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:0804.2291 · Journal of Physics A Mathematical and Theoretical

Classification of the entangled states of 2 ×N×N

Shuo Cheng, Junli Li, Cong-Feng Qiao

Published Jan 12, 2010• 9 citations
arXivHTMLPDF

Abstract

We develop a novel method in classifying the multipartite entanglement state of 2 × N × N under stochastic local operation and classical communication. In this method, all inequivalent classes of true entangled state can be sorted directly without knowing the classification information of lower dimension ones for any given dimension N .

DOI
10.1088/1751-8113/43/5/055303

Related papers

Classification of the Entangled States of 2×N ×N

Shuo Cheng, Junli Li, Cong‐Feng Qiao

We develop a novel method in classifying the multipartite entanglement state of 2×N ×N under stochastic local operations and classical communication (SLOCC). In this method, all inequivalent classes of a genuine entangled state can be assorted directly without knowing the classification information of lower dimension ones for any given dimension N. It also gives a nature explanation for the non-local parameters remaining in the entanglement classes when N ≥ 4. Entanglement is at the heart of the quantum information theory (QIT) and is now thought as a physical resource to realize quantum information tasks, such as quantum cryptography [1, 2], superdense coding [3, 4], and quantum computation [5]. The study of entanglement improves our further understanding of quantum non-locality [6] and also is of particular interest in QIT. One such case is the classification of multipartite entanglement. Two quantum states can be employed to implement the same tasks in QIT and are thought to be equivalent while they are mutually convertible by Stochastic Local Operations and Classical Communication (SLOCC) [7]. Nevertheless, in practice the classification of multipartite entanglement and in high dimensions in the Hilbert space seems to be a formidable task [8]. Matrix decomposition keeps to be a useful tool as in the two-partite case [9, 10]. A widely adopted philosophy in dealing with this issue is first to know the classification of the state in lower dimension (or less partite) and then extend to the higher dimension case [11, 12] (or more partite [13, 14]) in an inductive way. However, nontrivial aspect emerges as the dimension increases, i.e. some non-local parameters may nest in the entangled states [15, 16]. In literature, in

•Aug 4, 2013•11 citations

Classification of the Entangled States of $2 \times L \times M \times N$

Liang-Liang Sun, Junli Li, Cong‐Feng Qiao

We present a practical entanglement classification scheme for pure state in form of $2\times L\times M\times N$ under the stochastic local operation and classical communication (SLOCC), where every inequivalent class of the entangled quantum states may be sorted out according to its standard form and the corresponding transformation matrix. This provides a practical method for determining the interconverting matrix between two SLOCC equivalent entangled states, and classification examples for some $2\times 4\times M\times N$ systems are also presented.

arXiv:1401.6609•Jan 26, 2014•0 citations

Classification of multipartite entanglement in $2\times{M}\times{N}$ quantum system: range criterion and a systematic method

Lin Chen, Yixin Chen

We develop a range criterion which is a sufficient and necessary condition satisfied between two states transformable to each other under reversible stochastic local operations assisted by classical communication (SLOCC). Based on this criterion, we can effectively discriminate any two states in $2\\times{M}\\times{N}$ quantum system and we propose a systematic method for seeking all kinds of states in any true $2\\times{M}\\times{N}$ system. This method can greatly reduce the influence of parameters and it shows the potentially regular structure of inequivalent classes. The classification in $2\\times3\\times{N}$ system is explicitly described.

•Jun 18, 2005•0 citations

Classification of the Entangled States of $2\times L\times M\times N\times H$

Jia, Kang-Kang, Junli Li, Cong‐Feng Qiao

In this work we propose a practical entanglement classification scheme for pure states of $2\times L\times M\times N\times H$, under the stochastic local operation and classical communication (SLOCC), which generalizes the method explored in the entanglement classification of $2\times L\times M\times N$ to the five-partite system. The entangled states of $2\times L\times M\times N\times H$ system are first classified into different coarse-grained standard forms using matrix decompositions, and then fine-grained identification of two inequivalent entangled states with the same standard form are completed by using the matrix realignment technique. As an practical example, entanglement classes of the five-qubit system of $2\times 2\times 2\times 2\times 2$ are presented.

arXiv:1505.02007•May 8, 2015•0 citations

Inductive entanglement classification of four qubits under stochastic local operations and classical communication

Lucas Lamata, Juan León, David Cienfuegos Salgado +1 more

Using an inductive approach to classify multipartite entangled states under stochastic local operations and classical communication introduced recently by the authors [Phys. Rev. A 74, 052336 (2006)], we give the complete classification of four-qubit entangled pure states. Apart from the expected degenerate classes, we show that there exist eight inequivalent ways to entangle four qubits. In this respect, permutation symmetry is taken into account and states with a structure differing only by parameters inside a continuous set are considered to belong to the same class.

arXiv:quant-ph/0610233•Physical Review A•Feb 21, 2007•130 citations

Inductive classification of multipartite entanglement under stochastic local operations and classical communication

