© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2510.16561

A necessary and sufficient condition for genuinely entangled n-qubit states with six non-zero coefficients

Dafa Li

Published Oct 18, 2025• 0 citations

Abstract

In [Science 340, 1205, 7 June (2013)], via polytopes Michael Walter et al. proposed a sufficient condition detecting the genuinely entangled pure states. In this paper, assume that a state with six non-zero coefficients is not a trivially separable state. Then the state is separable if and only if its six basis states consist of the three partially complementary pairs and the corresponding coefficient matrix has proportional rows. The contrapositive of this result reads that the state is genuinely entangled if and only if its six basis states do not consist of the three partially complementary pairs or though the six basis states consist of the three partially complementary pairs, the corresponding coefficient matrix does not have proportional rows. We propose four corresponding coefficient 2 by 3 matrices and show that if the four coefficient matrices don't have proportional rows, then the state is genuinely entangled. It is trivial to know if two rows of a 2 by 3 coefficient matrix are proportional. The difference from the previous articles is that the structure of the basis states is used to detect entanglement in this paper. One can see that Osterloh and Siewert's states of five and six qubits are genuinely entangled because two rows for any one of the four corresponding coefficient 2 by 3 matrices are not proportional. These states were distinguished as the maximal entangled states by the complicated filters before. Keywords: entanglement, separability, entangled states, separable states, qubits.

DOI: 10.48550/arxiv.2510.16561

Related papers

Mathematical framework for describing multipartite entanglement in terms of rows or columns of coefficient matrices

Yi Huang, Huapeng Yu, Fang Miao +2 more

In this paper, we develop a mathematical framework for describing entanglement quantitatively and qualitatively for multipartite qudit states in terms of rows or columns of coefficient matrices. More specifically, we propose an entanglement measure and separability criteria based on rows or columns of coefficient matrices. This entanglement measure has an explicit mathematical expression by means of exterior products of all pairs of rows or columns in coefficient matrices. It is introduced via our result that the [Formula: see text]-concurrence coincides with the entanglement measure based on two-by-two minors of coefficient matrices. Depending on our entanglement measure, we obtain the separability criteria and maximal entanglement criteria in terms of rows or columns of coefficient matrices. Our conclusions show that just like every two-by-two minor in a coefficient matrix of a multipartite pure state, every pair of rows or columns can also exhibit its entanglement properties, and thus can be viewed as its smallest entanglement contribution unit too. The great merit of our entanglement measure and separability criteria is two-fold. First, they are very practical and convenient for computation compared to other methods. Second, they have clear geometric interpretations.

International Journal of Quantum Information•Dec 13, 2021•2 citations

Identification of three-qubit entanglement

Ming‐Jing Zhao, Ting-Gui Zhang, Xianqing Li‐Jost +1 more

We present a way of identifying all kinds of entanglement for three-qubit pure states in terms of the expectation values of Pauli operators. The necessary and sufficient conditions to classify the fully separable, biseparable, and genuine entangled states are explicitly given. The approach can be generalized to multipartite high-dimensional cases. For three-qubit mixed states, we propose two kinds of inequalities in terms of the expectation values of complementary observables. One inequality has advantages in entanglement detection of the quantum state with positive partial transpositions, and the other is able to detect genuine entanglement. The results give an effective method for experimental entanglement identification.

arXiv:1301.4317•Physical Review A•Jan 17, 2013•28 citations

A simple entanglement criterion of two-qubit system

Chao Zhao, Qizhi Guo, Weihan Tan

We present a novel method to judge the entanglement of two-qubit system, the density matrix of a two-qubit system can be constituted by the principle density matrix [Formula: see text] and the separable density matrices [Formula: see text]–[Formula: see text]. The necessary and sufficient conditions for the two-qubit system, the three involved coefficients p [Formula: see text] 0, p [Formula: see text] 0, p [Formula: see text] 0, and the principal density matrix [Formula: see text] being separable, then the two-qubit system is separable, otherwise the two-qubit system is entangled. Finally, the criterion for several known density matrices have been verified in this way.

