The fidelity of the noisy quantum channel is a measure of how closely the output of a quantum channel resembles the input, and is therefore a measure of how well a quantum channel preserves information. If a pure state ψ ψ is sent into a quantum channel emerges as the state represented by density matrix p , the fidelity of transmission is defined as F = ⟨ ⟨ ψ ψ | p | ψ ψ ⟩ ⟩ .