Lucas Lamata, Juan León, David Cienfuegos Salgado +1 more

We propose an inductive procedure to classify $N$-partite entanglement under stochastic local operations and classical communication provided such a classification is known for $N\ensuremath{-}1$ qubits. The method is based upon the analysis of the coefficient matrix of the state in an arbitrary product basis. We illustrate this approach in detail with the well-known bipartite and tripartite systems, obtaining as a by-product a systematic criterion to establish the entanglement class of a given pure state without resourcing to any entanglement measure. The general case is proved by induction, allowing us to find an upper bound for the number of $N$-partite entanglement classes in terms of the number of entanglement classes for $N\ensuremath{-}1$ qubits.

arXiv:quant-ph/0603243•Physical Review A•Nov 28, 2006•77 citations
We develop a novel method in classifying the multipartite entanglement state of 2 × N × N under stochastic local operation and classical communication. In this method, all inequivalent classes of true entangled state can be sorted directly without knowing the classification information of lower dimension ones for any given dimension N .
We develop a novel method in classifying the multipartite entanglement state of 2×N ×N under stochastic local operations and classical communication (SLOCC). In this method, all inequivalent classes of a genuine entangled state can be assorted directly without knowing the classification information of lower dimension ones for any given dimension N. It also gives a nature explanation for the non-local parameters remaining in the entanglement classes when N ≥ 4. Entanglement is at the heart of the quantum information theory (QIT) and is now thought as a physical resource to realize quantum information tasks, such as quantum cryptography [1, 2], superdense coding [3, 4], and quantum computation [5]. The study of entanglement improves our further understanding of quantum non-locality [6] and also is of particular interest in QIT. One such case is the classification of multipartite entanglement. Two quantum states can be employed to implement the same tasks in QIT and are thought to be equivalent while they are mutually convertible by Stochastic Local Operations and Classical Communication (SLOCC) [7]. Nevertheless, in practice the classification of multipartite entanglement and in high dimensions in the Hilbert space seems to be a formidable task [8]. Matrix decomposition keeps to be a useful tool as in the two-partite case [9, 10]. A widely adopted philosophy in dealing with this issue is first to know the classification of the state in lower dimension (or less partite) and then extend to the higher dimension case [11, 12] (or more partite [13, 14]) in an inductive way. However, nontrivial aspect emerges as the dimension increases, i.e. some non-local parameters may nest in the entangled states [15, 16]. In literature, in
Classification of the Entangled States of 2×L×M×N2 \times L \times M \times N2×L×M×N
We present a practical entanglement classification scheme for pure state in form of 2×L×M×N2\times L\times M\times N2×L×M×N under the stochastic local operation and classical communication (SLOCC), where every inequivalent class of the entangled quantum states may be sorted out according to its standard form and the corresponding transformation matrix. This provides a practical method for determining the interconverting matrix between two SLOCC equivalent entangled states, and classification examples for some 2×4×M×N2\times 4\times M\times N2×4×M×N systems are also presented.
Classification of multipartite entanglement in 2×M×N2\times{M}\times{N}2×M×N quantum system: range criterion and a systematic method
We develop a range criterion which is a sufficient and necessary condition satisfied between two states transformable to each other under reversible stochastic local operations assisted by classical communication (SLOCC). Based on this criterion, we can effectively discriminate any two states in 2timesMtimesN2\\times{M}\\times{N}2timesMtimesN quantum system and we propose a systematic method for seeking all kinds of states in any true 2timesMtimesN2\\times{M}\\times{N}2timesMtimesN system. This method can greatly reduce the influence of parameters and it shows the potentially regular structure of inequivalent classes. The classification in 2times3timesN2\\times3\\times{N}2times3timesN system is explicitly described.
Classification of the Entangled States of 2×L×M×N×H2\times L\times M\times N\times H2×L×M×N×H
In this work we propose a practical entanglement classification scheme for pure states of 2×L×M×N×H2\times L\times M\times N\times H2×L×M×N×H, under the stochastic local operation and classical communication (SLOCC), which generalizes the method explored in the entanglement classification of 2×L×M×N2\times L\times M\times N2×L×M×N to the five-partite system. The entangled states of 2×L×M×N×H2\times L\times M\times N\times H2×L×M×N×H system are first classified into different coarse-grained standard forms using matrix decompositions, and then fine-grained identification of two inequivalent entangled states with the same standard form are completed by using the matrix realignment technique. As an practical example, entanglement classes of the five-qubit system of 2×2×2×2×22\times 2\times 2\times 2\times 22×2×2×2×2 are presented.
Using an inductive approach to classify multipartite entangled states under stochastic local operations and classical communication introduced recently by the authors [Phys. Rev. A 74, 052336 (2006)], we give the complete classification of four-qubit entangled pure states. Apart from the expected degenerate classes, we show that there exist eight inequivalent ways to entangle four qubits. In this respect, permutation symmetry is taken into account and states with a structure differing only by parameters inside a continuous set are considered to belong to the same class.
We propose an inductive procedure to classify NNN-partite entanglement under stochastic local operations and classical communication provided such a classification is known for $N\ensuremath{-}1$ qubits. The method is based upon the analysis of the coefficient matrix of the state in an arbitrary product basis. We illustrate this approach in detail with the well-known bipartite and tripartite systems, obtaining as a by-product a systematic criterion to establish the entanglement class of a given pure state without resourcing to any entanglement measure. The general case is proved by induction, allowing us to find an upper bound for the number of NNN-partite entanglement classes in terms of the number of entanglement classes for $N\ensuremath{-}1$ qubits.