Exa 3

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ A.3}$

a) Show that the tensor product of a pure distribution is pure.

b) Show that any distribution is a linear combination of pure distributions. Conclude that the set of distributions on a finite set $A$ is convex.

c) Show that any pure distribution on a joint system $A\times B$ is uncorrelated.

d) A distribution is said to be extremal if it cannot be written as a linear combination of other distributions. Show that the extremal distributions are exactly the pure distributions.