Structural physical approximations and entanglement witnesses

Abstract

The structural physical approximation (SPA) to a positive map is considered to be one of the most important method to detect entanglement in the real physical world. We first show that an arbitrary entanglement witness (EW) $W$ can be constructed from a separable density matrix $\sigma$ in the form of $W=\sigma -c_{\sigma }I$, where $c_{\sigma }$ is a non-negative number and $I$ is the identity matrix. Following the general form of EWs from separable states, we show a sufficient condition and a sufficient and necessary condition in low dimensions of that SPAs to positive maps do not define entanglement-breaking channels. We show that either the SPA of an EW or the SPA of the partial transposition of the EW in low dimensions is an entanglement-breaking channel. We give sufficient conditions of violating the SPA conjecture [\emph{Phys. Rev. A}{\bf 78,} 062105 (2008)]. Our results indicate that the SPA conjecture is independent of whether or not positive maps are optimal.

arXiv comment7 pages, fixed some typos, thanks to the many comments received!

References in corpus (26)

• Quantum entanglement
• Separability Criterion for Density Matrices
• Separability of Mixed States: Necessary and Sufficient Conditions
• Entanglement detection
• On the volume of the set of mixed entangled states
• Bell Inequalities and the Separability Criterion
• Optimization of entanglement witnesses
• Robustness of entanglement
• Classification of mixed three-qubit states
• Separability of very noisy mixed states and implications for NMR quantum computing
• A complete family of separability criteria
• Largest separable balls around the maximally mixed bipartite quantum state
• Direct detection of quantum entanglement
• Quantifying Entanglement with Witness Operators
• Schmidt number witnesses and bound entanglement
• Relations between Entanglement Witnesses and Bell Inequalities
• Necessary and sufficient conditions for bipartite entanglement
• Structural approximations to positive maps and entanglement breaking channels
• Constructing new optimal entanglement witnesses
• The geometry of entanglement witnesses and local detection of entanglement
• Constructing optimal entanglement witnesses. II
• When different entanglement witnesses detect entangled states simultaneously
• Structural physical approximations of unphysical maps and generalized quantum measurements
• On structural physical approximations and entanglement breaking maps
• The structural physical approximations and optimal entanglement witnesses
• Constructing all entanglement witnesses from density matrices

Cited by in corpus (9)

• Entanglement witnesses: construction, analysis and classification
• Zigzag approach to higher key rate of sending-or-not-sending twin field quantum key distribution with finite key effects
• Composable security for practical quantum key distribution with two way classical communication
• Geometric representation of two-qubit entanglement witnesses
• Disproving the conjecture on structural physical approximation to optimal decomposable entanglement witnesses
• Recurrent construction of optimal entanglement witnesses for 2N qubit systems
• Universal methods for extending any entanglement witness from the bipartite to the multipartite case
• Guaranteeing Completely Positive Quantum Evolution
• Characterization and properties of weakly optimal entanglement witnesses