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arXiv:math-ph/0604071 · Reviews in Mathematical Physics

ENTANGLEMENT, HAAG-DUALITY AND TYPE PROPERTIES OF INFINITE QUANTUM SPIN CHAINS

Michael Keyl, Taku Matsui, Dirk-M. Schlingemann, Reinhard F. Werner

Published Oct 1, 2006• 36 citations
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Abstract

We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φS provides a particular example for this type of entanglement.

DOI
10.1142/s0129055x0600284x

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We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φS provides a particular example for this type of entanglement.
Abstract We develop the theory of local operations and classical communication (LOCC) for bipartite quantum systems represented by commuting von Neumann algebras. Our central result is the extension of Nielsen’s Theorem, stating that the LOCC ordering of bipartite pure states is equivalent to the majorization of their restrictions to arbitrary factors. As a consequence, we find that in bipartite system modeled by commuting factors in Haag duality, (a) all states have infinite single-shot entanglement if and only if the local factors are not of type I, (b) type III factors are characterized by LOCC transitions of arbitrary precision between any two pure states, and c) the latter holds even without classical communication for type III1\hbox {III}_{1}III1​ III 1 factors. In the case of semifinite factors, the usual construction of pure state entanglement monotones carries over. Together with recent work on embezzlement of entanglement, this gives a one-to-one correspondence between the classification of factors into types and subtypes and operational entanglement properties. In the appendix, we provide a self-contained treatment of majorization on semifinite von Neumann algebras and σ\sigma σ σ -finite measure spaces.
In quantum systems with infinitely many degrees of freedom, states can be infinitely entangled across a pair of subsystems. But are there different forms of infinite entanglement? To answer this, we provide a rigorous framework for studying information-theoretic properties in infinite systems, formulated in terms of von Neumann algebras. In this framework, we show that there are operational tasks that can distinguish between different forms of infinite entanglement. We find that the type classification of von Neumann algebras (types I, II, III, and their respective subtypes) is in one-to-one correspondence with operational entanglement properties. For instance, the Connes classification of type III factors corresponds to the system’s capability at the operational task of “embezzling” entanglement. Our findings promote the type classification from algebraic bookkeeping to a classification of infinite quantum systems based on their operational entanglement properties.
We continue the analysis of nontrivial examples of quantum Markov processes. This is done by applying the construction of entangled Markov chains obtained from classical Markov chains with infinite state–space. The formula giving the joint correlations arises from the corresponding classical formula by replacing the usual matrix multiplication by the Schur multiplication. In this way, we provide nontrivial examples of entangled Markov chains on , F being any infinite dimensional type I factor, J a finite interval of , and the bar the von Neumann tensor product between von Neumann algebras. We then have new nontrivial examples of quantum random walks which could play a rôle in quantum information theory. In view of applications to quantum statistical mechanics too, we see that the ergodic type of an entangled Markov chain is completely determined by the corresponding ergodic type of the underlying classical chain, provided that the latter admits an invariant probability distribution. This result parallels the corresponding one relative to the finite dimensional case. Finally, starting from random walks on discrete ICC groups, we exhibit examples of quantum Markov processes based on type II 1 von Neumann factors.
We continue the analysis of nontrivial examples of quantum Markov processes. This is done by applying the construction of entangled Markov chains obtained from classical Markov chains with infinite state–space. The formula giving the joint correlations arises from the corresponding classical formula by replacing the usual matrix multiplication by the Schur multiplication. In this way, we provide nontrivial examples of entangled Markov chains on , F being any infinite dimensional type I factor, J a finite interval of , and the bar the von Neumann tensor product between von Neumann algebras. We then have new nontrivial examples of quantum random walks which could play a rôle in quantum information theory. In view of applications to quantum statistical mechanics too, we see that the ergodic type of an entangled Markov chain is completely determined by the corresponding ergodic type of the underlying classical chain, provided that the latter admits an invariant probability distribution. This result parallels the corresponding one relative to the finite dimensional case. Finally, starting from random walks on discrete ICC groups, we exhibit examples of quantum Markov processes based on type II 1 von Neumann factors.