On the structure of mirrored operators obtained from optimal entanglement witnesses

Abstract

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.

13 pages, 3 figures, 1 table, comments are welcome

References in corpus (12)

• Entanglement detection
• Optimal entanglement criterion for mixed quantum states
• The state space for two qutrits has a phase space structure in its core
• Structural approximations to positive maps and entanglement breaking channels
• The geometry of bipartite qutrits including bound entanglement
• Free versus Bound Entanglement: Machine learning tackling a NP-hard problem
• How many mutually unbiased bases are needed to detect bound entangled states?
• Comparing bound entanglement of bell diagonal pairs of qutrits and ququarts
• Almost complete solution for the NP-hard separability problem of Bell diagonal qutrits
• A class of Bell diagonal entanglement witnesses in ${\mathbb{C}}^4\otimes {\mathbb{C}}^4$: optimization and the spanning property
• Is bound entanglement Lorentz invariant?
• A class of optimal positive maps in $M_n$

Cited by in corpus (4)

• Bipartite Bound Entanglement
• A mirrored pair of optimal non-decomposable entanglement witnesses for two qudits does exist
• Mirrored Entanglement Witnesses for Multipartite and High-Dimensional Quantum Systems
• Spectral characterizations of entanglement witnesses