Average pure-qubit fidelity of a bit-flip channel ($⟨F^2⟩=1-2p/3$)

For pure input $|\psi ⟩$, squared quantum fidelity after the bit-flip channel is $(1-p)+p|⟨\psi |X|\psi ⟩{|}^2$. The last factor is the square of the input's Bloch $x$ component. Rotational symmetry of the uniform pure-qubit average gives $⟨r_x^2⟩=1/3$, because $r_x^2+r_y^2+r_z^2=1$. Hence the average squared fidelity is $1-2p/3$. Averaging an unsquared fidelity or over a different ensemble gives a different quantity.

// Target: quantum-mechanics.bigb