[转载]Quantum Logic and Probability Theory

Origin: https://plato.stanford.edu/entries/qt-quantlog/

Quantum Logic and Probability Theory

First published Mon Feb 4, 2002; substantive revision Tue Aug 10, 2021

Mathematically, quantum mechanics can be regarded as a non-classical probability calculus resting upon a non-classical propositional logic. More specifically, in quantum mechanics each probability-bearing proposition of the form “the value of physical quantity A lies in the range B” is represented by a projection operator on a Hilbert space H. These form a non-Boolean—in particular, non-distributive—orthocomplemented lattice. Quantum-mechanical states correspond exactly to probability measures (suitably defined) on this lattice.

What are we to make of this? Some have argued that the empirical success of quantum mechanics calls for a revolution in logic itself. This view is associated with the demand for a realistic interpretation of quantum mechanics, i.e., one not grounded in any primitive notion of measurement. Against this, there is a long tradition of interpreting quantum mechanics operationally, that is, as being precisely a theory of measurement. On this latter view, it is not surprising that a “logic” of measurement-outcomes, in a setting where not all measurements are compatible, should prove not to be Boolean. Rather, the mystery is why it should have the particular non-Boolean structure that it does in quantum mechanics. A substantial literature has grown up around the programme of giving some independent motivation for this structure—ideally, by deriving it from more primitive and plausible axioms governing a generalized probability theory.

1. Quantum Mechanics as a Probability Calculus
   
   1.1 Quantum Probability in a Nutshell
   
   1.2 The “Logic” of Projections
   
   1.3 Probability Measures and Gleason’s Theorem
   
   1.4 The Reconstruction of QM
2. Interpretations of Quantum Logic
   
   2.1 Realist Quantum Logic
   
   2.2 Operational Quantum Logic
3. Generalized Probability Theory
   
   3.1 Discrete Classical Probability Theory
   
   3.2 Test Spaces
   
   3.3 Kolmogorovian Probability Theory
   
   3.4 Quantum Probability Theory
4. Logics associated with probabilistic models
   
   4.1 Operational Logics
   
   4.2 Orthocoherence
   
   4.3 Lattices of Properties
5. Piron’s Theorem
   
   5.1 Conditioning and the Covering Law
6. Classical Representations
   
   6.1 Classical Embeddings
   
   6.2 Contextual Hidden Variables
7. Composite Systems
   
   7.1 The Foulis-Randall Example
   
   7.2 Aerts’ Theorem
   
   7.3 Ramifications
8. Effect Algebras
   
   8.1 Quantum effects and Naimark’s Theorem
   
   8.2 Sequential Effect Algebras
   
   Bibliography
   
   Academic Tools
   
   Other Internet Resources
   
   Related Entries
9. Quantum Mechanics as a Probability Calculus
   
   It is uncontroversial (though remarkable) that the formal apparatus of quantum mechanics reduces neatly to a generalization of classical probability in which the role played by a Boolean algebra of events in the latter is taken over by the “quantum logic” of projection operators on a Hilbert space.[1] Moreover, the usual statistical interpretation of quantum mechanics asks us to take this generalized quantum probability theory quite literally—that is, not as merely a formal analogue of its classical counterpart, but as a genuine doctrine of chances. In this section, I survey this quantum probability theory and its supporting quantum logic.[2]

[For further background on Hilbert spaces, see the entry on quantum mechanics. For further background on ordered sets and lattices, see the supplementary document: The Basic Theory of Ordering Relations. Concepts and results explained these supplements will be used freely in what follows.]

1.1 Quantum Probability in a Nutshell

The quantum-probabilistic formalism, as developed by von Neumann [1932], assumes that each physical system is associated with a (separable) Hilbert space H, the unit vectors of which correspond to possible physical states of the system. Each “observable” real-valued random quantity is represented by a self-adjoint operator A on H, the spectrum of which is the set of possible values of A. If u is a unit vector in the domain of A, representing a state, then the expected value of the observable represented by A in this state is given by the inner product