Is absolute separability determined by the partial transpose?

Abstract

The absolute separability problem asks for a characterization of the quantum states $\rho \in M_m\otimes M_n$ with the property that $U\rho U^{†}$ is separable for all unitary matrices $U$. We investigate whether or not it is the case that $\rho$ is absolutely separable if and only if $U\rho U^{†}$ has positive partial transpose for all unitary matrices $U$. In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer-Hall map. We also show that these two properties coincide for the family of isotropic states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.

Two of our results were corrected since v2; the primary results of interest remain unchanged

References in corpus (7)

• Entanglement detection
• A complete family of separability criteria
• Optimal entanglement criterion for mixed quantum states
• Unifying several separability conditions using the covariance matrix criterion
• Simpler semidefinite programs for completely bounded norms
• Positive reduction from spectra
• Norms and Cones in the Theory of Quantum Entanglement

Cited by in corpus (1)

• Positive reduction from spectra