Quantum Error-Correcting Codes Need Not Completely Reveal the Error Syndrome

TL;DR

A code which does not find the complete error syndrome and can be used for reliable transmission of quantum information through channels which add more than one bit of entropy per transmitted bit.

Abstract

Quantum error-correcting codes so far proposed have not worked in the presence of noise which introduces more than one bit of entropy per qubit sent through a quantum channel, nor can any code which identifies the complete error syndrome. We describe a code which does not find the complete error syndrome and can be used for reliable transmission of quantum information through channels which add more than one bit of entropy per transmitted bit. In the case of the depolarizing channel our code can be used in a channel of fidelity .8096. The best existing code worked only down to .8107.

Figures (3)

• Figure 1: The first stage of the $k=4$ code, showing the overall purification view.
• Figure 2: Bob's view of the conditional $M$'s as the BXORs are done in sequence.
• Figure 3: The yield of distillable $| \Phi^+ \rangle$ states by purification or the fraction of transmitted bits which can be protected from noise as a function of channel fidelity for various values of $k$. Note that the curves are all in order from $k=1$ to $k=7$ along the right side of the graph.