International Journal of Modern Physics B•Jul 20, 2019•3 citations

A novel entanglement criterion of two-qubit system

Chao Zhao, Qizhi Guo, Weihan Tan

The “separability problem” in quantum information theory is a quite important and well-known hard problem. The low-dimensional system satisfies the PPT criterion. However, the high-dimensional system problem has been shown to be NP-hard problem. In general, it is very difficult to find the analytic solution of the density matrix for the high-dimensional system. Therefore, getting an analytic solution for two-qubit system is an interesting and useful problem. We propose a novel criterion for separability and entanglement-verification of two-qubit system. We expressed the density matrix by a sum of a principal density matrix and six separable density matrices. The necessary and sufficient conditions for the two-qubit system include that if the four involved coefficients [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and the principal density matrix [Formula: see text] are separable, then the two-qubit system is separable, otherwise the two-qubit system is entangled. Finally, our criterion results in a totally different conclusion compared to Horodecki’s criterion. We believe that the new criterion is more stringent than existing PPT methods.

International Journal of Modern Physics B•Feb 17, 2020•1 citation

Entanglement criterion of N qubit system

Weihan Tan, Chao Zhao, Qizhi Guo

In previous paper ( 2019 Int. J. Mod. Phys. B 33 1950197 ; 2020 Int. J. Mod. Phys. B 34 2050022 ), we presented a method to judge the entanglement of 2-qubit system. The necessary and sufficient conditions for the 2qubit system being separable are that if the relevant coefficients is positive and the principal density matrix is separable, then the system is separable, otherwise it is entangled. Now in this paper, we try to generalize this criterion to a 3-qubit system, and then, we further generalize the criterion of 3-qubit system to an N-qubit system. This is a complicated and interesting issue.

Acta Physica Sinica•Sep 29, 2022•0 citations

Genuinely multipartite entangled states in higher dimensions: a generalization of balancedness

Andreas Osterloh

I generalize the concept of balancedness to qudits with arbitrary dimension d . It is an extension of the concept of balancedness by Osterloh and Siewert (2010 New J. Phys. 12 075025 ). At first, I define maximally entangled states as being the stochastic states (with local reduced density matrices for a d -dimensional local Hilbert space) that are not product states and show that every so-defined maximal genuinely multi-qudit entangled state is balanced. Furthermore, all irreducibly balanced states are genuinely multi-qudit entangled and are locally equivalent with respect to SL( d ) transformations (i.e. the local filtering transformations) to a maximally entangled state. In particular the concept given here gives a condition that all maximal genuinely multi-qudit entangled states for general local Hilbert space dimension d have to satisfy. A general genuinely multi-qudit entangled state is an element of the partly balanced SU( d )-orbits.

arXiv:1406.0361•Journal of Physics A Mathematical and Theoretical•Nov 12, 2014•1 citation

In [Science 340, 1205, 7 June (2013)], via polytopes Michael Walter et al. proposed a sufficient condition detecting the genuinely entangled pure states. In this paper, assume that a state with six non-zero coefficients is not a trivially separable state. Then the state is separable if and only if its six basis states consist of the three partially complementary pairs and the corresponding coefficient matrix has proportional rows. The contrapositive of this result reads that the state is genuinely entangled if and only if its six basis states do not consist of the three partially complementary pairs or though the six basis states consist of the three partially complementary pairs, the corresponding coefficient matrix does not have proportional rows. We propose four corresponding coefficient 2 by 3 matrices and show that if the four coefficient matrices don't have proportional rows, then the state is genuinely entangled. It is trivial to know if two rows of a 2 by 3 coefficient matrix are proportional. The difference from the previous articles is that the structure of the basis states is used to detect entanglement in this paper. One can see that Osterloh and Siewert's states of five and six qubits are genuinely entangled because two rows for any one of the four corresponding coefficient 2 by 3 matrices are not proportional. These states were distinguished as the maximal entangled states by the complicated filters before. Keywords: entanglement, separability, entangled states, separable states, qubits.

