A Symplectic Proof of the Quantum Singleton Bound

Abstract

We present a symplectic linear-algebraic proof of the Quantum Singleton Bound for stabiliser quantum error-correcting codes together with a Lean4 formalisation of the linear-algebraic argument. The proof is formulated in the language of finite-dimensional symplectic vector spaces modelling Pauli operators and relies on distance-based erasure correctability and the cleaning lemma. Using a dimension-counting argument within the symplectic stabiliser framework, we derive the bound $k+2(d-1)\le n$ for any $[[n,k,d]]$ stabiliser code. This approach isolates the algebraic structure underlying the bound and avoids the heavier analytic machinery that appears in entropy-based proofs, while remaining well-suited to formal verification.

6 pages