  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.
