# Tensor norms and their applications in quantum information

Yaoyun Shi
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The Quantum Information Processing Reading (QIP) Seminar is
an informal forum on QIP, organized by members of the
(https://ctools.umich.edu/portal/site/1092952194789-1524055).
We welcome the participation of anyone interested in QIP.

A norm on the tensor product of two Banach spaces is called
a tensor norm (or a crossnorm) if $\|A \otimes B\|=\|A\|*\|B\|$,
for all $A$ and $B$.
Tensor norms have been studied for decades but their applications
in quantum information are only seen in recent years. For example,
an elegant theorem due to O. Rudolph (J. Phys. A: Math. Gen. 33 3951-3955) says that a bipartite mixed
quantum state is separable (i.e. not entangled) if and only if
its greatest tensor norm (a particular tensor norm) is precisely 1.
My recent work (STOC 2005) relates another tensor norm to the complexity of
the classical simulation of quantum communication.

In this talk I will first introduce some basics about tensor
norms, then present Rudolph's result with a simpler proof than the
original, and finally discuss directions for more applications of tensor
norms in quantum information processing.