  Extracting classical information from quantum systems is an essential step of
many quantum algorithms. However, this information could be corrupted as the
systems are prone to quantum noises, and its distortion under quantum dynamics
has not been adequately investigated. In this work, we introduce a systematic
framework to study how well we can retrieve information from noisy quantum
states. Given a noisy quantum channel, we fully characterize the range of
recoverable classical information. This condition allows a natural measure
quantifying the information recoverability of a channel. Moreover, we resolve
the minimum information retrieving cost, which, along with the corresponding
optimal protocol, is efficiently computable by semidefinite programming. As
applications, we establish the limits on the information retrieving cost for
practical quantum noises and employ the corresponding protocols to mitigate
errors in ground state energy estimation. Our work gives the first full
characterization of information recoverability of noisy quantum states from the
recoverable range to the recovering cost, revealing the ultimate limit of
probabilistic error cancellation.
