# Worked Problems in Physics

## Nielsen and Chuang Exercise 2.31

Posted in Nielsen/Chuang by rpmuller on March 6, 2010

Exercise 2.31 Show that ${A\otimes B}$ is positive if ${A,B}$ are positive.

$\displaystyle A\otimes B = (U\Lambda U^\dagger)\otimes(U'\Lambda'U'^\dagger) = (U\otimes U')(\Lambda\otimes\Lambda')(U^\dagger\otimes U'^\dagger)$

The final decomposition represents an eigendecomposition in the tensor space. ${U\otimes U'}$ is unitary (ex 2.29), and ${U^\dagger\otimes U'^\dagger}$ is its adjoint. ${\Lambda\otimes\Lambda'}$ is a diagonal matrix with nonnegative values. Thus ${A\otimes B}$ is positive.