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Physica Scripta

Inertia of decomposable entanglement witnesses

Lin Chen, Bohan Jiang

Published Nov 21, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ad94ae

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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.
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.
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.
We present a systematic investigation of the spectral properties of entanglement witnesses (EWs). Specifically, we analyze the infimum and supremum of the largest eigenvalue, the smallest eigenvalue, the negativity (defined as the absolute value of the sum of negative eigenvalues), and the squared Frobenius norm of a unit-trace (normalized) entanglement witness, along with the conditions under which these values are attained. Our study provides distinct characterizations for decomposable (DEWs) and nondecomposable entanglement witnesses (NDEWs). While these two classes share many spectral similarities, we reveal a fundamental divergence by proving that the infimum of the smallest eigenvalue can be attained by DEWs, yet remains strictly unattainable for all NDEWs. We apply the results to provide necessary conditions for an EW to possess a mirrored EW. Furthermore, we demonstrate the superior detection capability of NDEWs by proving that any non-positive-partial-transpose state beyond the two-qubit and qubit-qutrit systems can be detected by an NDEW.
We investigate various properties of multipartite block-positive operators, decomposable EWs (DEWs), and non-decomposable EWs. We provide a necessary and sufficient condition to construct a special DEW using the multipartite GHZ state. For multipartite DEW, we explicitly characterize the supremum and infimum of maximum and minimum eigenvalues, as well as the trace of its square. We also derive other results concerning NDEW, eigenvalues of 2×n2 \times n2×n EWs and the corresponding physical implications. Furthermore, we investigate the tightness of inequalities of these eigenvalues with examples.
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.