Deutsch algorithm with pure dephasing ($P_{\mathrm{correct}}=\frac{1}{2}(1+v\cos \alpha )$)

If the input qubit of the Deutsch algorithm acquires conditional environment states with overlap $⟨e_0|e_1⟩=v{e}^{i\alpha }$ between its two Hadamard gates, its fixed zero-for-constant, one-for-balanced rule succeeds with probability $(1+v\cos \alpha )/2$. A known phase can be compensated before the last Hadamard gate, raising the success probability to $(1+v)/2$. An orthogonal pair of environment states gives $v=0$ and destroys all information in the input qubit about the answer; unit overlap modulus preserves coherence up to a known phase.