No-Go Theorems for Quantum Resource Purification: New Approach and Channel Theory

It is recently shown that there exist fundamental limits to the accuracy and efficiency of purifying generic noisy resource states in any well-behaved quantum resource theory [Fang/Liu, Phys. Rev. Lett. 125, 060405 (2020)]. Here, we develop a novel and powerful method for analyzing the limitations of quantum resource purification, which not only leads to improved bounds that rule out exact purification for a broader range of noisy states and are tight in certain cases, but also allow us to establish a no-purification theory for quantum channel (dynamical) resources. More specifically, we employ the new method to derive universal bounds on the error and cost of transforming generic noisy channels (multiple instances can be used adaptively, in contrast to the state theory) to some unitary resource channel under any free channel-to-channel map. We address several cases of practical interest in more concrete terms, and discuss the connections and applications of our general results to quantum error correction, quantum Shannon theory, and quantum circuit synthesis.