Quantum error detection

We show that the probability of undetected error for a quantum code on the depolarizing channel can be expressed via code's weight enumerators. We prove that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive a lower bound on this exponent. To derive upper bounds we formulate a linear programming problem and present two feasible programs for it. The asymptotic upper and lower bounds coincide in a certain interval of code rates close to 1.