Exercise 3.13 Discussion

$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 3.13}$

a) Show that the four-qubit state $\ket\psi = \frac{1}{2}(\ket{00} + \ket{11} + \ket{22} + \ket{33})$ of Example 3.2.3 is entangled with respect to the decomposition into two two-qubit subsystems consisting of the first and second qubits and the third and fourth qubits.
b) For the four decompositions into two subsystems consisting of one and three qubits, say whether $\ket\psi$ is entangled or unentangled with respect to each of these decompositions.