Consider the set of states

$$\left|\beta_{z x}\right\rangle:=\frac{1}{\sqrt{2}}\left[|0 x\rangle+(-1)^{z}|1 \bar{x}\rangle\right],$$

where $x, z \in\{0,1\}$ and $\bar{x}=x \oplus 1$ (addition modulo 2 ).

(i) Show that

$$(H \otimes \mathbb{I}) \circ \mathrm{CX}\left|\beta_{z x}\right\rangle=|z x\rangle \quad \forall z, x \in\{0,1\},$$

where $H$ denotes the Hadamard gate and CX denotes the controlled- $X$ gate.

(ii) Show that for any $z, x \in\{0,1\}$,

$$\tag{*} \left(Z^{z} X^{x} \otimes \mathbb{I}\right)\left|\beta_{00}\right\rangle=\left|\beta_{z x}\right\rangle .$$

[Hint: For any unitary operator $U$, we have $(U \otimes \mathbb{I})\left|\beta_{00}\right\rangle=\left(\mathbb{I} \otimes U^{T}\right)\left|\beta_{00}\right\rangle$, where $U^{T}$ denotes the transpose of $U$ with respect to the computational basis.]

(iii) Suppose Alice and Bob initially share the state $\left|\beta_{00}\right\rangle$. Show using (*) how Alice can communicate two classical bits to Bob by sending him only a single qubit.