In this paper, we develop a mathematical framework for describing entanglement quantitatively and qualitatively for multipartite qudit states in terms of rows or columns of coefficient matrices. More specifically, we propose an entanglement measure and separability criteria based on rows or columns of coefficient matrices. This entanglement measure has an explicit mathematical expression by means of exterior products of all pairs of rows or columns in coefficient matrices. It is introduced via our result that the [Formula: see text]-concurrence coincides with the entanglement measure based on two-by-two minors of coefficient matrices. Depending on our entanglement measure, we obtain the separability criteria and maximal entanglement criteria in terms of rows or columns of coefficient matrices. Our conclusions show that just like every two-by-two minor in a coefficient matrix of a multipartite pure state, every pair of rows or columns can also exhibit its entanglement properties, and thus can be viewed as its smallest entanglement contribution unit too. The great merit of our entanglement measure and separability criteria is two-fold. First, they are very practical and convenient for computation compared to other methods. Second, they have clear geometric interpretations.

We present a way of identifying all kinds of entanglement for three-qubit pure states in terms of the expectation values of Pauli operators. The necessary and sufficient conditions to classify the fully separable, biseparable, and genuine entangled states are explicitly given. The approach can be generalized to multipartite high-dimensional cases. For three-qubit mixed states, we propose two kinds of inequalities in terms of the expectation values of complementary observables. One inequality has advantages in entanglement detection of the quantum state with positive partial transpositions, and the other is able to detect genuine entanglement. The results give an effective method for experimental entanglement identification.

We present a novel method to judge the entanglement of two-qubit system, the density matrix of a two-qubit system can be constituted by the principle density matrix [Formula: see text] and the separable density matrices [Formula: see text]–[Formula: see text]. The necessary and sufficient conditions for the two-qubit system, the three involved coefficients p [Formula: see text] 0, p [Formula: see text] 0, p [Formula: see text] 0, and the principal density matrix [Formula: see text] being separable, then the two-qubit system is separable, otherwise the two-qubit system is entangled. Finally, the criterion for several known density matrices have been verified in this way.

The “separability problem” in quantum information theory is a quite important and well-known hard problem. The low-dimensional system satisfies the PPT criterion. However, the high-dimensional system problem has been shown to be NP-hard problem. In general, it is very difficult to find the analytic solution of the density matrix for the high-dimensional system. Therefore, getting an analytic solution for two-qubit system is an interesting and useful problem. We propose a novel criterion for separability and entanglement-verification of two-qubit system. We expressed the density matrix by a sum of a principal density matrix and six separable density matrices. The necessary and sufficient conditions for the two-qubit system include that if the four involved coefficients [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and the principal density matrix [Formula: see text] are separable, then the two-qubit system is separable, otherwise the two-qubit system is entangled. Finally, our criterion results in a totally different conclusion compared to Horodecki’s criterion. We believe that the new criterion is more stringent than existing PPT methods.

In previous paper ( 2019 Int. J. Mod. Phys. B 33 1950197 ; 2020 Int. J. Mod. Phys. B 34 2050022 ), we presented a method to judge the entanglement of 2-qubit system. The necessary and sufficient conditions for the 2qubit system being separable are that if the relevant coefficients is positive and the principal density matrix is separable, then the system is separable, otherwise it is entangled. Now in this paper, we try to generalize this criterion to a 3-qubit system, and then, we further generalize the criterion of 3-qubit system to an N-qubit system. This is a complicated and interesting issue.

I generalize the concept of balancedness to qudits with arbitrary dimension d . It is an extension of the concept of balancedness by Osterloh and Siewert (2010 New J. Phys. 12 075025 ). At first, I define maximally entangled states as being the stochastic states (with local reduced density matrices for a d -dimensional local Hilbert space) that are not product states and show that every so-defined maximal genuinely multi-qudit entangled state is balanced. Furthermore, all irreducibly balanced states are genuinely multi-qudit entangled and are locally equivalent with respect to SL( d ) transformations (i.e. the local filtering transformations) to a maximally entangled state. In particular the concept given here gives a condition that all maximal genuinely multi-qudit entangled states for general local Hilbert space dimension d have to satisfy. A general genuinely multi-qudit entangled state is an element of the partly balanced SU( d )-orbits.