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.

翻译：从量子系统中提取经典信息是许多量子算法中的关键步骤。然而，由于系统易受量子噪声影响，这些信息可能被破坏，且其量子动力学下的畸变尚未得到充分研究。本文引入了一个系统性框架，用于研究从噪声量子态中恢复信息的能力。针对给定的噪声量子信道，我们完整刻画了可恢复经典信息的范围。该条件自然地定义了一种量化信道信息可恢复性的度量。此外，我们解决了最小信息恢复成本问题，该成本及其对应的最优协议可通过半正定规划高效计算。作为应用，我们确定了实际量子噪声下信息恢复成本的极限，并利用相应协议缓解基态能量估计中的误差。本文首次从可恢复范围到恢复成本完整刻画了噪声量子态的信息可恢复性，揭示了概率误差消除的终极极限。