Norm duality and the cross norm criteria for quantum entanglement

Abstract

We use duality techniques to prove and generalize the cross norm and computable cross norm criteria for separability of quantum states. While the original proof of the cross norm criterion is long and involved, our new proof is short and elementary. Furthermore, our proof generalizes naturally to arbitrary Schmidt number. We also use these techniques to generalize the computable cross norm criterion to arbitrary Schmidt number and prove some results of independent interest along the way.

Keywords:

• norms
• duality
• quantum entanglement
• Schmidt number

AMS Subject Classifications:

• 81P40
• 15A60
• 47A30
• 47A80

Acknowledgments

The author was supported by the University of Guelph Brock Scholarship and an NSERC Postdoctoral Fellowship. The author is grateful to their Ph.D. advisor, David W. Kribs, for helpful conversations and work related to that contained in this paper. Parts of this paper made up part of the author’s doctoral thesis.