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arXiv:1902.08853

A Characterization For Entangled Vectors

Stanley Gudder

Published Feb 23, 2019• 2 citations
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Abstract

This article presents a simple characterization for entangled vectors in a finite dimensional Hilbert space $H$. The characterization is in terms of the coefficients of an expansion of the vector relative to an orthonormal basis for $H$. This simple necessary and sufficient condition contains a restriction and we also present a more complicated condition that is completely general. Although these characterizations apply to bipartite systems, we show that generalizations to multipartite systems are also valid.

DOI
10.48550/arxiv.1902.08853

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This article presents a simple characterization for entangled vectors in a finite dimensional Hilbert space HHH. The characterization is in terms of the coefficients of an expansion of the vector relative to an orthonormal basis for HHH. This simple necessary and sufficient condition contains a restriction and we also present a more complicated condition that is completely general. Although these characterizations apply to bipartite systems, we show that generalizations to multipartite systems are also valid.
We introduce a simple sufficient criterion allowing to tell whether a subspace of a bipartite or multipartite Hilbert space is entangled. The main ingredient of our criterion is a bound on the minimal entanglement of a subspace in terms of entanglement of vectors spanning that subspace expressed for geometrical measures of entanglement. The criterion is applicable to both completely and genuinely entangled subspaces. We explore its applicability in several important scenarios. Further, an entanglement criterion for mixed states following directly from the condition is stated. As an auxiliary result we provide a formula for the generalized geometric measure of entanglement of the ddd--level Dicke states.
An entangled basis with a fixed Schmidt number k (EBk) is a set of orthonormal basis states with the same Schmidt number k in a product Hilbert space . It is a generalization of both the product basis and the maximally entangled basis. We show here that for any , EBk exists in for any d and d '. Consequently, general methods of constructing SEBk (EBk with the same Schmidt coefficients) and EBk (but not SEBk) are proposed. Moreover, we extend the concept of EBk to multipartite cases and find out that the multipartite EBk can be constructed similarly.
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.
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Abstract Entanglement is a defining feature of quantum mechanics, producing correlations between the results of spacelike-separated measurements which cannot be explained by any local, single-world theory of nature. In this paper we employ a rigorous mathematical approach to explore the classification of bases for quantum systems by their entanglement properties. For bipartite systems, a basis is said to have type if it contains p entangled states and q separable states. Our main result is to prove a long-standing conjecture of Eakins and Jaroszkiewicz stating that no type orthonormal (o.n.) basis exists: an o.n. basis containing an entangled state is guaranteed to contain at least one other. Furthermore, we prove that o.n. bases of all other types do exist, before moving on to discuss the multipartite case. Our results show that there are fundamental constraints on the emergence of nonlocality in general orthogonal measurement.