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

Physical Review A

Entanglement classification and invariant-based entanglement measures

Xiangrong Li, Dafa Li

Published Jan 7, 2015• 6 citations
Publisher

Abstract

We propose a method of classifying $n$-qubit states into stochastic local operations and classical communication inequivalent families in terms of the rank of the square matrix $C{(i{\ensuremath{\sigma}}_{y})}^{\ensuremath{\bigotimes}k}{C}^{T}$, where $C$ is the rectangular coefficient matrix of the state and ${\ensuremath{\sigma}}_{y}$ is the Pauli operator. The rank of the square matrix $C{(i{\ensuremath{\sigma}}_{y})}^{\ensuremath{\bigotimes}k}{C}^{T}$ is capable of distinguishing between $n$-qubit Greenberger-Horne-Zeilinger and $W$ states. The determinant of the matrix gives rise to a family of polynomial invariants for $n$ qubits which include as special cases well-known polynomial invariants in the literature. In addition, explicit expressions can be given for these polynomial invariants and this allows us to investigate the properties of entanglement measures built upon the absolute values of polynomial invariants for product states.

DOI
10.1103/physreva.91.012302

Related papers

Polynomial invariants of degree 4 for even- n qubits and their applications in entanglement classification

Xiangrong Li, Dafa Li

We develop a simple method for constructing polynomial invariants of degree 4 for even-$n$ qubits and give explicit expressions for these polynomial invariants. We demonstrate the invariance of the polynomials under stochastic local operations and classical communication and exemplify the use of the invariance in classifying entangled states. The absolute values of these polynomial invariants are entanglement monotones, thereby allowing entanglement measures to be built. Finally, we discuss the properties of these entanglement measures.

arXiv:1308.2436•Physical Review A•Aug 8, 2013•9 citations

Method for classifying multiqubit states via the rank of the coefficient matrix and its application to four-qubit states

Xiangrong Li, Dafa Li

We construct coefficient matrices of size ${2}^{\ensuremath{\ell}}$ by ${2}^{n\ensuremath{-}\ensuremath{\ell}}$ associated with pure $n$-qubit states and prove the invariance of the ranks of the coefficient matrices under stochastic local operations and classical communication (SLOCC). The ranks give rise to a simple way of partitioning pure $n$-qubit states into inequivalent families and distinguishing degenerate families from one another under SLOCC. Moreover, the classification scheme via the ranks of coefficient matrices can be combined with other schemes to build a more refined classification scheme. To exemplify we classify the nine families of four qubits introduced by Verstraete et al. [Phys. Rev. A 65, 052112 (2002)] further into inequivalent subfamilies via the ranks of coefficient matrices, and as a result, we find 28 genuinely entangled families and all the degenerate classes can be distinguished up to permutations of the four qubits. We also discuss the completeness of the classification of four qubits into nine families.

arXiv:1201.2229•Physical Review A•Oct 23, 2012•25 citations

Classification of General n -Qubit States under Stochastic Local Operations and Classical Communication in Terms of the Rank of Coefficient Matrix

Xiangrong Li, Dafa Li

We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for general n-qubit states. For two arbitrary pure n-qubit states connected via local operations, we establish an equation between the two coefficient matrices associated with the states. The rank of the coefficient matrix is preserved under SLOCC and gives rise to a simple way of partitioning all the pure states of n qubits into different families of entanglement classes, as exemplified here. When applied to the symmetric states, this approach reveals that all the Dicke states |ℓ,n> with ℓ=1,…,[n/2] are inequivalent under SLOCC.

arXiv:1106.6105•Physical Review Letters•May 3, 2012•54 citations

Polynomial invariants for discrimination and classification of four-qubit entanglement

Oliver Viehmann, Christopher Eltschka, Jens Siewert

The number of entanglement classes in stochastic local operations and classical communication (SLOCC) classifications increases with the number of qubits and is already infinite for four qubits. Criteria for explicitly discriminating and classifying pure states of four and more qubits are highly desirable and therefore at the focus of intense theoretical research. We develop a general criterion for the discrimination of pure $N\text{\ensuremath{-}}$partite entangled states in terms of polynomial $\mathrm{SL}(d,\mathrm{C}){}^{\ensuremath{\bigotimes}N}$ invariants. By means of this criterion, existing SLOCC classifications of four-qubit entanglement are reproduced. Based on this we propose a polynomial classification scheme in which entanglement types are identified through ``tangle patterns.'' This scheme provides a practicable way to classify states of arbitrary multipartite systems. Moreover, the use of polynomials induces a corresponding quantification of the different types of entanglement.

arXiv:1101.5558•Physical Review A•May 31, 2011•65 citations

Classification of Multipartite Entanglement of All Finite Dimensionality

Gilad Gour, Nolan R. Wallach

We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such a SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Our construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, we provide an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits we also provide much simpler expressions.

arXiv:1304.7259•Physical Review Letters•Aug 8, 2013•84 citations

Classifying entanglement in the superposition of Greenberger-Horne-Zeilinger and W states

Xuan Wang, Xian Shi

In this paper, we study the entanglement classification of the two-dimensional subspace spanned by $n$-qubit Greenberger-Horne-Zeilinger (GHZ) and $W$ states under stochastic local operations and classical communication (SLOCC). We show that for $n\ensuremath{\ge}4$ such a system can have infinite SLOCC inequivalent classes. Specifically, two states, ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+x{|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+y{|W\ensuremath{\rangle}}_{n}$, from the subspace spanned by $n$-qubit ${|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}$ are SLOCC equivalent if and only if ${(\frac{x}{y})}^{n}=1$.

