Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1709.05916 · Journal of Mathematical Physics

Coarse-grained entanglement classification through orthogonal arrays

Luigi Seveso, Dardo Goyeneche, Karol Życzkowski

Published Jul 1, 2018• 4 citations
arXivHTMLPDF

Abstract

Classification of entanglement in multipartite quantum systems is an open problem solved so far only for bipartite systems and for systems composed of three and four qubits. We propose here a coarse-grained classification of entanglement in systems consisting of N subsystems with an arbitrary number of internal levels each, based on the properties of orthogonal arrays with N columns. In particular, we investigate in detail a subset of highly entangled pure states which contains all states defining maximum distance separable codes. To illustrate the methods presented, we analyze systems of four and five qubits, as well as heterogeneous tripartite systems consisting of two qubits and one qutrit or one qubit and two qutrits.

DOI
10.1063/1.5006890

Related papers

Multipartite entanglement in heterogeneous systems

Dardo Goyeneche, Jakub Bielawski, Karol Życzkowski

Heterogeneous bipartite quantum pure states, composed of two subsystems with a different number of levels, cannot have both reductions maximally mixed. In this work, we demonstrate the existence of a wide range of highly entangled states of heterogeneous multipartite systems consisting of $N>2$ parties such that every reduction to one and two parties is maximally mixed. Two constructions of generating genuinely multipartite maximally entangled states of heterogeneous systems for an arbitrary number of subsystems are presented. Such states are related to quantum error correction codes over mixed alphabets and mixed orthogonal arrays. Additionally, we show the advantages of considering heterogeneous systems in practical implementations of multipartite steering.

arXiv:1602.08064•Physical Review A•Jul 29, 2016•60 citations

Genuinely multipartite entangled states and orthogonal arrays

Dardo Goyeneche, Karol Życzkowski

A pure quantum state of $N$ subsystems with $d$ levels each is called $k$-multipartite maximally entangled state, which we call a $k$-uniform state, if all its reductions to $k$ qudits are maximally mixed. These states form a natural generalization of $N$-qudit Greenberger-Horne-Zeilinger states which belong to the class 1-uniform states. We establish a link between the combinatorial notion of orthogonal arrays and $k$-uniform states and prove the existence of several classes of such states for $N$-qudit systems. In particular, known Hadamard matrices allow us to explicitly construct 2-uniform states for an arbitrary number of $N>5$ qubits. We show that finding a different class of 2-uniform states would imply the Hadamard conjecture, so the full classification of 2-uniform states seems to be currently out of reach. Furthermore, we establish links between the existence of $k$-uniform states and classical and quantum error correction codes and provide a graph representation for such states.

arXiv:1404.3586•Physical Review A•Aug 15, 2014•157 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 Sinica•Jan 1, 2007•1 citation

A new method for constructing entangled states via orthogonal arrays

Xiaoyuan Yu, Zha Xin-wei, CHE JunLing

A pure quantum state of $N$ subsystems with $d$ levels each is called $k$-multipartite maximally entangled state, which we call a $k$-uniform state, if all its reductions to $k$ qubits are maximally mixed. Recently, Goyeneche and Zyczkowski proposed a method of constructing $k$-uniform states via orthogonal arrays, by which, we will also propose a new method to construct quantum states with orthogonal arrays in this paper, which is the method of increasing phase parameters. We can find the quantum states constructed via our new method by using the same orthogonal array, for four-qubit quantum states, not only are they 1-uniform states, also may become maximally entangled states; for five-qubit quantum states, we find that it can make the original 1-uniform states into 2-uniform states. In particular, for four-qubit quantum states, we can construct a series of new forms of maximally entangled states via our new method.

Zhongguo kexue. Wulixue Lixue Tianwenxue•Sep 21, 2017•3 citations

Partially separable criterion and classification of states in multipartite systems with an arbitrary finite number of qubits

An Min Wang

After introducing the partially separable concept, we proved the equivalence between the partial separability of a given $m$-partite subsystem with $m$ qubits and the purity of states of this $m$-partite subsystem for a pure state in multipartite systems with arbitrary finite $n(>m)$ qubits. Furthermore, we give out the operational realizations (corollaries) of our theorem, which are the sufficient and necessary criterions of partial separability of states and can be used to classification of states. Our results are helpful to understand and describe quantum entanglement in multipartite systems.

arXiv:quant-ph/0305016•May 5, 2003•0 citations

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:1302.4654•Feb 19, 2013•6 citations
Classification of entanglement in multipartite quantum systems is an open problem solved so far only for bipartite systems and for systems composed of three and four qubits. We propose here a coarse-grained classification of entanglement in systems consisting of N subsystems with an arbitrary number of internal levels each, based on the properties of orthogonal arrays with N columns. In particular, we investigate in detail a subset of highly entangled pure states which contains all states defining maximum distance separable codes. To illustrate the methods presented, we analyze systems of four and five qubits, as well as heterogeneous tripartite systems consisting of two qubits and one qutrit or one qubit and two qutrits.
Heterogeneous bipartite quantum pure states, composed of two subsystems with a different number of levels, cannot have both reductions maximally mixed. In this work, we demonstrate the existence of a wide range of highly entangled states of heterogeneous multipartite systems consisting of N>2N>2N>2 parties such that every reduction to one and two parties is maximally mixed. Two constructions of generating genuinely multipartite maximally entangled states of heterogeneous systems for an arbitrary number of subsystems are presented. Such states are related to quantum error correction codes over mixed alphabets and mixed orthogonal arrays. Additionally, we show the advantages of considering heterogeneous systems in practical implementations of multipartite steering.
A pure quantum state of NNN subsystems with ddd levels each is called kkk-multipartite maximally entangled state, which we call a kkk-uniform state, if all its reductions to kkk qudits are maximally mixed. These states form a natural generalization of NNN-qudit Greenberger-Horne-Zeilinger states which belong to the class 1-uniform states. We establish a link between the combinatorial notion of orthogonal arrays and kkk-uniform states and prove the existence of several classes of such states for NNN-qudit systems. In particular, known Hadamard matrices allow us to explicitly construct 2-uniform states for an arbitrary number of N>5N>5N>5 qubits. We show that finding a different class of 2-uniform states would imply the Hadamard conjecture, so the full classification of 2-uniform states seems to be currently out of reach. Furthermore, we establish links between the existence of kkk-uniform states and classical and quantum error correction codes and provide a graph representation for such states.
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.
A pure quantum state of NNN subsystems with ddd levels each is called kkk-multipartite maximally entangled state, which we call a kkk-uniform state, if all its reductions to kkk qubits are maximally mixed. Recently, Goyeneche and Zyczkowski proposed a method of constructing kkk-uniform states via orthogonal arrays, by which, we will also propose a new method to construct quantum states with orthogonal arrays in this paper, which is the method of increasing phase parameters. We can find the quantum states constructed via our new method by using the same orthogonal array, for four-qubit quantum states, not only are they 1-uniform states, also may become maximally entangled states; for five-qubit quantum states, we find that it can make the original 1-uniform states into 2-uniform states. In particular, for four-qubit quantum states, we can construct a series of new forms of maximally entangled states via our new method.
After introducing the partially separable concept, we proved the equivalence between the partial separability of a given mmm-partite subsystem with mmm qubits and the purity of states of this mmm-partite subsystem for a pure state in multipartite systems with arbitrary finite n(>m)n(>m)n(>m) qubits. Furthermore, we give out the operational realizations (corollaries) of our theorem, which are the sufficient and necessary criterions of partial separability of states and can be used to classification of states. Our results are helpful to understand and describe quantum entanglement in multipartite systems.
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.