Fully Quantum Computational Entropies

Abstract: Quantum information theory, particularly its entropic formulations, has made remarkable strides in characterizing quantum systems and tasks. However, a critical dimension remains underexplored: computational efficiency. While classical computational entropies integrate complexity and feasibility into information measures, analogous concepts have yet to be rigorously developed in the quantum setting. In this work, we lay the basis for a new quantum computational information theory. Such a theory will allow studying efficient -- thus relevant in practice -- manipulation of quantum information. We introduce two innovative entropies: quantum computational min- and max-entropies (along with their smooth variants). Our quantum computational min-entropy is both the fully quantum counterpart of the classical unpredictability entropy, as well as the computational parallel to the quantum min-entropy. We establish a series of essential properties for this new entropy, including data processing and a chain rule. The quantum computational max-entropy is defined via a duality relation and gains operational meaning through an alternative formulation that we derive. Notably, it captures the efficiency of entanglement distillation with the environment, restricted to local quantum circuits of bounded size. With the introduction of our computational entropies and their study, this work marks a critical step toward a quantum information theory that incorporates computational elements.