  The robustness of quantum memory against physical noises is measured by two
methods: the exact and approximate quantum error correction (QEC) conditions
for error recoverability, and the decoder-dependent error threshold which
assesses if the logical error rate diminishes with system size. Here we unravel
their relations and propose a unified framework to extract an intrinsic error
threshold from the approximate QEC condition, which could upper bound other
decoder-dependent error thresholds. Our proof establishes that relative
entropy, effectively measuring deviations from exact QEC conditions, serves as
the order parameter delineating the transition from asymptotic recoverability
to unrecoverability. Consequently, we establish a unified framework for
determining the error threshold across both exact and approximate QEC codes,
addressing errors originating from noise channels as well as those from code
space imperfections. This result sharpens our comprehension of error thresholds
across diverse QEC codes and error models.
