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AIP conference proceedings

The role of identity and entanglement in quantum mechanics

G. C. Ghirardi

Published Jan 1, 2000• 1 citation
Publisher

Abstract

We review the main aspects of entanglement and we prove some general theorems concerning entangled states. In order to satisfy the symmetry requirements for systems containing identical constituents one is unavoidably led to consider entangled states. One could then naively conclude that the identity gives rise, by itself, to peculiar and problematic aspects of physical systems. This is by no means the case: the symmetrization or antisymmetrization of factorized states is an absolutely benign procedure and does not pose any specific conceptual problem apart from the one of the impossibility of attributing their elemetary constituents to one or another of two far-away systems. We briefly reconsider the same problem within the context of recently proposed models of dynamical reduction based on nonlinear and stochastic modifications of the standard evolution.

DOI
10.1063/1.1337709

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We review the main aspects of entanglement and we prove some general theorems concerning entangled states. In order to satisfy the symmetry requirements for systems containing identical constituents one is unavoidably led to consider entangled states. One could then naively conclude that the identity gives rise, by itself, to peculiar and problematic aspects of physical systems. This is by no means the case: the symmetrization or antisymmetrization of factorized states is an absolutely benign procedure and does not pose any specific conceptual problem apart from the one of the impossibility of attributing their elemetary constituents to one or another of two far-away systems. We briefly reconsider the same problem within the context of recently proposed models of dynamical reduction based on nonlinear and stochastic modifications of the standard evolution.
We argue that in the case of identical particles the most natural identification of separability, that is of absence of non-classical correlations, is via the factorization of mean values of commuting observables. It thus follows that separability and entanglement depend both on the state and on the choice of observables and are not absolute notions. We compare this point of view with a recent novel approach to the entanglement of identical particles, which allows for the definition of an entanglement entropy from a suitably defined reduced particle density matrix, without the need of labelling the system constituents. We contrast this figure of merit with the aforementioned lack of an absolute notion of entanglement by considering few paradigmatic examples.
Abstract Various topics concerning the entanglement of composite quantum systems are considered with particular emphasis concerning the strict relations of such a problem with the one of attributing objective properties to the constituents. In particular we will focus our attention to composite quantum systems composed of identical constituents, with the purpose of dealing with subtle issues, which have never been adequately discussed in the literature, originating from the true indistinguishability of the subsystems involved.
While entanglement in the case of distinguishable particles is clearly understood, in the case of indistinguishable particles, there is still a wide debate regarding basic conceptual issues. Here, I will address two such issues. First, the debate almost always rests on a differentiated treatment of the cases of distinguishability and indistinguishability, even with respect to the definition of the concept of entanglement. Second, the fact that symmetrized and antisymmetrized states of indistinguishable particles are non-factorizable leads to certain perplexities about the very nature of entanglement. In the present work, both issues will be addressed from the perspective of an ontology of properties, which finds its natural expression in the algebraic formalism of quantum mechanics. The final goal will be to show that the proposed ontology allows a unified treatment of entanglement in the distinguishability and indistinguishability cases and offers a straightforward way out to perplexities. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.
The report presents an exhaustive review of the recent attempt to overcome the difficulties that standard quantum mechanics meets in accounting for the measurement (or macro-objectification) problem, an attempt based on the consideration of nonlinear and stochastic modifications of the Schroedinger equation. The proposed new dynamics is characterized by the feature of not contradicting any known fact about microsystems and of accounting, on the basis of a unique, universal dynamical principle, for wavepacket reduction and for the classical behavior of macroscopic systems. We recall the motivations for the new approach and we briefly review the other proposals to circumvent the above mentioned difficulties which appeared in the literature. In this way we make clear the conceptual and historical context characterizing the new approach. After having reviewed the mathematical techniques (stochastic differential calculus) which are essential for the rigorous and precise formulation of the new dynamics, we discuss in great detail its implications and we stress its relevant conceptual achievements. The new proposal requires also to work out an appropriate interpretation; a procedure which leads us to a reconsideration of many important issues about the conceptual status of theories based on a genuinely Hilbert space description of natural processes. We also discuss the possibility and the problems one meets in trying to develop an analogous formalism for the relativistic case. Finally we discuss the experimental implications of the new dynamics for various physical processes which should allow, in principle, to test it against quantum mechanics.
The impossibility of ascribing definite states to the constituents of an entangled system restricts the scope of Pauli's principle in this context. We analyze the conceptual and physical aspects of the problem by studying the actual scope of the principle and the possibility of extending the concept of exclusion in correlated systems. When the entanglement is weak, as in the archetypical case of the Helium atom, the principle can be applied in an approximated way. The concept of non-complete set of properties plays a crucial role in the argument, which also clarifies the physical meaning of the atomic quantum numbers; they are a multi-particle property, not an one-electron one. In contrast, for strong entanglement the excluded states are independent of the principle. We describe some of these states, many times determined by symmetries of the multi-fermion state.