Ex10 24

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 10.24}$

Maximal connectedness of cluster states.

a) Show by induction that for the qubits corresponding to the ends of the chain in the cluster state $\ket{\phi_n}$ for the $1\times n$ lattice, there is a sequence of single-qubit measurements that place these qubits in a Bell pair.

b) Show that for any two qubits $q_1$ and $q_2$ in a graph state, there exists a sequence of single-qubit measurements that leave these qubits as the end qubits of a cluster state of a $1\times r$ lattice. Conclude that graph
states are maximally connected.