© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Modern Physics Letters A

Spectral characterization of entanglement witnesses in 2×n and 3×n systems

Changchun Feng, Laifa Tao, Lin Chen

Published Dec 12, 2025• 0 citations

Abstract

In this paper, we investigate the inertia (i.e. the eigenvalue triplet counting negative, zero, and positive eigenvalues) of entanglement witnesses arising from the partial transposes of non-positive partial transpose (NPT) states across bipartite ([Formula: see text] and [Formula: see text]) and tripartite (three-qubit) systems. For any [Formula: see text] NPT state [Formula: see text] whose partial transpose [Formula: see text] has inertia [Formula: see text], we prove that [Formula: see text] can be converted into a [Formula: see text] NPT subsystem via stochastic local operations and classical communication (SLOCC). For any [Formula: see text] NPT state [Formula: see text] whose partial transpose [Formula: see text] has inertia [Formula: see text], we prove that [Formula: see text] can always be projected into a [Formula: see text] NPT subsystem via local operations. Through extensive numerical simulations generating random sparse positive semidefinite matrices, we catalog all possible inertia trios [Formula: see text] for three-qubit NPT states. We identify 72 distinct inertia trios, revealing asymmetric distributions and co-occurrence constraints. These findings establish fundamental links between inertia patterns, distillability properties, and the geometric structure of entangled states, providing new spectral tools for characterizing entanglement in quantum information processing.

DOI: 10.1142/s0217732325502347

Related papers

Inertias of entanglement witnesses

Yi Shen, Lin Chen, Li-Jun Zhao

Abstract Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We study the inertias of EWs, i.e., the triplet of the numbers of negative, zero, and positive eigenvalues respectively. We focus on the EWs constructed by the partial transposition of states with non-positive partial transposes. We provide a method to generate more inertias from a given inertia by the relevance between inertias. Based on that we exhaust all the inertias for EWs in each qubit–qudit system. We apply our results to propose a separability criterion in terms of the rank of the partial transpose of state. We also connect our results to tripartite genuinely entangled states and the classification of states with non-positive partial transposes. Additionally, the inertias of EWs constructed by X -states are clarified.

arXiv:2007.13041•Journal of Physics A Mathematical and Theoretical•Oct 6, 2020•13 citations

Inertia of decomposable entanglement witnesses

Lin Chen, Bohan Jiang

Abstract We investigate the inertia (i.e., the array of numbers of negative, zero and positive eigenvalues of an Hermitian matrix) of decomposable entanglement witnesses (EWs). We show that the 2 × n and two-qutrit decomposable EWs have the same inertias as those of non-positive-transpose (NPT) EWs. We also show that if an m × n EW W has inertia (p, a p , mn − p − a p ) with p≥1, then for every integer b ∈ [0, a p ], then we can find an EW W b such that InW b = (p, b, mn − p − b). If W is a decomposable (resp. NPT) EW, then we can choose W b as also a decomposable (resp. NPT) EW. We further show that the m × n decomposable EW with the maximum number of negative eigenvalues can be chosen as an NPT EW. Then we explicitly characterize the 2 × 3 EWs, and decomposable EWs P Γ + Q with positive semidefinite matrices P of rank one and Q. We also show that a 2 × 4 non-decomposable EW has no inertia (3, 2, 3). Then we show some properties of a 2 × 4 non-decomposable EW of inertia (2, 3, 3), if it exists.

Physica Scripta•Nov 21, 2024•0 citations

Inertia of two-qutrit entanglement witnesses

Changchun Feng, Lin Chen, Chang Liang Xu +1 more

Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We investigate the inertia of bipartite EWs constructed by the partial transpose of NPT states. Furthermore, we find out most of the inertia of the partial transpose of the two-qutrit bipartite NPT states. As an application, we extend our results to high-dimensional states.

arXiv:2204.09252•Linear and Multilinear Algebra•Dec 25, 2022•5 citations

Inertia of entanglement witness

Yi Shen, Lin Chen, Li-Jun Zhao

Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We study the matrix inertias of EWs with a focus on the EWs constructed by the partial transposition of states with non-positive partial transposes. We provide a method to generate more inertias from a given inertia by the relevance between inertias. Based on that we exhaust all the inertias for EWs in each qubit-qudit system. We apply our results to propose a separability criterion in terms of the rank of the partial transpose of state. We also connect our results to tripartite genuinely entangled states and the classification of states with non-positive partial transposes. Additionally, the inertias of EWs constructed by X-states are clarified.

•Jul 26, 2020•0 citations

Construction of three-qubit positive-partial-transpose entangled states of rank four

Yonggang Cheng, Lin Chen

Multiqubit positive-partial-transpose (PPT) entangled states play an important role in quantum information theory. We characterize such states of minimum rank in three-qubit system, namely rank four. Depending on whether the Lorentz invariant is zero, we classify such states into two types. The PPT entangled states constructed by unextendible product bases (UPB) have nonzero invariants, which belong to type I. We provide a method to effectively determine whether a state can be constructed from UPB. For states with zero invariant, which belong to type II, we provide an explicit expression up to equivalence of stochastic local operations and classical communications (SLOCC). It turns out that we can represent them with only one complex parameter. We further study SLOCC-equivalence relation within the expression. We also investigate the Lorentz invariants of multiqubit states with rank less than three and analyze their range.

