Quantum shadow enumerators

In a recent paper [7], Shor and Laflamme define two “weight enumerators” for quantum error correcting codes, connected by a MacWilliams transform, and use them to give a linear-programming bound for quantum codes. We extend their work by introducing another enumerator, based on the classical theory of shadow codes, that tightens their bounds significantly. In particular, nearly all of the codes known to be optimal among additive quantum codes (codes derived from orthogonal geometry ([1])) can be shown to be optimal among all quantum codes. We also use the shadow machinery to extend a bound on self-dual additive codes ([6]) to general codes, obtaining as a consequence that any pure code of length n can correct at most ⌊ 6 ⌋ errors. Introduction One of the basic problems in the theory of quantum error correcting codes (henceforth abbreviated QECCs) is that of giving good upper bounds on the minimum distance of a QECC. The strongest technique to date for this problem is the linear programming bound introduced by Shor and Laflamme ([7]). Their bound involves the definition of two “weight enumerators” for a QECC; the two enumerators satisfy certain inequalities (e.g., nonnegative coefficients), and are related by MacWilliams identities. This allows linear programming to be applied, just as for classical error correcting codes ([4]). Linear programming was first applied to bounds for quantum codes in [1], which gave bounds only for codes of the type introduced in that paper (henceforth denoted “additive” codes). The linear programming bound given there essentially consists of three families of inequalities. Two of these were generalized to arbitrary quantum codes in [7]; the current paper generalizes the third. Consequently, in the table of upper bounds given in [1], all but 10 apply in general; it follows that nearly all of the codes known to be optimal among additive codes are optimal among QECCs in general. A quick note on terminology: We will be using the terms “pure” and “impure” in place of the somewhat cumbersome terms “nondegenerate” and “degenerate”; that is, a pure code is one in which all low weight errors act nontrivially on the codewords. Typeset by AMS-TEX 1