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The fidelity between two quantum states and , expressed as density matrices, is commonly defined as: [1] [2] (,) = ().The square roots in this expression are well-defined because both and are positive semidefinite matrices, and the square root of a positive semidefinite matrix is defined via the spectral theorem.
Formally, this correcting procedure corresponds to the application of the following map to the output of the channel: = + =. Note that, while this procedure perfectly corrects the output when zero or one flips are introduced by the channel, if more than one qubit is flipped then the output is not properly corrected.
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