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Conditional Probability, Three-Slit Experiments, and the Jordan Algebra Structure of Quantum Mechanics

Gerd Niestegge

Most quantum logics do not allow for a reasonable calculus of conditional probability. However, those ones which do so provide a very general and rich mathematical structure, including classical probabilities, quantum mechanics, and Jordan algebras. This structure exhibits some similarities with Alfsen and Shultz's noncommutative spectral theory, but these two mathematical approaches are not identical. Barnum, Emerson, and Ududec adapted the concept of higher-order interference, introduced by Sorkin in 1994, into a general probabilistic framework. Their adaption is used here to reveal a close link between the existence of the Jordan product and the nonexistence of interference of third or higher order in those quantum logics which entail a reasonable calculus of conditional probability. The complete characterization of the Jordan algebraic structure requires the following three further postulates: a Hahn-Jordan decomposition property for the states, a polynomial functional calculus for the observables, and the positivity of the square of an observable. While classical probabilities are characterized by the absence of any kind of interference, the absence of interference of third (and higher) order thus characterizes a probability calculus which comes close to quantum mechanics but still includes the exceptional Jordan algebras.Most quantum logics do not allow for a reasonable calculus of conditional probability. However, those ones which do so provide a very general and rich mathematical structure, including classical probabilities, quantum mechanics, and Jordan algebras. This structure exhibits some similarities with Alfsen and Shultz's noncommutative spectral theory, but these two mathematical approaches are not identical. Barnum, Emerson, and Ududec adapted the concept of higher-order interference, introduced by Sorkin in 1994, into a general probabilistic framework. Their adaption is used here to reveal a close link between the existence of the Jordan product and the nonexistence of interference of third or higher order in those quantum logics which entail a reasonable calculus of conditional probability. The complete characterization of the Jordan algebraic structure requires the following three further postulates: a Hahn-Jordan decomposition property for the states, a polynomial functional calculus for the observables, and the positivity of the square of an observable. While classical probabilities are characterized by the absence of any kind of interference, the absence of interference of third (and higher) order thus characterizes a probability calculus which comes close to quantum mechanics but still includes the exceptional Jordan algebras.

arXiv:0912.0203Advances in Mathematical PhysicsJan 1, 201224 citations

A Logic for Probability in Quantum Systems

Ron van der Meyden, Manas K. Patra

Lecture notes in computer scienceJan 1, 200318 citations

The Qutrit Anthropics Problem: A Rigorous Theoretical Framework for SU(3) Observer-Selection Effects

Trent Slade

Classical anthropic reasoning relies on a binary observer model that treats existence as a yes/no variable, generating persistent paradoxes in self-locating uncertainty, indexical probability, and observer-selection effects. This work introduces Qutrit Anthropics, a triadic framework that models the observer as an SU(3)-symmetric quantum-geometric entity embedded in the complex projective plane CP². Using the Fubini–Study metric, symplectic volume, and the algebraic structure of SU(3), the paper resolves classical paradoxes such as Sleeping Beauty, the Doomsday Argument, observer-moment duplication, and SIA inflation. The framework demonstrates that anthropic paradoxes arise from forcing binary logic onto a system whose natural geometry is triadic, curved, and compact. Qutrit Anthropics reframes observerhood as a bundle section over CP², establishing a unified quantum-geometric basis for probability, existence, and indexical reasoning.Classical anthropic reasoning relies on a binary observer model that treats existence as a yes/no variable, generating persistent paradoxes in self-locating uncertainty, indexical probability, and observer-selection effects. This work introduces Qutrit Anthropics, a triadic framework that models the observer as an SU(3)-symmetric quantum-geometric entity embedded in the complex projective plane CP². Using the Fubini–Study metric, symplectic volume, and the algebraic structure of SU(3), the paper resolves classical paradoxes such as Sleeping Beauty, the Doomsday Argument, observer-moment duplication, and SIA inflation. The framework demonstrates that anthropic paradoxes arise from forcing binary logic onto a system whose natural geometry is triadic, curved, and compact. Qutrit Anthropics reframes observerhood as a bundle section over CP², establishing a unified quantum-geometric basis for probability, existence, and indexical reasoning.

Dec 9, 20250 citations

Quantum Logic and Quantum Probability

Enrico G. Beltrametti

Springer eBooksJan 1, 20044 citations

Dynamic Logic for Quantum Probability

Tomoaki Kawano

Lecture notes in computer scienceSep 23, 20250 citations

Reasoning About Imperative Quantum Programs

Rohit Chadha, Paulo Mateus, Amı́lcar Sernadas

A logic for reasoning about states of basic quantum imperative programs is presented. The models of the logic are ensembles obtained by attaching probabilities to pairs of quantum states and classical states. The state logic is used to provide a sound Hoare-style calculus for quantum imperative programs. The calculus is illustrated by proving the correctness of the Deutsch algorithm.

Electronic Notes in Theoretical Computer ScienceMay 1, 200670 citations

Probability Logics for Reasoning About Quantum Observations

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