Distance of a quantum error-correcting code ($[[n,k,d]]$)

An $[[n,k,d]]$ quantum error correction code is a $2^k$-dimensional subspace of $n$ qubits. Its distance $d$ is the minimum weight of a Pauli operator $E$ for which $PEP$ is not a scalar multiple of the code projector $P$. Thus every operator supported on fewer than $d$ qubits has scalar compression to the code, by expansion in the Pauli basis. This definition includes degenerate codes; a low-weight operator acting as a scalar does not reduce the distance.

The guaranteed radius is the largest integer $t$ for which all unknown-location Pauli operators of weight at most $t$ can be corrected together. Pairwise products have weight at most $2t$, so $2t<d$ suffices by the Knill--Laflamme condition. Conversely, split a minimum-weight undetectable Pauli operator between two subsets of its support. If $d\le 2t$, this gives two errors of weight at most $t$ whose pairwise product violates that condition. Consequently $d$ is either $2t+1$ or $2t+2$; parity cannot be recovered from $t$ alone.

For a code carrying $k>0$ logical qubits, two disjoint subsets $A,B$ of $d-1$ physical qubits are each correctable erasures. Purify the maximally mixed logical input using a reference $R$, and call the remaining physical subsystem $C$. Erasure decoupling gives $S(RA)=S(R)+S(A)$ and $S(RB)=S(R)+S(B)$. Purity and Subadditivity of Von Neumann entropy imply $2S(R)\le 2S(C)$, hence $k\le n-2(d-1)$. If two such disjoint subsets would not fit, partition the entire physical system into two correctable subsets; purity and decoupling then force $S(R)=0$, a contradiction. Thus the partition used in the proof is justified independently of the bound.

Errors on a known subset of $m$ qubits are correctable by a distance-$d$ code whenever $m<d$. Every product $E_a^{†}{E}_b$ of errors supported on that subset still has support in the same subset, so its compression is scalar. The Knill--Laflamme condition therefore supplies one recovery for every channel on that subset. Equivalently, erasing the subset leaks no logical information to it: if a reference is maximally entangled with the encoded logical system, its joint density operator with the erased subsystem is a product state.