(a) Define what it means for a 2-qubit state $|\psi\rangle_{A B}$ of a composite quantum system $A B$ to be entangled.

Consider the 2-qubit state

$$|\alpha\rangle=\frac{1}{\sqrt{3}}(2|00\rangle-H \otimes H|11\rangle)$$

where $H$ is the Hadamard gate. From the definition of entanglement, show that $|\alpha\rangle$ is an entangled state.

(b) Alice and Bob are distantly separated in space. Initially they each hold one qubit of the 2-qubit entangled state

$$\left|\phi^{+}\right\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$$

They are able to perform local quantum operations (unitary gates and measurements) on quantum systems they hold. Alice wants to communicate two classical bits of information to Bob. Explain how she can achieve this (within their restricted operational resources) by sending him a single qubit.