Law of Total Probability in Quantum Theory and Its Application in Wigner's Friend Scenario
arXiv:2204.12285 · doi:10.3390/e24070903
Abstract
It is well-known that the law of total probability does not hold in general in quantum theory. However, the recent arguments on some of the fundamental assumptions in quantum theory based on the extended Wigner's Friend scenario show a need to clarify how the law of total probability should be formulated in quantum theory and under what conditions it still holds. In this work, the definition of conditional probability in quantum theory is extended to POVM measurements. Rule to assign two-time conditional probability is proposed for incompatible POVM operators, which leads to a more general and precise formulation of the law of total probability. Sufficient conditions under which the law of total probability holds are identified. Applying the theory developed here to analyze several quantum no-go theorems related to the extended Wigner's friend scenario reveals logical loopholes in these no-go theorems. The loopholes exist as a consequence of taking for granted the validity of the law of total probability without verifying the sufficient conditions. Consequently, the contradictions in these no-go theorems only reconfirm the invalidity of the law of total probability in quantum theory, rather than invalidating the physical statements that the no-go theorems attempt to refute.
arXiv commentClosely match the published version. 13 pages, 1 figure
References in corpus (23)
- Decoherence, einselection, and the quantum origins of the classical
- Relational Quantum Mechanics
- How Quantum Mechanics can consistently describe the use of itself
- Quantum theory cannot consistently describe the use of itself
- Quantum states and generalized observables: a simple proof of Gleason's theorem
- A strong no-go theorem on the Wigner's friend paradox
- The Trinity of Relational Quantum Dynamics
- Relaxed Bell inequalities and Kochen-Specker theorems
- Relational EPR
- A no-go theorem for observer-independent facts
- Quantum Time
- Experimental test of local observer-independence
- Translation of Lueders' "Uber die Zustandsanderung durch den Messprozess"
- Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements
- "Space is blue and birds fly through it"
- Stable Facts, Relative Facts
- Generalized probability rules from a timeless formulation of Wigner's friend scenarios
- Physics and Metaphysics of Wigner's Friends: Even performed pre-measurements have no results
- Quantum Mechanics from Relational Properties, Part I: Basic Formulation
- The Conditional Probability Interpretation of the Hamiltonian Constraint
- Decoherence framework for Wigner's friend experiments
- Relational Quantum Mechanics and Probability
- Quantum Mechanics from Relational Properties, Part III: Path Integral Implementation