# Is the quantum Singleton bound compatible with the Toric Code?

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?