ABSTRACT
In this final chapter we present a number of bounds on the minimum dis tance of a quantum error correcting code (QECC). Sections 7.1-7.3 derive the quantum Hamming, Gilbert-Varshamov, and Singleton bounds, respectively. The corresponding classical bounds were derived in Section 1.2.5. Section 7.4 introduces quantum generalizations of the classical MacWilliams identities, and of weight and shadow enumerators [1,2]. With these tools in hand, the problem of finding an optimal upper bound on the code distance is reduced to solving a problem in linear programming [3-5]. Finally, we close in Section 7.5 by showing how QECCs can be connected to the process of entanglement pu rification [6,7]. This final section allows our study of QECCs to segue into the developing field of quantum information theory [8,9], and provides a good place to bring the discussion in this book to a close.
