transactions - Royal Society. Mathematical, Physical and engineering sciences/Philosophical transactions - Royal Society. Mathematical, physical and engineering sciences,
05/2016, Volume:
374, Issue:
2068
Journal Article
Peer reviewed
Open access
Wikipedia has claimed for over 3 years now that John von Neumann was the 'first quantum Bayesian'. In context, this reads as stating that von Neumann inaugurated QBism, the approach to quantum theory ...promoted by Fuchs, Mermin and Schack. This essay explores how such a claim is, historically speaking, unsupported.
Introducing the Qplex: a novel arena for quantum theory Appleby, Marcus; Fuchs, Christopher A.; Stacey, Blake C. ...
The European physical journal. D, Atomic, molecular, and optical physics,
07/2017, Volume:
71, Issue:
7
Journal Article
Peer reviewed
Open access
We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, “Unperformed experiments have no results.” The tools of quantum information theory, and in particular the ...symmetric informationally complete (SIC) measurements, provide a concise expression of how exactly Peres’s dictum holds true. That expression is a constraint on how the probability distributions for outcomes of different, hypothetical and mutually exclusive experiments ought to mesh together, a type of constraint not foreseen in classical thinking. Taking this as our foundational principle, we show how to reconstruct the formalism of quantum theory in finite-dimensional Hilbert spaces. The central variety of mathematical entity in our reconstruction is the
qplex,
a very particular type of subset of a probability simplex. Along the way, by closely studying the symmetry properties of qplexes, we derive a condition for the existence of a
d
-dimensional SIC.
Graphical abstract
Symmetric informationally complete quantum measurements, or SICs, are mathematically intriguing structures, which in practice have turned out to exhibit even more symmetry than their definition ...requires. Recently, Zhu classified all the SICs whose symmetry groups act doubly transitively. I show that lattices of integers in the complex numbers, the quaternions and the octonions yield the key parts of these symmetry groups.
Complex systems display behavior at a range of scales. Large-scale behaviors can emerge from the correlated or dependent behavior of individual small-scale components. To capture this observation in ...a rigorous and general way, we introduce a formalism for multiscale information theory. Dependent behavior among system components results in overlapping or shared information. A system’s structure is revealed in the sharing of information across the system’s dependencies, each of which has an associated scale. Counting information according to its scale yields the quantity of scale-weighted information, which is conserved when a system is reorganized. In the interest of flexibility we allow information to be quantified using any function that satisfies two basic axioms. Shannon information and vector space dimension are examples. We discuss two quantitative indices that summarize system structure: an existing index, the complexity profile, and a new index, the marginal utility of information. Using simple examples, we show how these indices capture the multiscale structure of complex systems in a quantitative way.
An unexpected connection exists between compatibility criteria for quantum states and Symmetric Informationally Complete quantum measurements (SIC-POVMs). Beginning with Caves, Fuchs and Schack’s ..."Conditions for compatibility of quantum state assignments", I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics.
We describe a general procedure for associating a minimal informationally complete quantum measurement 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 scenarios—while 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 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 different 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.
► We present a complex networks perspective on large interfirm organizational networks. ► We propose two micro-dynamic models that nicely reproduce the actual observations. ► Spatial locations of ...firms are positively correlated with the population density. ► Firm competition is governed by cumulative advantage rules and geographic distance. ► Competition network formation and firm size dynamics are closely intertwined.
A substantial number of studies have extended the work on universal properties in physical systems to complex networks in social, biological, and technological systems. In this paper, we present a complex networks perspective on interfirm organizational networks by mapping, analyzing and modeling the spatial structure of a large interfirm competition network across a variety of sectors and industries within the United States. We propose two micro-dynamic models that are able to reproduce empirically observed characteristics of competition networks as a natural outcome of a minimal set of general mechanisms governing the formation of competition networks. Both models, which utilize different approaches yet apply common principles to network formation give comparable results. There is an asymmetry between companies that are considered competitors, and companies that consider others as their competitors. All companies only consider a small number of other companies as competitors; however, there are a few companies that are considered as competitors by many others. Geographically, the density of corporate headquarters strongly correlates with local population density, and the probability two firms are competitors declines with geographic distance. We construct these properties by growing a corporate network with competitive links using random incorporations modulated by population density and geographic distance. Our new analysis, methodology and empirical results are relevant to various phenomena of social and market behavior, and have implications to research fields such as economic geography, economic sociology, and regional economic development.
Von Neumann was not a Quantum Bayesian Stacey, Blake C.
Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences,
05/2016, Volume:
374, Issue:
2068
Journal Article
Peer reviewed
Wikipedia has claimed for over 3 years now that John von Neumann was the 'first quantum Bayesian'. In context, this reads as stating that von Neumann inaugurated QBism, the approach to quantum theory ...promoted by Fuchs, Mermin and Schack. This essay explores how such a claim is, historically speaking, unsupported.
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