Abstract:

We consider four norms on tensor product spaces that have appeared in quantum information theory and demonstrate duality relationships between them. We show that the product numerical radius is dual to the robustness of entanglement, and we similarly show that the S(k)-norm is dual to the projective tensor norm. We show that, analogous to how the product numerical radius and the S(k)-norm characterize k-block positivity of operators, there is a natural version of the projective tensor norm that characterizes Schmidt number. In this way we obtain an elementary new proof of the cross norm criterion for separability, and we also generalize both the cross norm and realignment criteria to the case of arbitrary Schmidt number.

Authors:

• Nathaniel Johnston
• David Kribs

Download:

• Official publication from Houston Journal of Mathematics
• Preprint from arXiv:1304.2328 [quant-ph]
• Local preprint [pdf]
• Slideshow presentation [pdf]

Cite as:

• N. Johnston and D. W. Kribs. Duality of entanglement norms. Houston Journal of Mathematics, 41(3):831–847, 2015.

Related publications:

• Norm Duality and the Cross Norm Criteria for Quantum Entanglement – a conference proceedings paper with many related results