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.

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

References in corpus (10)

• Entanglement detection
• A complete family of separability criteria
• Quantifying Entanglement with Witness Operators
• Necessary and sufficient conditions for bipartite entanglement
• Structural approximations to positive maps and entanglement breaking channels
• Constructing optimal entanglement witnesses. II
• When different entanglement witnesses detect entangled states simultaneously
• 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