Born's rule as a quantum extension of Bayesian coherence

Abstract

The subjective Bayesian interpretation of probability asserts that the rules of the probability calculus follow from the normative principle of Dutch-book coherence: A decision-making agent should not assign probabilities such that a series of monetary transactions based on those probabilities would lead them to expect a sure loss. Similarly, the subjective Bayesian interpretation of quantum mechanics (QBism) asserts that the Born rule is a normative rule in analogy to Dutch-book coherence, but with the addition of one or more empirically based assumptions -- i.e., the "only a little more" that connects quantum theory to the particular characteristics of the physical world. Here we make this link explicit for a conjectured representation of the Born rule which holds true if symmetric informationally complete POVMs (or SICs) exist for every finite dimensional Hilbert space. We prove that an agent who thinks they are gambling on the outcomes of measurements on a sufficiently quantum-like system, but refuses to use this form of the Born rule when placing their bets is vulnerable to a Dutch book. The key property for being sufficiently quantum-like is that the system admits a symmetric reference measurement, but that this measurement is not sampling any hidden variables.

arXiv comment13 pages, 35 references, 3 appendices, 1 miserable year finally over and done with; v2: title change to match journal version

References in corpus (22)

• Symmetric Informationally Complete Quantum Measurements
• On the reality of the quantum state
• Quantum probabilities as Bayesian probabilities
• An Introduction to QBism with an Application to the Locality of Quantum Mechanics
• Einstein, incompleteness, and the epistemic view of quantum states
• Quantum Mechanics as Quantum Information (and only a little more)
• Quantum-Bayesian Coherence: The No-Nonsense Version
• Quantum states and generalized observables: a simple proof of Gleason's theorem
• The SIC Question: History and State of Play
• QBism and the Greeks: why a quantum state does not represent an element of physical reality
• Respecting One's Fellow: QBism's Analysis of Wigner's Friend
• Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements
• Introducing the Qplex: A Novel Arena for Quantum Theory
• Symmetric Informationally Complete Measurements Identify the Irreducible Difference between Classical and Quantum Systems
• SICs and Algebraic Number Theory
• The Lie Algebraic Significance of Symmetric Informationally Complete Measurements
• Dimension of physical systems, information processing, and thermodynamics
• Discrete Wigner Functions from Informationally Complete Quantum Measurements
• Entanglement Breaking Rank and the existence of SIC POVMs
• Quantum Theory as Symmetry Broken by Vitality
• SICs: Some explanations
• Ideas Abandoned en Route to QBism

Cited by in corpus (7)

• QBism and Relational Quantum Mechanics compared
• Uniqueness of quantum state over time function
• Ozawa's Intersubjectivity Theorem as objection to QBism individual agent perspective
• Quantum and Classical Bayesian Agents
• On Relationalist Reconstructions of Quantum Theory
• Can QBism exist without Q? Morphophoric measurements in generalised probabilistic theories
• Bayesian rational agents in iterated quantum games