11 Quantum Singleton Bound
11.1 Overview
This chapter proves the quantum Singleton bound: for a stabilizer code with logical dimension \(k\) and distance \(d\) on \(n\) qudits over a prime field \(\mathbb {F}_p\),
The proof uses the symplectic vector space \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) with the standard symplectic form, and the key idea of erasure correctability.
11.2 Symplectic vector space
For \(u = (x,z), v = (x',z') \in V = \mathbb {F}_p^n \times \mathbb {F}_p^n\),
\(\omega \) is bilinear: additive and scalar-homogeneous in each argument.
\(\omega (u,v) = -\omega (v,u)\).
If \(\omega (u,v) = 0\) for all \(v\), then \(u = 0\).
\(\omega \) packaged as a .
11.3 Support, weight, and isotropic subspaces
The support \(\mathrm{supp}(v) \subseteq \mathrm{Fin}\, n\) consists of coordinates where either component of \(v\) is nonzero; the weight is \(\mathrm{wt}(v) = |\mathrm{supp}(v)|\).
\(\mathrm{wt}(v) \le n\) for all \(v \in V\).
For \(C \subseteq \mathrm{Fin}\, n\), the support submodule \(V_C = \{ v \in V \mid \mathrm{supp}(v) \subseteq C\} \).
\(\dim _{\mathbb {F}_p}(V_C) = 2|C|\).
For a submodule \(S \le V\), \(S^{\perp _\omega } = \{ v \in V \mid \forall s \in S,\; \omega (v,s) = 0\} \).
\(\dim (S^{\perp _\omega }) = 2n - \dim (S)\).
\(S\) is isotropic if \(S \le S^{\perp _\omega }\), i.e. \(\omega (u,v) = 0\) for all \(u,v \in S\).
11.4 Code parameters and erasure correctability
\(d(S) = \min \{ \mathrm{wt}(v) \mid v \in S^{\perp _\omega } \setminus S,\; \mathrm{wt}(v) \neq 0\} \) (or \(0\) if no such \(v\) exists).
\(d(S) \le n\).
An erasure set \(E \subseteq \mathrm{Fin}\, n\) is correctable for \(S\) if every \(v \in V_E \cap S^{\perp _\omega }\) is already in \(S\).
If \(|E| \lt d(S)\), then \(E\) is correctable.
\(k(S) = n - \dim (S)\).
11.5 Key lemma: two disjoint correctable sets
Let \(S\) be isotropic. If \(A,B \subseteq \mathrm{Fin}\, n\) are disjoint and both correctable, then
11.6 Quantum Singleton bound
For any isotropic submodule \(S \le V\),