A family of norms with applications in quantum information theory
Abstract
We consider a family of vector and operator norms defined by the Schmidt decomposition theorem for quantum states. We use these norms to tackle two fundamental problems in quantum information theory: the classification problem for kpositive linear maps and entanglement witnesses, and the existence problem for nonpositive partial transpose bound entangled states. We begin with an analysis of the norms, showing that the vector norms can be explicitly calculated, and we derive several inequalities in order to bound the operator norms and compute them in special cases. We then use the norms to establish what appears to be the most general spectral test for kpositivity currently available, showing how it implies several other known tests as well as some new ones. Building on this work, we frame the nonpositive partial transpose bound entangled problem as a concrete problem on a specific limit, specifically that a particular entangled Werner state is bound entangled if and only if a certain norm inequality holds on a given family of projections.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 August 2010
 DOI:
 10.1063/1.3459068
 arXiv:
 arXiv:0909.3907
 Bibcode:
 2010JMP....51h2202J
 Keywords:

 quantum entanglement;
 03.67.Mn;
 03.65.Ud;
 Entanglement production characterization and manipulation;
 Entanglement and quantum nonlocality;
 Quantum Physics
 EPrint:
 Original posting "Schmidt norms for quantum states" evolved into two papers, this one and arXiv:1006.0898. Proposition 4.5 is corrected in v4