The Classification of Decoherence Functionals: An Analogue of Gleason's Theorem
arXiv:gr-qc/9406015 · doi:10.1063/1.530679
Abstract
Gell-Mann and Hartle have proposed a significant generalisation of quantum theory with a scheme whose basic ingredients are `histories' and decoherence functionals. Within this scheme it is natural to identify the space $\UP$ of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where $\UP$ is the lattice of projectors $\PV$ in some Hilbert space $\V$; in effect we seek the history analogue of Gleason's famous theorem in standard quantum theory. In the present paper we present the solution to this problem for the case where $\V$ is finite-dimensional. In particular, we show that every decoherence functional $d(\a,\b)$, $\a,\b\in\PV$ can be written in the form $d(\a,\b)=\tr_{\V\otimes\V}(\a\otimes\b X)$ for some operator on the tensor product space $\V\otimes\V$.
18 Pages, Imperial/TP/93--94/42, DAMTP 94--44
References in corpus (4)
Cited by in corpus (46)
- Consistent Histories and Quantum Reasoning
- Continuous Histories and the History Group in Generalised Quantum Theory
- Structural Issues in Quantum Gravity
- Observable measure of bipartite quantum correlations
- Topos Theory and Consistent Histories: The Internal Logic of the Set of all Consistent Sets
- Comparing Formulations of Generalized Quantum Mechanics for Reparametrization-Invariant Systems
- The action operator for continuous-time histories
- Continuous Time and Consistent Histories
- Decoherent Histories Approach to the Arrival Time Problem
- Consistent probabilities in loop quantum cosmology
- Poincare' invariance for continuous-time histories
- Continuous-time histories: observables, probabilities, phase space structure and the classical limit
- Spacetime Quantum Actions
- A Bell Inequality Analog in Quantum Measure Theory
- Diffeomorphisms as Symplectomorphisms in History Phase Space: Bosonic String Model
- Symmetries of Decoherence Functionals
- Quantum Measure Theory: A New Interpretation
- Symmetry and History Quantum Theory: An analogue of Wigner's Theorem
- On the Consistent Effect Histories Approach to Quantum Mechanics
- Entropy of Classical Histories
- Quantum Fields in Nonstatic background: A Histories Perspective
- The consistent histories approach to loop quantum cosmology
- Homogeneous decoherence functionals in standard and history quantum mechanics
- Completeness of Decoherence Functionals
- Spectral Geometry and Causality
- A Bayesian Account of Quantum Histories
- Theoretical insights on measuring quantum correlations
- Path Integrals from Spacetime Quantum Actions
- A Bayesian Analogue of Gleason's Theorem
- A model of quantum reduction with decoherence
- On Tracial Operator Representations of Quantum Decoherence Functionals
- Compatible Quantum Theory
- Weak Values and Relational Generalisations
- Pedestrian notes on quantum mechanics
- On the connection between the theorems of Gleason and of Kochen and Specker
- Bayesian Probabilities and the Histories Algebra
- Affine Histories in Quantum Gravity: Introduction and the Representation for a Cosmological Model
- Maximum Information and Quantum Prediction Algorithms
- Local quantum information dynamics
- A Consistent Histories approach to the Unruh effect
- Temporal Quantum Theory
- All non-Gaussian states are advantageous for channel discrimination: Robustness of non-convex continuous variable quantum resources
- Spacetime quantum mechanics for bosonic and fermionic systems
- Phase Space Representations and Perturbation Theory for Continuous-time Histories
- BRST Quantisation of Histories Electrodynamics
- Symmetries and conservation laws in histories-based theories