quant-pharXiv:2605.19530•May 20, 2026•0 citations

Detection and Classification of Bipartite and Multipartite Entangled States

Anu Kumari

The detection and classification of entanglement properties in a two-qubit and a multi-qubit system is a topic of great interest. This topic has been extensively studied, and as a result, we discovered various approaches for detecting and classifying multi-qubit, in particular three-qubit entangled states. The emphasis of this work is on a formalism of methods for the detection and classification of bipartite as well as multipartite quantum systems. We have used the method of structural physical approximation of partially transposed matrix (SPA-PT) for the detection of entangled states in arbitrary dimensional bipartite quantum systems. Also, we have proposed criteria for the classification of all possible stochastic local operations and classical communication (SLOCC) inequivalent classes of a pure and mixed three-qubit state using the SPA-PT map. To quantify entanglement, we have defined a new measure of entanglement based on the method of SPA-PT, which we named as "structured negativity". We have shown that this measure can be used to quantify entanglement for negative partial transposed entangled states (NPTES). Since the methods for detection, classification and quantification of entanglement, defined in this thesis are based on SPA-PT, they may be realized in an experiment.

arXiv:2305.16643•May 26, 2023•0 citations

In this paper, we investigate the inertia (i.e. the eigenvalue triplet counting negative, zero, and positive eigenvalues) of entanglement witnesses arising from the partial transposes of non-positive partial transpose (NPT) states across bipartite ([Formula: see text] and [Formula: see text]) and tripartite (three-qubit) systems. For any [Formula: see text] NPT state [Formula: see text] whose partial transpose [Formula: see text] has inertia [Formula: see text], we prove that [Formula: see text] can be converted into a [Formula: see text] NPT subsystem via stochastic local operations and classical communication (SLOCC). For any [Formula: see text] NPT state [Formula: see text] whose partial transpose [Formula: see text] has inertia [Formula: see text], we prove that [Formula: see text] can always be projected into a [Formula: see text] NPT subsystem via local operations. Through extensive numerical simulations generating random sparse positive semidefinite matrices, we catalog all possible inertia trios [Formula: see text] for three-qubit NPT states. We identify 72 distinct inertia trios, revealing asymmetric distributions and co-occurrence constraints. These findings establish fundamental links between inertia patterns, distillability properties, and the geometric structure of entangled states, providing new spectral tools for characterizing entanglement in quantum information processing.

Abstract Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We study the inertias of EWs, i.e., the triplet of the numbers of negative, zero, and positive eigenvalues respectively. We focus on the EWs constructed by the partial transposition of states with non-positive partial transposes. We provide a method to generate more inertias from a given inertia by the relevance between inertias. Based on that we exhaust all the inertias for EWs in each qubit–qudit system. We apply our results to propose a separability criterion in terms of the rank of the partial transpose of state. We also connect our results to tripartite genuinely entangled states and the classification of states with non-positive partial transposes. Additionally, the inertias of EWs constructed by X -states are clarified.

Abstract We investigate the inertia (i.e., the array of numbers of negative, zero and positive eigenvalues of an Hermitian matrix) of decomposable entanglement witnesses (EWs). We show that the 2 × n and two-qutrit decomposable EWs have the same inertias as those of non-positive-transpose (NPT) EWs. We also show that if an m × n EW W has inertia (p, a p , mn − p − a p ) with p≥1, then for every integer b ∈ [0, a p ], then we can find an EW W b such that InW b = (p, b, mn − p − b). If W is a decomposable (resp. NPT) EW, then we can choose W b as also a decomposable (resp. NPT) EW. We further show that the m × n decomposable EW with the maximum number of negative eigenvalues can be chosen as an NPT EW. Then we explicitly characterize the 2 × 3 EWs, and decomposable EWs P Γ + Q with positive semidefinite matrices P of rank one and Q. We also show that a 2 × 4 non-decomposable EW has no inertia (3, 2, 3). Then we show some properties of a 2 × 4 non-decomposable EW of inertia (2, 3, 3), if it exists.

Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We study the matrix inertias of EWs with a focus on the EWs constructed by the partial transposition of states with non-positive partial transposes. We provide a method to generate more inertias from a given inertia by the relevance between inertias. Based on that we exhaust all the inertias for EWs in each qubit-qudit system. We apply our results to propose a separability criterion in terms of the rank of the partial transpose of state. We also connect our results to tripartite genuinely entangled states and the classification of states with non-positive partial transposes. Additionally, the inertias of EWs constructed by X-states are clarified.

Multiqubit positive-partial-transpose (PPT) entangled states play an important role in quantum information theory. We characterize such states of minimum rank in three-qubit system, namely rank four. Depending on whether the Lorentz invariant is zero, we classify such states into two types. The PPT entangled states constructed by unextendible product bases (UPB) have nonzero invariants, which belong to type I. We provide a method to effectively determine whether a state can be constructed from UPB. For states with zero invariant, which belong to type II, we provide an explicit expression up to equivalence of stochastic local operations and classical communications (SLOCC). It turns out that we can represent them with only one complex parameter. We further study SLOCC-equivalence relation within the expression. We also investigate the Lorentz invariants of multiqubit states with rank less than three and analyze their range.

The detection and classification of entanglement properties in a two-qubit and a multi-qubit system is a topic of great interest. This topic has been extensively studied, and as a result, we discovered various approaches for detecting and classifying multi-qubit, in particular three-qubit entangled states. The emphasis of this work is on a formalism of methods for the detection and classification of bipartite as well as multipartite quantum systems. We have used the method of structural physical approximation of partially transposed matrix (SPA-PT) for the detection of entangled states in arbitrary dimensional bipartite quantum systems. Also, we have proposed criteria for the classification of all possible stochastic local operations and classical communication (SLOCC) inequivalent classes of a pure and mixed three-qubit state using the SPA-PT map. To quantify entanglement, we have defined a new measure of entanglement based on the method of SPA-PT, which we named as "structured negativity". We have shown that this measure can be used to quantify entanglement for negative partial transposed entangled states (NPTES). Since the methods for detection, classification and quantification of entanglement, defined in this thesis are based on SPA-PT, they may be realized in an experiment.