Exercises for Appendix A

Exercises are from QUANTUM COMPUTING: A GENTLE INTRODUCTION, by Eleanor Rieffel and Wolfgang Polak, published by The MIT Press.

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

Show that an independent joint distribution is the tensor product of its marginals.

$\mathbf{Exercise\ A.2}$

Show that a general distribution cannot be reconstructed from its marginals. Exhibit three distinct distributions with the same marginals.

$\mathbf{Exercise\ A.3}$

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.

$\mathbf{Exercise\ A.4}$

Show that the probability distributions $\mu$ whose corresponding operators $M_\mu$ are projectors are exactly the pure distributions.

$\mathbf{Exercise\ A.5}$

For each of the states $\ket 0$, $\ket -$, and $\ket \i = \frac{1}{\sqrt 2}(\ket 0 + \i\ket 1)$, give the matrix for the corresponding density operator in the standard basis, and write each of these states as a probability distribution over pure states. For which of these states is this distribution unique?

$\mathbf{Exercise\ A.6}$

a) Give an example of three density operators no two of which can be simultaneously diagonalized in that there does not exist a basis with respect to which both are diagonal.

b) Show that if a set of density operators commute then they can be simultaneously diagonalized.

$\mathbf{Exercise\ A.7}$

Show that the binary operator $f\otimes g: (a,b) \mapsto f(a)g(b)$ for $f\in {\bf R}^A$ and $g\in {\bf R}^B$ satisfies the relations defining a tensor product structure on ${\bf R}^{A\times B}$ given in Section 3.1.2.

$\mathbf{Exercise\ A.8}$

Show that a separable pure state must be uncorrelated.

$\mathbf{Exercise\ A.9}$
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 $W$.