Quantum Error Detection, I: Statement of the Problem; II: Bounds

This 2-part paper is devoted to the problem of error detection with quantum codes. In the first part, we examine possible problem settings for quantum error detection. Our goal is to derive a functional that describes the probability of undetected error under natural physical assumptions concerning transmission with error detection with quantum codes. We discuss possible transmission protocols with stabilizer and unrestricted quantum codes. The set of results proved in Part I shows that in all the cases considered the average probability of undetected error for a given code is essentially given by one and the same function of its weight enumerators. This enables us to give a consistent definition of the undetected error event. In the final section of this part, we examine polynomial invariants of quantum codes and show that Rains's "unitary weight enumerators" are known for classical codes under the name of binomial moments of the distance distribution. As in the classical situation, they provide an alternative expression for the probability of undetected error. In Part II, 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.