Ex10 6

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 10.6}$

Geometry of Bloch Regions.

a) Show that the Bloch region, the set $S$ of mixed states of an $n$-qubit system, can be parametrized by $2^{2n} -1$ real parameters.

b) Show that $S$ is a convex set.

c) Show that the set of pure states of an $n$ qubit system can be parametrized by $2^{n+1} - 2$ real parameters, and therefore the set of density matrices corresponding to pure states can be parameterized in this way also.

d) Explain why for $n > 2$ the boundary of the set of mixed states must consist of more than just pure states.

e) Show that the extremal points, those that are not convex linear combinations of other points, are exactly the pure states.

f) Characterize the non-extremal states which are on the boundary of the Bloch region.

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