Information Rates Achievable With Algebraic Codes on Quantum Discrete Memoryless Channels

Abstract

The highest information rate at which quantum error-correction schemes work reliably on a channel, which is called the quantum capacity, is proven to be lower bounded by the limit of the quantity termed coherent information maximized over the set of input density operators which are proportional to the projections onto the code spaces of symplectic stabilizer codes. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a completely positive linear map on a Hilbert space of finite dimension, and the codes that are proven to have the desired performance are symplectic stabilizer codes. On the depolarizing channel, this work's bound is actually the highest possible rate at which symplectic stabilizer codes work reliably. KeywordsCompletely positive (CP) linear maps, fidelity, symplectic geometry, the method of types, quantum capacity, quantum error-correcting codes.

Notes on the Fidelity of Symplectic Quantum Error-Correcting Codes

“…A selfcontained exposition of symplectic codes, as well as proofs of the lemmas in this section, can be found in Appendix A, which is a recast of Section III of Ref. 19, except for the proof of Theorem 3.…”

“…19) as x runs through F 2n d (and N −1 x R (0,t) N x = R (s,t) as can be checked easily), this means that the entanglement fidelity F e (π C (s) , R (s,t) A) of the symplectic code averaged over all code subspaces C (s) , s ∈ F n−k d , is given by P A (x(t) + L), as promised.…”

“…applies to this lemma if we replace X thereof by X m . Alternatively, the proof in Ref 19. works if we assume the inner code of the concatenated code thereof to be the identity map.Lemma 9.…”

Error Correction of Quantum Reference Frame Information

“…The construction presented in this section provides codes in which the Hilbert spaces can be exponentially large in |G|. However, it is known that in many cases random codes give near optimal error correcting schemes with good parameters [27][28][29][30][31]. In appendix B, we show that choosing a random covariant isometry yields approximate error correcting codes for which the dimension of each mode is just |G|.…”