Quantum capacity

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.
Hashing bound for Pauli channels.
The LSD theorem states that the coherent information of a quantum channel is an achievable rate for reliable quantum communication. For a Pauli channel, the coherent information has a simple form and the proof that it is achievable is particularly simple as well. We prove the theorem for this special case by exploiting random stabilizer codes and correcting only the likely errors that the channel produces.
Theorem (hashing bound). There exists a stabilizer quantum error-correcting code that achieves the hashing limit formula_1 for a Pauli channel of the following form:
where formula_3 and formula_4 is the entropy of this probability vector.
Proof. Consider correcting only the typical errors. That is, consider defining the
typical set of errors as follows:
where formula_6 is some sequence consisting of the letters formula_7 and formula_8 is the probability
that an IID Pauli channel issues some tensor-product error formula_9. This typical set consists of the likely errors in the sense that
for all formula_11 and sufficiently large formula_12. The error-correcting
conditions for a stabilizer code formula_13 in this case are that formula_14 is a correctable set of errors if
for all error pairs formula_16 and formula_17 such that formula_18 where formula_19 is the normalizer of formula_13. Also, we consider the expectation of the error probability under a random choice of a stabilizer code.
Proceed as follows:
The first equality follows by definition—formula_22 is an indicator function equal to one if formula_16 is uncorrectable under formula_13 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 formula_32 is the normalizer of
formula_13. The first inequality follows by ignoring any potential degeneracy in the code—we consider an error uncorrectable if it lies in the normalizer formula_32 and the probability can only be larger because formula_35. 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 formula_36 not equal to the identity commuting with
the stabilizer operators of a random stabilizer can be upper bounded as follows:
formula_37
The reasoning here is that the random choice of a stabilizer code is equivalent to
fixing operators formula_38, ..., formula_39 and performing a uniformly random
Clifford unitary. The probability that a fixed operator commutes with
formula_40, ..., formula_41 is then just the number of
non-identity operators in the normalizer (formula_42) divided by the total number of non-identity operators (formula_43). After applying the above bound, we then exploit the following typicality bounds:
We conclude that as long as the rate formula_46, 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.