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We derive a relationship between two different notions of fidelity (entanglement fidelity and average fidelity) for a completely depolarizing quantum channel. This relationship gives rise to a quantum analog of the MacWilliams identities in classical coding theory. These identities relate the weight enumerator of a code to the one of its dual and, with linear programming techniques, provide a powerful tool to investigate the possible existence of codes. The same techniques can be adapted to the quantum case. We give examples of their power.
Shor et al. (Mon,) studied this question.