Spectral characterizations of entanglement witnesses

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-transpose (NPT) state beyond the two-qubit and qubit-qutrit systems can be detected by an NDEW.

arXiv comment17 pages, no figure

References in corpus (44)

• Quantum entanglement
• Separability Criterion for Density Matrices
• Entanglement detection
• Separability criterion and inseparable mixed states with positive partial transposition
• Optimization of entanglement witnesses
• Robustness of entanglement
• Universal Quantum Estimator
• Direct estimations of linear and non-linear functionals of a quantum state
• Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement
• Largest separable balls around the maximally mixed bipartite quantum state
• Maximally entangled mixed states of two qubits
• Schmidt number witnesses and bound entanglement
• Geometry of entangled states
• Entanglement witnesses: construction, analysis and classification
• Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps
• Characterization of separable states and entanglement witnesses
• Nonlinear entanglement witnesses
• Relations between Entanglement Witnesses and Bell Inequalities
• Geometry of quantum systems: density states and entanglement
• Negative eigenvalues of partial transposition of arbitrary bipartite states
• Spectral conditions for positive maps
• Separability from Spectrum for Qubit-Qudit States
• PPT from spectra
• A Family of Norms With Applications In Quantum Information Theory
• Non-Positive Partial Transpose Subspaces Can be as Large as Any Entangled Subspace
• A (5,5) and (6,6) PPT edge state
• Distillability and PPT entanglement of low-rank quantum states
• Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
• Unextendible product bases and extremal density matrices with positive partial transpose
• Non-positive-partial-transpose quantum states of rank four are distillable
• Spectral properties of entanglement witnesses
• Spectral conditions for entanglement witnesses vs. bound entanglement
• Entanglement Witness $2.0$: Compressed/Mirrored Entanglement Witnesses
• Description of rank four PPT entangled states of two qutrits
• Determining eigenvalues of a density matrix with minimal information in a single experimental setting
• Fundamental Limitation on the Detectability of Entanglement
• The Inverse Eigenvalue Problem for Entanglement Witnesses
• Constructive counterexamples to additivity of minimum output Rényi entropy of quantum channels for all p>2
• Properties and construction of extreme bipartite states having positive partial transpose
• Inertias of entanglement witnesses
• On extremal positive maps acting between type I factors
• Equivalence classes and canonical forms for two-qutrit entangled states of rank four having positive partial transpose
• On the structure of mirrored operators obtained from optimal entanglement witnesses
• A mirrored pair of optimal non-decomposable entanglement witnesses for two qudits does exist