Ex4 15

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 4.15}$

a) Show that the operator $B$ of Example 4.3.4 is of the form $Q\otimes I$ where $Q$ is a $(2\times 2)$-Hermitian operator.

b) Show that any operator of the form $Q\otimes I$, where $Q$ is a $(2\times 2)$-Hermitian operator and $I$ is the $(2\times 2)$-identity operator, specifies a measurement of a two-qubit system. Describe the subspace decomposition associated with such an operator.

c) Describe the subspace decomposition associated with an operator of the form $I\otimes Q$ where $Q$ is a $(2\times 2)$-Hermitian operator and $I$ is the $(2\times 2)$-identity operator, and give a high-level description of such measurements.