A Symplectic Proof of the Quantum Singleton Bound
For quantum error correction researchers, this provides a cleaner, algebraically-focused proof of a known bound that is amenable to formal verification, but the result itself is not new.
The paper presents a symplectic linear-algebraic proof of the Quantum Singleton Bound for stabiliser codes, deriving $k + 2(d-1) \\le n$ for any $[[n, k, d]]$ code, and provides a Lean4 formalisation of the argument.
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.