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.
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.
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.
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.
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.
Experimental detection of entanglement of an arbitrary state of a given bipartite system is crucial for exploring many areas of quantum information processing. But such a detection should be made in a device-independent way if the preparation process of the state is considered to be faithful, in order to avoid detection of a separable state as an entangled one. The recently developed scheme of detecting bipartite entanglement in a measurement-device-independent way [Phys. Rev. Lett. 110, 060405 (2013)] does require information about the state. Here, by using Auguisiak et al.'s universal entanglement witness scheme for two-qubit states [Phys. Rev. A 77, 030301 (2008)], we provide a universal entanglement detection scheme for two-qubit states in a measurement-device-independent way. We also provide a set of universal witness operators for detecting NPT-ness (negative under partial transpose) of two-qudit states in a measurement-device-independent way. We conjecture that no such universal entanglement witness scheme exists for PPT (positive under partial transpose) entangled states. We also analyze the robustness of some of the experimental schemes---for detecting entanglement in a measurement-device-independent way---under the influence of noise in the inputs (from the referee) as well as in the measurement operator.
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.