An operational approach to quantum probability

In order to provide a mathmatical framework for the process of making repeated measurements on continuous observables in a statistical system we make a mathematical definition of an instrument, a concept which generalises that of an observable and that of an operation. It is then possible to develop such notions as joint and conditional probabilities without any of the commutation conditions needed in the approach via observables. One of the crucial notions is that of repeatability which we show is implicitly assumed in most of the axiomatic treatments of quantum mechanics, but whose abandonment leads to a much more flexible approach to measurement theory.At present on leave from the University of Oxford, Research supported by N.S.F. grant GP-7952X and A.F.O.S.R. contract no. F 44620-67-C-0029.At present on leave from Brasenose College, Oxford. Research supported by A.F.O.S.R. grant AF-AFOFR-69-1712.     

Quantum probability and operational statistics

We develop the concept of quantum probability based on ideas of R. Feynman. The general guidelines of quantum probability are translated into rigorous mathematical definitions. We then compare the resulting framework with that of operational statistics. We discuss various relationship between measurements and define quantum stochastic processes. It is shown that quantum probability includes both conventional probability theory and traditional quantum mechanics. Discrete quantum systems, transition amplitudes, and discrete Feynman amplitudes are treated. We close with some examples that illustrate previously defined concepts.     

Propensity,probability, and quantum physics

Popper's idea of propensities constituting the physical background of predictable probabilities is reviewed and developed by introducing a suitable formalism compatible with standard probability calculus and with its frequency interpretation. Quantum statistical ensembles described as pure cases (eigenstates) are shown to be necessarily not homogeneous if propensities are actually at work in nature. An extension of the theory to EPR experiments with local propensities leads to a new and more general proof of Bell's theorem. No joint probabilities for incompatible observables need to be introduced.     

An operational approach for testing the postulate of measurement in quantum theory

We interpret the (formal) postulates of measurement in quantum theory in terms of measurement procedures that can be done in the laboratory (at least in principle).     

Entropy change and information gain in photon counting measurement

Entropy change in a quantum state of cavity photons is investigated in the continuous measurement of photon number that is characterized by the quantum Markov process. It is shown that the average value of the entropy change in the quantum state of cavity photons is equal to the Shannon mutual information obtained from the outcome exhibited by the photodetector. Furthermore it is found that the entropy change reflects the photon statistics of the quantum state of cavity photons.     

Trajectories and causal phase-space approach to relativistic quantum mechanics

We analyze phase-space approaches to relativistic quantum mechanics from the viewpoint of the causal interpretation. In particular, we discuss the canonical phase space associated with stochastic quantization, its relation to Hilbert space, and the Wigner-Moyal formalism. We then consider the nature of Feynman paths, and the problem of nonlocality, and conclude that a perfectly consistent relativistically covariant interpretation of quantum mechanics which retains the notion of particle trajectory is possible.     

Photon entanglement in the phase-space Q representation of quantum optics

Several examples of photon entanglement are studied in the Q representation of quantum optics. In particular, the entangled states produced in parametric downconversion are studied in detail, and we determine the conditions for the violation of Bell's inequality. Our approach shows that photon entanglement is related to the existence of correlations between the quantum fluctuations of the electromagnetic field associated to different modes. Received 10 August 2002 / Received in final form 7 November 2002 Published online 4 February 2003     

Propensity and probability in quantum mechanics

We give in this letter an operational definition of quantum mechanical propensity based on a realistic measuring process. We show that such a propensity can be defined by a state to be measured, a filter or a reference state and a group of possible “motions” of such filter. Such a propensity exhibits the tendency of the measured state to take up certain states of the filter accessible by its “motion”. We give two examples of such propensities for space-momentum and spin-orientation measurements.     

Properties and operational propositions in quantum mechanics

In orthodox quantum mechanics, it has virtually become the custom to identify properties of a physical system with operationally testable propositions about the system. The causes and consequences of this practice are explored mathematically in this paper. Among other things, it is found that such an identification imposes severe constraints on the admissible states of the physical system.     

Combined systems in quantum probability

A framework for quantum probability is developed and combinations of systems are studied within this framework. In particular, we consider the horizontal sum, the direct sum, and the Cartesian product of quantum probability systems. The relations between these combinations and the concepts of interference and independence of measurements are derived. We also consider the amplitude superselection structure of these combinations.     

A precaution needed in using the phase-space formulation of quantum mechanics

We explain the origin of the apparent discrepancy recently reported between results obtained by the phase-space formulation of quantum mechanics and conventional (Schrödinger) quantum mechanics. We show how to arrive at a complete agreement.     

Free probability and quantum electrodynamics

The stochastic limit of a free particle coupled to the quantum electromagnetic field without dipole approximation leads to many new features such as: interacting Fock space, Hilbert module commutation relations, disappearance of the crossing diagrams, etc. In the present paper we begin to study how the situation is modified if a free particle is replaced by a particle in a potential which is the Fourier transform of a bounded measure.We prove that the stochastic limit procedure converges and that the overall picture is similar to the free case with the important difference that the structure of the limit Hilbert module is strongly dependent on the wave operator of the particle.     

Consistent histories and operational quantum theory

In this work a generalization of the consistent histories approach to quantum mechanics is presented. We first critically review the consistent histories approach to nonrelativistic quantum mechanics in a mathematically rigorous way and give some general comments about it. We investigate to what extent the consistent histories scheme is compatible with the results of the operational formulation of quantum mechanics. According to the operational approach, nonrelativistic quantum mechanics is most generally formulated in terms of effects, states, and operations. We formulate a generalized consistent histories theory using the concepts and the terminology which have proven useful in the operational formulation of quantum mechanics. The logical rule of the logical interpretation of quantum mechanics is generalized to the present context. The algebraic structure of the generalized theory is studied in detail.     

Quantum logic and operational quantum mechanics

We characterize the class of the μ-complete F-spaces with unit corresponding to the observables of a quantum logic. We show that, conversely, every μ-complete F-space satisfying Axiom I and Axiom II corresponds to a quantum logic. The latter class of F-spaces generalizes that of “spectral F-spaces” introduced by Alfsen and Shultz and by Edwards.     

Convex structures and operational quantum mechanics

A general mathematical framework called a convex structure is introduced. This framework generalizes the usual concept of a convex set in a real linear space. A metric is constructed on a convex structure and it is shown that mappings which preserve the structure are contractions. Convex structures which are isomomorphic to convex sets are characterized and for such convex structures it is shown that the metric is induced by a norm and that structure preserving mappings can be extended to bounded linear operators.Convex structures are shown to give an axiomatization of the states of a physical system and the metric is physically motivated. We demonstrate how convex structures give a generalizing and unifying formalism for convex set and operational methods in axiomatic quantum mechanics.