Abstract:

We consider the inverse eigenvalue problem for entanglement witnesses, which asks for a characterization of their possible spectra (or equivalently, of the possible spectra resulting from positive linear maps of matrices). We completely solve this problem in the two-qubit case and we derive a large family of new necessary conditions on the spectra in arbitrary dimensions. We also establish a natural duality relationship with the set of absolutely separable states, and we completely characterize witnesses (i.e., separating hyperplanes) of that set when one of the local dimensions is 2.

Authors:

• Nathaniel Johnston
• Everett Patterson

Download:

• Published version from Linear Algebra and Its Applications
• Preprint from arXiv:1708.05901 [quant-ph]
• Local preprint [pdf]
• Slideshow presentation (version 1) [pdf]
• Slideshow presentation (version 2) [pdf]

Cite as:

• N. Johnston and E. Patterson. The inverse eigenvalue problem for entanglement witnesses. Linear Algebra and Its Applications, 550:1–27, 2018.

Supplementary material:

• The Spectrum of the Partial Transpose of a Density Matrix – a related blog post from 2013