Quantum error detection II: Bounds

In Part I of this paper we formulated the problem of error detection with quantum codes on the depolarizing channel and gave an expression for the probability of undetected error via the weight enumerators of the code. In this part we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent, The lower (existence) bound is proved for stabilizer codes by a counting argument for classical self-orthogonal quaternary codes. Upper bounds are proved by linear programming. First we formulate two linear programming problems that are convenient for the analysis of specific short codes, Next we give a relaxed formulation of the problem in terms of optimization on the cone of polynomials in the Krawtchouk basis. We present two general solutions of the problem, Together they give an upper bound on the exponent of undetected error, The upper and lower asymptotic bounds coincide for a cel tain interval of code rates close to 1.