In the theory of quantum communication, the quantum capacity is the highest rate at which quantum information can be communicated over many independent uses of a noisy quantum channel from a sender to a receiver. It is also equal to the highest rate at which entanglement can be generated over the channel, and forward classical communication cannot improve it. The quantum capacity theorem is important for the theory of quantum error correction, and more broadly for the theory of quantum computation. The theorem giving a lower bound on the quantum capacity of any channel is colloquially known as the LSD theorem, after the authors Lloyd,Shor, and Devetak who proved it with increasing standards of rigor.
Theorem (Hashing Bound). There exists a stabilizer quantum error-correcting code that achieves the hashing limit for a Pauli channel of the following form:
where and is the entropy of this probability vector.
Proof. We consider correcting only the typical errors. That is, consider defining the typical set of errors as follows:
where is some sequence consisting of the letters and is the probability that an IID Pauli channel issues some tensor-product error . This typical set consists of the likely errors in the sense that
for all and sufficiently large . The error-correcting conditions for a stabilizer code in this case are that is a correctable set of errors if
for all error pairs and such that where is the normalizer of . Also, we consider the expectation of the error probability under a random choice of a stabilizer code.
We proceed as follows:
The first equality follows by definition— is an indicator function equal to one if is uncorrectable under and equal to zero otherwise. The first inequality follows, since we correct only the typical errors because the atypical error set has negligible probability mass. The second equality follows by exchanging the expectation and the sum. The third equality follows because the expectation of an indicator function is the probability that the event it selects occurs. Continuing, we have
The first equality follows from the error-correcting conditions for a quantum stabilizer code, where is the normalizer of . The first inequality follows by ignoring any potential degeneracy in the code—we consider an error uncorrectable if it lies in the normalizer and the probability can only be larger because . The second equality follows by realizing that the probabilities for the existence criterion and the union of events are equivalent. The second inequality follows by applying the union bound. The third inequality follows from the fact that the probability for a fixed operator not equal to the identity commuting with the stabilizer operators of a random stabilizer can be upper bounded as follows:
The reasoning here is that the random choice of a stabilizer code is equivalent to fixing operators , ..., and performing a uniformly random Clifford unitary. The probability that a fixed operator commutes with , ..., is then just the number of non-identity operators in the normalizer () divided by the total number of non-identity operators (). After applying the above bound, we then exploit the following typicality bounds:
We conclude that as long as the rate , the expectation of the error probability becomes arbitrarily small, so that there exists at least one choice of a stabilizer code with the same bound on the error probability.