Exa 9

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ A.9}$

Show that if a state $\rho \in V\otimes W$ is uncorrelated with respect to the tensor decomposition $V\otimes W$ then it is the tensor product of its partial traces with respect to $V$ and $W$.