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## "Tradeoffs for reliable quantum information storage in 2D systems", Sergey Bravyi, David Poulin, Barbara Terhal, 2010

*Reviewed February 24, 2024*

*Citation:* Bravyi, Sergey, David Poulin, and Barbara Terhal. "Tradeoffs for reliable quantum information storage in 2D systems." Physical review letters 104.5 (2010): 050503.

*Web:* https://arxiv.org/abs/0909.5200

*Tags:* Information-theory, No-go, Error-correcting-codes

This paper establishes a fundamental no-go theorem for quantum information.
Namely, it gives a dimension-dependent tradeoff between the amount of information
that can be in the codespace of a code and the distance of that code. Or, more simply,
a tradeoff between the density of information stored and the reliability of that information.
For stability in the thermodynamic limit one must have a code distance which tends to infinity.
In this special case, a corollary of the bound is that the density of encoded
information must tend to zero. This paper pairs nicely with the bounds in

> Bravyi, Sergey, and Barbara Terhal. "A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes." New Journal of Physics 11.4 (2009): 043029.

which upper bound the distance of a quantum code, in a dimension dependent way.
One of the conclusions from these works is that the toric code is in a sense quite optimal.

The proof strategy follows a similar pattern as to that of the computation
of topological entanglement entropy, and other similar results. One first breaks
up the region into several cleverly chosen subregions. Then, comparing entropy
between the different regions, one concludes the result. Implicit
in this proof strategy is the connection between code distance
and entropy.