Theorem: Standard qubit attainment with sequential encoding

The encoding and readout satisfy, for every $x,j$,

Hence

The same encoding is obtained from sequentially supplied bits without retaining a classical copy of the earlier bit:

1. initialize the qubit in $|0⟩$;

2. apply

   $$R_y\left((-1{)}^{x_1}\frac{\pi }{4}\right),R_y(\theta )=e^{-\mathrm{i}\theta {\sigma }_y/2};$$

3. apply ${\sigma }_x$ if $x_2=1$.

Each operation uses only its current bit and the qubit.

Proof. Born's rule gives

The first rotation maps the initial $+Z$ Bloch vector to

Conjugation by ${\sigma }_x$ preserves the $X$ component and reverses the $Z$ component. The final vector is exactly that of (eq:encoding).