Arithmetic of qudit dimensions governs erasure correction on the Fano plane and stabilizer-state entanglement

Abstract

Qudit codes often inherit the parameters of qubit codes, but not necessarily their correctable erasures. We show that a code encoding one qudit of dimension D into seven such qudits that corrects every erasure Steane's code corrects exists exactly when D is a power of 2; for the qutrit color code, exactly when D is odd. This needs no stabilizer assumption: these codes are, up to local unitaries, stabilizer codes of qubits, respectively prime-power qudits. The two patterns differ in one line, which completes the Fano plane only where 1+1=0. As ideal secret sharing schemes, these codes exist exactly where classical ones do: entanglement does not help, unlike for Euler's 36 officers. The same arithmetic makes the entropy cones of qubit and odd-dimensional stabilizer states incomparable from six parties on. At five, the stabilizer cone is strictly smaller than the quantum linear rank cone, answering an open question.

arXiv commentv2: revised and extended version with a new title. Main text 20 pages, 3 figures; Supplementary Information (41 pages) appended. Data, certificates and code: https://doi.org/10.5281/zenodo.23841282