## 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 k-positive linear maps and entanglement witnesses, and the existence problem for non-positive 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 k-positivity currently available, showing how it implies several other known tests as well as some new ones. Building on this work, we frame the NPPT 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.

**Authors:**

- Nathaniel Johnston
- David Kribs

** ****Download:**

- Official publication from JMP
- Preprint from arXiv:0909.3907 [quant-ph]
- Local preprint [pdf]
- Slideshow presentation (short version) [pdf]
- Slideshow presentation (long version) [pdf]

**Status:**

- Published in Journal of Mathematical Physics in August 2010.
- Selected for the Virtual Journal of Quantum Information.

** ****Cite as:**

- N. Johnston and D. W. Kribs,
*A Family of Norms With Applications In Quantum Information Theory*. Journal of Mathematical Physics**51**, 082202 (2010).

**Supplementary Material:**

- An Introduction to Schmidt Norms (blog post)
- Approximating the Distribution of Schmidt Vector Norms (blog post)

**Related Publications:**

- Schmidt Operator Norms and Entanglement Theory (conference proceedings)
- A Family of Norms With Applications In Quantum Information Theory II (sequel publication)