Minimal quantum logic with merged implications

Kenji Tokuo

Hirokazu Nishimura

Kenji Tokuo

We study the spectral sign implication for quantum logic, defined by the nonnegative spectral projection of the operator obtained by subtracting the antecedent projection from the consequent projection. The construction agrees with classical material implication on commuting projections and compares arbitrary pairs of projections through the spectral structure of their difference. It satisfies Hardegree's four minimal implicative conditions, his law of contraposition, and a falsity condition. It differs from the standard polynomial implications in that its value need not belong to the ortholattice generated by its arguments. Within a uniform class of Borel constructions for pairs of projections, the operations satisfying entailment, contraposition, and the falsity condition correspond exactly to measurable choices of spectral branch. In the continuous subclass, these three conditions determine the spectral sign implication uniquely. In finite dimensions, the operation is also the largest among the acceptance projections of optimal projective tests for Helstrom discrimination between subspace states with priors proportional to rank. These results connect quantum implication with the operator theory of two projections and binary quantum discrimination.

Sonja Smets

Chiara Manfriani, Roberto Giuntini, Richard J. Greechie

List of Figures. List of Tables. Preface. Acknowledgements. I: Mathematical and Physical Background. Introduction. 1. The mathematical scenario of quantum theory and von Neumann's axiomatization. 1.1. Algebraic structures. 1.2. The geometry of quantum theory. 1.3. The axiomatization of orthodox QT. 1.4. The 'logic' of the quantum events. 1.5. The logico-algebraic approach to QT. 2. Abstract axiomatic foundations of sharp QT. 2.1. Mackey's minimal axiomatization of QT. 2.2. Events. 2.3. Event-state systems. 2.4. Event-state systems and preclusivity spaces. 3. Back to Hilbert space. 3.1. Events as closed subspaces. 3.2. Events as projections. 3.3. Hilbert event-state systems. 3.4. From abstract orthoposets of events to Hilbert lattices. 4. The emergence of fuzzy events in Hilbert space quantum theory. 4.1. The notion of effect. 4.2. Effect-Brouwer Zadeh posets. 4.3. MacNeille completions. 4.4. Unsharp preclusivity spaces. 5. Effect algebras and quantum MV algebras. 5.1. Effect algebras and Brouwer Zadeh effect algebras. 5.2. The Lukasiewwicz operations. 5.3. MV algebras and QMV algebras. 5.4. Quasi-linear QMV algebras and effect algebras. 6. Abstract axiomatic foundations of unsharp quantum theory. 6.1. A minimal axiomatization of unsharp QT. 6.2. The algebraic structure of abstract effects. 6.3. The sharply dominating principle. 6.4. Abstract unsharp preclusivity spaces. 6.5. Sharp and unsharp abstract quantum theory. 7. To what extent is quantum ambiguity ambiguous? 7.1. Algebraic notions of 'sharp'. 7.2. Probabilistic definitions of 'sharpness'. II: Quantum Logics as Logic. Introduction. 8. Sharp quantum logics. 8.1. Algebraic and Kripkean semantics for sharp quantum logics. 8.2. Algebraic and Kripkean realizations of Hilbert event-state systems. 8.3. The implication problem in quantum logic. 8.4. Five polynomial conditions. 8.5. The quantum logical conditional as a counterfactual conditional. 8.6. Implication-connectives. 9. Metalogical properties and anomalies of quantum logic. 9.1. The failure of the Lindenbaum properety. 9.2. A modal interpretation of sharp quantum logics. 10. An axiomatization of OL and OQL. 10.1. The calculi of OL and OQL. 10.2. The soundness and completeness theorems. 11. The metalogical intractability of orthomodularity. 11.1. Orthomodularity is not elementary. 11.2. The embeddability problem. 11.3. Hilbert quantum logic and the orthomodular law. 12. First-order quantum logics and quantum set theories. 12.1. First-order semantics. 12.2. Quantum set theories. 13. Partial classical logic, the Lindenbaum property and the hidden variable problem. 13.1. Partial classical logic. 13.2. Partial classical logic and the Lindenbaum property. 13.3. States on partial Boolean algebras. 13.4. The Lindenbaum property and the hidden variable problem. 14. Unsharp quantum logics. 14.1.

Dov M. Gabbay, Kurt Engesser, Daniel J. Lehmann