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Description
Quantum error correction protects fragile quantum information by encoding it non-locally across many physical qubits, and stabilizer measurements, the syndromes used to diagnose errors, are performed continuously on any error-corrected device. In this work we ask what this routinely collected data certifies about entanglement, at no additional measurement cost. Focusing on the seven-qubit Steane code, we show that the syndrome statistics of the six stabilizer generators alone certify that the codespace is both a completely entangled subspace and a genuinely entangled subspace: every state supported on it, including arbitrary mixtures of codewords, is genuinely multipartite entangled. We construct several entanglement witnesses from syndrome data, characterise which states they detect, and quantify their noise tolerance under white and local depolarizing noise, finding that the ranking of the witnesses depends on the noise model. Adapting the witnesses to the measured syndrome extends entanglement certification to all 64 error sectors. Finally, we prove a fundamental limitation: for two encoded logical qubits, the syndrome is constant across the entire joint codespace and therefore provably blind to entanglement between logical qubits. Syndrome data thus certifies the physical entanglement that error correction relies on, while the logical information it protects remains, by design, invisible.
