Quantum bit-flip channel (${\mathcal{E}}_p(\rho )=(1-p)\rho +pX\rho X$)

A bit-flip channel applies the Pauli X gate with probability $p$ and does nothing otherwise. If the classical error outcome is not recorded, its output is the convex mixture ${\mathcal{E}}_p(\rho )=(1-p)\rho +pX\rho X$, not a coherent superposition of the two alternatives.

// Target: quantum-information-theory.bigb

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