# Bit threads as noisy quantum channels

In literature on the holographic duality, recently a new way of looking at the Ryu-Takayanagi formula has been presented, where the minimal surface area has been replaced by the maximum number of "bit-threads" that can emanate from a boundary region.

If the entanglement entropy of the boundary region $$A$$ is $$S(A)$$, then the new paper says there will be $$S(A)$$ bit-threads emanating from the region and ending on the rest of the boundary. Each bit-thread carries one qubit of information.

So if we think of a bit-thread as a noiseless qubit channel, we have that $$S(A)$$ uses of the noiseless qubit channel are needed to carry information between the boundary region $$A$$ and the rest of the boundary. This, being essentially the statement of Schumacher's quantum data compression theorem, made me wonder: if for some reason we only had noisy qubit channels, in which case we know that we need more than $$S(A)$$ uses to carry information on the state of $$A$$ to the rest of the boundary, what effect on the metric of space would there be?

To summarize: Can we think of these bit-threads as noiseless qubit channels traversing the bulk? If so, what effect is there on the metric of space if these channels became noisy?

• If you allow me to speculate alongside you-- as you said, noisy qubit channels means more than $S(A)$ is needed, which effectively increases $S(A)$ for all regions. Which means all minimal surfaces have more area than the noiseless case. This is a rescaling, it can't change the symmetries of the metric, so I'd guess it simply increases $\ell_{AdS} \propto S(A)$. – Dwagg Jan 9 '20 at 23:39
• Possibly silly follow-up question: what if some of the bit-threads are noisy and some of them aren't, depending on which part of the bulk they travel in? – user120404 Feb 9 '20 at 16:48