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Communications in Mathematical Physics

Entanglement and Non-Separability of CAR Systems

Hajime Moriya

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

We study quantum entanglement of bipartite CAR systems. We show that triangle inequality of von Neumann entropy does not hold in these systems. Meanwhile, we show non-separability in the sense of Florig-Summers for our CAR systems. Entanglement degrees by Narnhofer are generalized to an arbitrary, that is, not-necessarily commuting pair of general C ∗-systems. These new entanglement degrees are investigated with some explicit calculations for certain pairs of our CAR systems. We introduce a notion named half-sided entanglement via our generalized entanglement degree. By using this notion, we describe those certain notable features of quantum entanglement which originate from CAR. I

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We study quantum entanglement of bipartite CAR systems. We show that triangle inequality of von Neumann entropy does not hold in these systems. Meanwhile, we show non-separability in the sense of Florig-Summers for our CAR systems. Entanglement degrees by Narnhofer are generalized to an arbitrary, that is, not-necessarily commuting pair of general C ∗-systems. These new entanglement degrees are investigated with some explicit calculations for certain pairs of our CAR systems. We introduce a notion named half-sided entanglement via our generalized entanglement degree. By using this notion, we describe those certain notable features of quantum entanglement which originate from CAR. I
A notion of entangled Markov chain was introduced by Accardi and Fidaleo in the context of quantum random walk. They proved that, in the finite-dimensional case, the corresponding states have vanishing entropy density, but they did not prove that they are entangled. In this note this entropy result is extended to the infinite-dimensional case under the assumption of finite speed of hopping. Then the entanglement problem is discussed for spin-1/2, entangled Markov chains generated by a binary symmetric channel with hopping probability 1-q. The von Neumann entropy of these states, restricted on a sublattice is explicitly calculated and shown to be independent of the size of the sublattice. This is a new, purely quantum, phenomenon. Finally the entanglement property between the sublattices [Formula: see text] and [Formula: see text] is investigated using the PPT criterium. It turns out that, for q≠ 0, 1, ½ the states are non-separable, thus truly entangled, while for q = 0, 1, ½, they are separable.
Abstract This chapter describes entanglement in bipartite systems. We define entanglement for pure states introducing the concept of Schmidt decomposition, and we show that entanglement can be quantified through the von Neumann entropy. We then present dense coding and teleportation, two quantum information protocols that use entanglement as a resource. We present the notions of locality and realism and derive the Clauser-Horne-Shimony-Holt inequality, an extended version of Bell’s inequalities. We then focus on mixed states and we present different techniques to solve the separability problem
In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and if it is, then how much entanglement is carried by that? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the issue of classification of multipartite entanglement. In this dissertation, after reviewing these three fundamental topics for finite dimensional Hilbert spaces, I present my contribution to knowledge. My main result is the elaboration of the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This problem is simple for pure states, however, for mixed states it has not been considered in full detail yet. I give not only the classification but also necessary and sufficient criteria for the classes, which make it possible to determine to which class a mixed state belongs. Moreover, these criteria are given by the vanishing of quantities measuring entanglement. Apart from these, I present some side results related to the entanglement of mixed states. These results are obtained in the learning phase of my studies and give some illustrations and examples.