Physical Review A•Oct 14, 2015•1 citation
We propose a method of classifying nnn-qubit states into stochastic local operations and classical communication inequivalent families in terms of the rank of the square matrix C(i\ensuremathσy)\ensuremath⨂kCTC{(i{\ensuremath{\sigma}}_{y})}^{\ensuremath{\bigotimes}k}{C}^{T}C(i\ensuremathσy​)\ensuremath⨂kCT, where CCC is the rectangular coefficient matrix of the state and \ensuremathσy{\ensuremath{\sigma}}_{y}\ensuremathσy​ is the Pauli operator. The rank of the square matrix C(i\ensuremathσy)\ensuremath⨂kCTC{(i{\ensuremath{\sigma}}_{y})}^{\ensuremath{\bigotimes}k}{C}^{T}C(i\ensuremathσy​)\ensuremath⨂kCT is capable of distinguishing between nnn-qubit Greenberger-Horne-Zeilinger and WWW states. The determinant of the matrix gives rise to a family of polynomial invariants for nnn qubits which include as special cases well-known polynomial invariants in the literature. In addition, explicit expressions can be given for these polynomial invariants and this allows us to investigate the properties of entanglement measures built upon the absolute values of polynomial invariants for product states.
We develop a simple method for constructing polynomial invariants of degree 4 for even-nnn qubits and give explicit expressions for these polynomial invariants. We demonstrate the invariance of the polynomials under stochastic local operations and classical communication and exemplify the use of the invariance in classifying entangled states. The absolute values of these polynomial invariants are entanglement monotones, thereby allowing entanglement measures to be built. Finally, we discuss the properties of these entanglement measures.
We construct coefficient matrices of size ${2}^{\ensuremath{\ell}}$ by ${2}^{n\ensuremath{-}\ensuremath{\ell}}$ associated with pure nnn-qubit states and prove the invariance of the ranks of the coefficient matrices under stochastic local operations and classical communication (SLOCC). The ranks give rise to a simple way of partitioning pure nnn-qubit states into inequivalent families and distinguishing degenerate families from one another under SLOCC. Moreover, the classification scheme via the ranks of coefficient matrices can be combined with other schemes to build a more refined classification scheme. To exemplify we classify the nine families of four qubits introduced by Verstraete et al. [Phys. Rev. A 65, 052112 (2002)] further into inequivalent subfamilies via the ranks of coefficient matrices, and as a result, we find 28 genuinely entangled families and all the degenerate classes can be distinguished up to permutations of the four qubits. We also discuss the completeness of the classification of four qubits into nine families.
We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for general n-qubit states. For two arbitrary pure n-qubit states connected via local operations, we establish an equation between the two coefficient matrices associated with the states. The rank of the coefficient matrix is preserved under SLOCC and gives rise to a simple way of partitioning all the pure states of n qubits into different families of entanglement classes, as exemplified here. When applied to the symmetric states, this approach reveals that all the Dicke states |ℓ,n> with ℓ=1,…,[n/2] are inequivalent under SLOCC.
The number of entanglement classes in stochastic local operations and classical communication (SLOCC) classifications increases with the number of qubits and is already infinite for four qubits. Criteria for explicitly discriminating and classifying pure states of four and more qubits are highly desirable and therefore at the focus of intense theoretical research. We develop a general criterion for the discrimination of pure $N\text{\ensuremath{-}}$partite entangled states in terms of polynomial $\mathrm{SL}(d,\mathrm{C}){}^{\ensuremath{\bigotimes}N}$ invariants. By means of this criterion, existing SLOCC classifications of four-qubit entanglement are reproduced. Based on this we propose a polynomial classification scheme in which entanglement types are identified through ``tangle patterns.'' This scheme provides a practicable way to classify states of arbitrary multipartite systems. Moreover, the use of polynomials induces a corresponding quantification of the different types of entanglement.
We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such a SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Our construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, we provide an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits we also provide much simpler expressions.
In this paper, we study the entanglement classification of the two-dimensional subspace spanned by nnn-qubit Greenberger-Horne-Zeilinger (GHZ) and WWW states under stochastic local operations and classical communication (SLOCC). We show that for $n\ensuremath{\ge}4$ such a system can have infinite SLOCC inequivalent classes. Specifically, two states, ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+x{|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+y{|W\ensuremath{\rangle}}_{n}$, from the subspace spanned by nnn-qubit ${|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}$ are SLOCC equivalent if and only if (xy)n=1{(\frac{x}{y})}^{n}=1(yx​)n=1.