Quantum capacity explained

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,^[1] Shor,^[2] and Devetak^[3] who proved it with increasing standards of rigor.^[4]

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

R =1- H\left( p \right) for a Pauli channel of the following form:$$\backslash rho\; \backslash mapsto\; p\_\backslash rho+p\_X\backslash rho\; X+p\_Y\backslash rho\; Y+p\_Z\backslash rho\; Z,$$where p = \left(p _{I} ,p _{X} ,p _{Y} ,p _{Z} \right) and H\left( p \right) is the entropy of this probability vector.

Proof. Consider correcting only the typical errors. That is, consider defining thetypical set of errors as follows:$$T\_^\backslash equiv\backslash left\backslash ,$$where

a ^{n} is some sequence consisting of the letters \left\{I,X,Y,Z\right\} and \Pr\left\{E _{a ^{n}} \right\} is the probability that an IID Pauli channel issues some tensor-product error E _{a ^{n}} \equiv E _{a _{1}} ⊗ … ⊗ E _{a _{n}} . This typical set consists of the likely errors in the sense that$$\backslash sum\_\backslash Pr\backslash left\backslash \; \backslash geq\; 1-\backslash epsilon,$$for all \epsilon> 0 and sufficiently large n . The error-correctingconditions^[5] for a stabilizer code l{S} in this case are that \{E _{a ^{n}} :a ^{n} \in T ^{p ^{n}} _{\delta} \} is a correctable set of errors if

$$E\_^E\_\backslash notin\; N\backslash left(\backslash mathcal\backslash right)\; \backslash backslash\; \backslash mathcal,$$for all error pairs

E _{a ^{n}} and E _{b ^{n}} such that a ^{n} ,b ^{n} \in T ^{p ^{n}} _{\delta} where N( l{S}) is the normalizer of l{S} . 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—

l{I} is an indicator function equal to one if E _{a ^{n}} is uncorrectable under l{S} 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:$$=\backslash sum\_\backslash Pr\backslash left\backslash \backslash Pr\_\backslash left\backslash$$

\leq\sum _{a ^{n} \in T ^{A ^{n}} _{\delta}} \Pr\left\{ E _{a ^{n}} \right\} \Pr _{l{S} }\left\ = \sum _{a ^{n} \in T ^{p ^{n}} _{\delta}} \Pr\left\{ E _{a ^{n}} \right\} \Pr _{l{S} }\left\ \leq\sum _{a ^{n} ,b ^{n} \in T ^{p ^{n}} _{\delta} , b ^{n} ≠ a ^{n}} \Pr\left\{ E _{a ^{n}} \right\} \Pr _{l{S} }\left\ \leq\sum _{a ^{n} ,b ^{n} \in T ^{p ^{n}} _{\delta} , b ^{n} ≠ a ^{n}} \Pr\left\{ E _{a ^{n}} \right\} 2 ^{-\left(} \leq 2 ^{2n\left[} 2 ^{-n\left[ H\left(} 2 ^{-\left(} =2 ^{-n\left[} . The first equality follows from the error-correcting conditions for a quantum stabilizer code, where N\left( l{S}\right) is the normalizer ofl{S} . The first inequality follows by ignoring any potential degeneracy in the code—we consider an error uncorrectable if it lies in the normalizer N\left( l{S}\right) and the probability can only be larger because N\left( l{S}\right) \backslash l{S}\in N\left( l{S}\right) . 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 E ^{\dagger} _{a ^{n}} E _{b ^{n}} not equal to the identity commuting withthe stabilizer operators of a random stabilizer can be upper bounded as follows:\Pr _{l{S} }\left\ =\frac\leq2^.The reasoning here is that the random choice of a stabilizer code is equivalent tofixing operators Z _{1} , ..., Z _{n-k} and performing a uniformly randomClifford unitary. The probability that a fixed operator commutes with\overline{Z} _{1} , ..., \overline{Z} _{n-k} is then just the number ofnon-identity operators in the normalizer (2 ^{n+k} -1 ) divided by the total number of non-identity operators (2 ^{2n} -1 ). After applying the above bound, we then exploit the following typicality bounds:$$\backslash forall\; a^\; \backslash in\; T\_^:\backslash Pr\backslash left\backslash \; \backslash leq2^,$$$$\backslash left\backslash vert\; T\_^\backslash right\backslash vert\; \backslash leq2^.$$We conclude that as long as the rate k/n =1- H\left( p \right) -4 \delta , 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.

See also

• Quantum computing

References

1. Physical Review A . 55 . 3 . 1613–1622 . 1997 . Seth Lloyd . Capacity of the noisy quantum channel . 10.1103/PhysRevA.55.1613 . quant-ph/9604015. 1997PhRvA..55.1613L . 5555850 .
2. Lecture Notes, MSRI Workshop on Quantum Computation . 2002 . Peter Shor . The quantum channel capacity and coherent information .
3. IEEE Transactions on Information Theory . 51 . 44–55 . 2005 . Igor Devetak . The private classical capacity and quantum capacity of a quantum channel . 10.1109/TIT.2004.839515 . quant-ph/0304127. 12246393 .
4. Book: Wilde, Mark M. . Quantum information theory . 2017 . 978-1-316-80997-6 . 2nd . Cambridge, UK . 972292559.
5. .