A Family of Norms With Applications In Quantum Information Theory

TL;DR

This paper introduces a family of vector and operator norms based on the Schmidt decomposition, applying them to classify quantum states and analyze entanglement, providing new spectral tests for k-positivity and bound entanglement.

Contribution

It develops explicit calculations and inequalities for these norms, establishing a general spectral test for k-positivity and framing the bound entangled state problem in terms of norm inequalities.

Findings

Explicit formulas for vector norms

A new spectral test for k-positivity

Norm-based criterion for bound entanglement

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.