Abstract:

The absolute separability problem asks for a characterization of the quantum states \(\rho\in M_m\otimes M_n\) with the property that \(U\rho U^\dagger\) is separable for all unitary matrices \(U\). We provide evidence that \(\rho\) is absolutely separable if and only if \(U\rho U^\dagger\) has positive partial transpose for all unitary matrices \(U\). In particular, we show that many well-known separability criteria are unable to detect entanglement in any such state, 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 families of isotropic and Werner states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.

Authors:

• Srinivasan Arunachalam
• Nathaniel Johnston
• Vincent Russo

Download:

• Preprint from arXiv:1405.5853 [quant-ph]
• Local preprint [pdf]
• Slideshow presentation [PowerPoint]

Cite as:

• S. Arunachalam, N. Johnston, and V. Russo. Is absolute separability determined by the partial transpose? Quantum Information & Computation, 15(7 & 8):0694–0720, 2015.

Supplementary material:

• Supplementary scripts for implementing separability from spectrum SDPs numerically – MATLAB scripts for implementing some of the semidefinite programs that appear throughout the paper.

Related publications:

• Separability from spectrum for qubit–qudit states – a paper that considers the same problem in systems where one of the local dimensions is 2