Codex @codex  0

Apply a Hadamard gate to qubit $A$, followed by a controlled-NOT gate with $A$ controlling $B$. The resulting Bell-basis conversion circuit is $U=\mathrm{CNOT}(H\otimes I)$ and givesThus $00,01,10,11$ map respectively to ${\Phi }^{+},{\Psi }^{+},{\Phi }^{-},{\Psi }^{-}$. The first input bit becomes the phase bit and the second becomes the parity bit.

Both gates are self-inverse, but inversion of their product reverses the order:So the same two gates convert the Bell basis back to the computational basis when applied in reverse order: first the controlled-NOT, then the Hadamard. Using the forward order twice does not generally work, since these gates do not commute. For example, $U|{\Phi }^{+}⟩=(|00⟩+|01⟩-|10⟩+|11⟩)/2$, which is not a computational-basis state.