Quantification of Multi-Particle Entanglement

Jens Eisert, H. J. Briegel

Published Jul 21, 2000• 3 citations

Abstract

We present a measure of quantum entanglement which is capable of quantifying the degree of entanglement of a multi-partite quantum system. This measure, which is based on a generalization of the Schmidt rank of a pure state, is defined on the full state space and is shown to be an entanglement monotone, that is, it cannot increase under local quantum operations with classical communication and under mixing. For a large class of mixed states this measure of entanglement can be calculated exactly, and it provides a detailed classification of mixed states. We also present a connection to the best separable approximation of an entangled state of a two-qubit system.

Related papers

Schmidt measure as a tool for quantifying multiparticle entanglement

Jens Eisert, Hans J. Briegel

We present a measure of quantum entanglement which is capable of quantifying the degree of entanglement of a multipartite quantum system. This measure, which is based on a generalization of the Schmidt rank of a pure state, is defined on the full state space, and is shown to be an entanglement monotone, that is, it cannot increase under local quantum operations with classical communication and under mixing. For a large class of mixed states this measure of entanglement can be calculated exactly, and it provides a detailed classification of mixed states.

arXiv:quant-ph/0007081Physical Review AJul 9, 2001249 citations

Quantifying the dimensionality of multiparticle entanglement via partition rank

Sophia Denker, Ismaël Septembre, Robin Krebs +1 more

The usefulness of entanglement as a resource in quantum technologies increases for larger systems, that is, if more particles or higher-dimensional quantum systems are considered. Yet, the interplay between dimensionality and multiparticle entanglement is not well understood. Only for two-particle systems an unambiguous and coherent notion of entanglement dimensionality, based on the Schmidt decomposition, is known. We introduce a concept to characterize the entanglement dimensionality of multiparticle states based on decompositions of pure states into superpositions of states without genuine multiparticle entanglement. We provide constructive methods to characterize the resulting partition rank for pure and mixed states. This allows the identification of novel maximally correlated states as well as a discrete classification of quantum states under stochastic local operations and classical communication. From a mathematical perspective, our approach can be formulated in terms of the slice rank and partition rank of tensors and our results allow to characterize these by connecting them to a generalized injective tensor norm.

quant-pharXiv:2608.23399Aug 25, 20260 citations

Quantum entanglement: geometric quantification and applications to multi-partite states and quantum phase transitions

Tzu-Chieh Wei

The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement is explored for bi-partite and multi-partite pure and mixed states. It is determined analytically for arbitrary two-qubit mixed states, generalized Werner, and isotropic states, and is also applied to certain multi-partite mixed states, including two distinct multi-partite bound entangled states. Moreover, the ground-state entanglement of the XY model in a transverse field is calculated and shown to exhibit singular behavior near the quantum critical line. Along the way, connections are pointed out between the geometric measure of entanglement, the Hartree approximation, entanglement witnesses, correlation functions, and the relative entropy of entanglement.

arXiv:0905.2467May 15, 20090 citations

Classification and quantification of qubit and qutrit multipartite pure state entanglement

Guoying Lu, Pan Feng, (1)辽宁师范大学物理系，大连 116029; (2)辽宁师范大学物理系，大连 116029;辽宁大学物理系，沈阳 110036

Classification and quantification of bipartite qubit, tripartite qubit, and bipartite qutrit entanglement are studied in detail based on the entanglement measure of multipartite pure states. The results show that different ways of pure state entanglement can be characterized by extremal values of the corresponding entanglement measure under the stochastic local operation and classical communication(SLOCC). Detailed analysis shows that there are three different kinds of entanglement under the SLOCC in both tripartite qubit and bipartite qutrit pure state systems. Comparison of the measure with the recently proposed entropy-product measure is made and briefly discussed.

Acta Physica SinicaJan 1, 20071 citation

Quantifying entanglement with probabilities

M. A. Cirone

We propose a new approach to the problem of defining the degree of entanglement between two particles in a pure state with Hilbert spaces of arbitrary finite dimensions. The central idea is that entanglement gives rise to correlations between the particles that do not occur in separable states. We individuate the contributions of these correlations to the joint and the conditional probabilities of local measurements outcomes. We use these probabilities to define the measure of entanglement. Our measure turns out to be proportional to the so-called 2-entropy and therefore satisfies the properties required for any measure of entanglement. We conclude with an outlook on the problem of extending our approach to the case of multipartite systems and mixed states.

arXiv:quant-ph/0110139Oct 24, 20011 citation

Quantum entanglement in finite-dimensional Hilbert spaces

Szilárd Szalay

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.

arXiv:1302.4654Feb 19, 20136 citations