Quantum Error Correction Beyond Completely Positive Maps

We present a generalized theory of quantum error correction (QEC) that applies to any linear map, in particular maps that are not completely positive (CP). This theory of ``linear quantum error correction'' is applicable in cases where the standard and restrictive assumption of a factorized initial system-bath state does not apply. For linear maps that preserve positivity and/or Hermiticity, we find that standard QEC based on CP recovery maps still applies. Other linear maps generally require non-CP recovery operations. We illustrate our findings with examples of QEC for non-CP maps.