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arXiv:2508.11308 · Journal of Mathematical Physics

Spectral characterizations of entanglement witnesses

Zhiwei Song, Lin Chen

Published Jul 1, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/5.0302887

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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.
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 versatile tool in the verification of entangled states. The framework of mirrored EW doubles the power of a given EW by introducing its twin-a mirrored EW-whereby two EWs related by mirroring can bound the set of separable states more efficiently. In this work, we investigate the relation between the EWs and its mirrored ones, and present a conjecture which claims that the mirrored operator obtained from an optimal EW is either a positive operator or a decomposable EW, which implies that positive-partial-transpose entangled states, also known as the bound entangled states, cannot be detected. This conjecture is reached by studying numerous known examples of optimal EWs. However, the mirrored EWs obtained from the non-optimal ones can be non-decomposable as well. We also show that mirrored operators obtained from the extremal decomposable witnesses are positive semi-definite. Interestingly, the witnesses that violate the well known conjecture of Structural Physical Approximation, do satisfy our conjecture. The intricate relation between these two conjectures is discussed and it reveals a novel structure of the separability problem.
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.
One aspect of complexity for quantum many-body systems resides in their ability to exhibit entanglement, uniquely quantum correlations central to quantum computation. To quantitatively explore the diminishing effect that quantum noise has on entanglement resources, we focus on the negativity witness, an entanglement measure based on local time reversal transformations that provides a perfect witness in simple physical contexts. Beyond calculation of the negativity, we analyze the associated eigenvalue spectra of partially transposed density matrices for qudit-qudit bipartitions, finding that higher negativity rank provides greater(reduced) robustness to targeted(depolarizing) noise. In particular, we extend the maximum number of separable states that can be arbitrarily mixed into an entangled pure ensemble while retaining negativity to be exactly equal to the original negativity rank. Furthermore, we report how understanding the role of the negativity rank with respect to the completely depolarizing channel allows direct calculation of (1) certain noise boundaries for the Absolutely Positive Partial Transpose (APPT) subspace beyond which no unitary operator is capable of generating negativity and (2) the maximum qutrit-qutrit APPT purity. Beyond proving tight bounds and contributing to the ongoing search for APPT entanglement, these results expand our understanding of the nuanced relationship between computation with entanglement and practical environments for operating quantum devices.