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Journal of Physics A Mathematical and General

Observables of asymptotically vanishing correlations, states at infinity and quantum separability

Kay-Kong Wan, R. G. McLean

Published Mar 11, 1984• 10 citations
Publisher

Abstract

The asymptotic behaviour of quantum mechanical states at large times has been discussed in two recent papers by Wan and McLean (1983). In a third paper a detailed study of the corresponding behaviour of observables was made. The mathematical properties of asymptotic operators obtained are applied in this paper to establish an algebraic formulation of quantum mechanics which has the characteristic of being asymptotically separable. The age-old non-locality problem in quantum mechanics can then be effectively tackled.

DOI
10.1088/0305-4470/17/4/024

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The asymptotic behaviour of quantum mechanical states at large times has been discussed in two recent papers by Wan and McLean (1983). In a third paper a detailed study of the corresponding behaviour of observables was made. The mathematical properties of asymptotic operators obtained are applied in this paper to establish an algebraic formulation of quantum mechanics which has the characteristic of being asymptotically separable. The age-old non-locality problem in quantum mechanics can then be effectively tackled.
This thesis explores the possibility of an algebraic formulation of non-relativistic quantum theory in which certain paradoxes associated with non-locality may be resolved. It is shown that the localisation of a free quantum mechanical wave function at large time coincides approximately with the localisation of an ensemble of classical particles having the same momentum range. This result is used to give a formal definition of spatially separating states and spatially separating particles. We then study certain C*-algebras on which expectation values converge in an infinite time limit. By considering such algebras which contain local observables it is possible to introduce states at infinity as limits of states described by wave functions. In such a state at infinity there is zero probability of a position measurement finding the system in any bounded region in configuration space. It is shown that a C*-algebra exists on which any coherent superposition of spatially separating states will converge in an infinite time limit to a mixture of disjoint states. This allows us to obtain an asymptotic resolution of de Broglie's paradox and the Einstein, Podolsy and Rosen paradox. These results are obtained for the simplest types of quantum systems i.e. a one particle system without spin having configuration space IRⁿ and a system consisting of two such particles which may be distinguished from each other.
This thesis investigates the description of localised states and observables in quantum mechanics.Motivated by the fact that actual experimental apparatus is of only finite size, a method of localising any observable ("selfadjoint operator on Hilbert space) is introduced, which is applicable to both bounded and unbounded operators.This method makes use of objects called localising isometries, which also prove useful in modifying the evolution of non-relativistic particles, so that they remain well localised at all times: this modification does not jeopardise the well tested predictions of standard quantum mechanics.It is mathematically more convenient when describing systems of well separated identical particles.Pairs of correlated particles, such as those considered in EPR-type thought experiments, undergo chronological disordering at large times, and the analysis of this phenomenon is used to argue physically in favour of asymptotic algebras, which do not contain two-particle observables at infinite times.The second part of the thesis investigates the description of the position of a spinless relativistic particle.Existing Lorentz covariant currents are non-local, i.e. they describe particles which are not strictly localised in a bounded region of space, and these currents are modified so as to describe strictly localised particles, which evolve causally.vi
An algebraic approach to quantum non-separability is applied to the case of two qubits. It is based on the partition of the algebra of observables into independent subalgebras and the tensor product structure of the Hilbert space is not exploited. Even in this simple case, such a general formulation has some advantages. Using algebraic formalism, we can explicitly show the relativity of the notion of entanglement to the observables measured in the system and characterize separable and non-separable pure states. As a universal measure of non-separability of pure states, we propose to take the so-called total correlation. This quantity depends on the state as well as on the algebraic partition. Its numerical value is given by the norm of the corresponding correlation matrix.
The purpose of this paper is threefold: to provide a mathematically rigorous formulation of the quantum-mechanical scattering problem from the time-independent point of view as has been done by Jauch from the time-dependent point of view, to establish a union between the two formulations, and to investigate the necessity of the asymptotic condition which occurs as a postulate in the time-dependent formulation. The formulation of the problem depends only on the ``total'' and ``free'' Hamiltonian operators. Under the conditions necessary for the time-dependent formulation, the wave operators defined by the asymptotic limits provide a unique solution of this problem. The possibility that solutions can exist when the asymptotic conditions are not valid is investigated by defining wave operators by an integral representation. The conditions sufficient for these to provide a unique solution are shown to be possibly weaker than the asymptotic conditions; there may be a class of Hamiltonian operators for which such solutions exist but for which the asymptotic limits do not. An explicit characterization of such a set of Hamiltonian operators is not achieved, but this question of the necessity of the asymptotic condition has been reduced to a specific mathematical problem. It is hoped that this paper will find a reader who is able to carry the mathematical investigation further.