Lower bound for the quantum capacity of a discrete memoryless quantum channel

Abstract

We generalize the random coding argument of stabilizer codes and derive a lower bound on the quantum capacity of an arbitrary discrete memoryless quantum channel. For the depolarizing channel, our lower bound coincides with that obtained by Bennett et al. We also slightly improve the quantum Gilbert-Varshamov bound for general stabilizer codes, and establish an analogue of the quantum GilbertVarshamov bound for linear stabilizer codes. Our proof is restricted to the binary quantum channels, but its extension of to l-adic channels is straightforward.

Noise tolerance of the BB84 protocol with random privacy amplification

“…By this procedure, we can securely share a key against general eavesdropping attacks with a linear code whose decoding error probability as a part of a CSS code is small over a BSC (Binary Symmetric Channel) [7,Lemmas 2,3].…”

“…In this section, we will prove that if we fix a rate R lower than 1 − H(p0)+H(p1) 2 and choose a code C ⊥ 2 of rate R at random with the condition (a ), with high probability the condition (c) will be satisfied. Some ideas used in the proof are borrowed from [3], [4].…”

Lower bounds on the quantum capacity and highest error exponent of general memoryless channels

“…Matsumoto and Uyematsu [27] proved Lemma 6 with A(x) replaced by {L ∈ A | x ∈ L ⊥ \ L} using the Witt lemma explicitly [43], [44]. The present definition of A(x) makes the argument easier.…”

Asymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH Codes

“…The asymptotic GV bound. From Theorem 5 and [37], it can be deduced the following asymptotic GV bound.…”

“…Sufficient (respectively, necessary) conditions for existence of (sometimes pure) quantum codes are given by the Gilbert-Varshamov bounds [13,17,28,37] (respectively, quantum singleton or Hamming bounds [4,24,28,39]).…”