# Entanglement-assisted hashing bound for asymmetric depolarizing channels

I reading the paper EXIT-Chart Aided Quantum Code Design Improves the Normalised Throughput of Realistic Quantum Devices, which proposes the use of QTCs in order to do quantum error correction for realistic quantum devices whose error model can be approximated by asymmetric depolarizing channels. Before presenting the error correcting methods, the authors present a discussion on the entanglement assisted hashing bound as a function of the asymmetry presented in the channel, which can be formulated as:

$$C_Q^{EA} = 1 + (1-p)\log_2(1-p)+p_x\log_2(p_x)+p_y\log_2(p_y)+p_z\log_2(p_z) + \theta E_{max}$$,

where $$p=p_x+p_y+p_z$$, $$\theta\in[0,1]$$ and $$E_{max}$$ is the maximum entanglement consumption.

At a first glance, the expression does not differ much from the EA hashing bounds of symmetric depolarizing channels, but a further observation of it presents the doubt I have about the topic. If one takes a look to the figures 5b, 5c or 5d of the paper, one can see that for channels presenting asymmetries, there are codes using entanglement-assistance with rates that do not present a noise limit $$p^*$$ for their performance.

I do not know if I am getting this correctly, but does that mean that we can construct codes that are error free? Does that mean that quantum information using such methods would be perfect?