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Note: Cross-posted on Physics SE.


The quantum Singleton bound states that for an error-correcting code with $n$ physical qubits and $k$ encoded qubits, and some subsystem $R$ of $m$ qubits that can 'access the entire quantum code', it is necessary that $m \ge \frac{n+k}{2}$.

As I understand (from section 4.3 of Harlow's TASI notes), one way to state the condition for 'accessing the entire code' is the Knill-Laflamme condition, which is the following.

Let $\bar{R}$ denote the complement of $R$, $\mathscr{H}_\bar{R}$ be the space of operators supported on $\bar{R}$, and $P$ denote the projection matrix onto the code subspace $\mathscr{H}_{code}$. Then for any operator $O_{\bar{R}} \in \mathscr{H}_\bar{R}$, $P O_{\bar{R}} P = \lambda P$, where $\lambda$ is some constant that depends on the operator $O_{\bar{R}}$

This means that operator supported on the complement region $\bar{R}$ has no effect on measurements on $\mathscr{H}_{code}$.

I'm confused because this does not seem compatible with the toric code. Because, it can be shown that in the toric code (where the number of encoded bits $k=2$), the Knill-Laflamme condition is satisfied for $\bar{R}$ being any contractible region of qubits, i.e. for $R$ containing the union of two distinct nontrivial cycles on the torus. In this case on a torus of length $L \times L$, we will have the number of physical qubits being $n = L^2$ and the number of bits needed to access being $m = 2L$. So, it seems that the singleton bound is explicitly violated.

Where does the logic I'm presenting fail, and why should the Toric Code be compatible with this?

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