Entanglement theory with limited computational resources

The precise quantification of the limits to manipulating quantum resources lies at the core of quantum information theory. However, standard information-theoretic analyses do not consider the actual computational complexity involved in performing certain tasks. Here we address this issue within the realm of entanglement theory, finding that accounting for computational efficiency substantially changes what can be achieved using entangled resources. We consider two key figures of merit: the computational distillable entanglement and the computational entanglement cost. These measures quantify the optimal rates of entangled bits that can be extracted from or used to dilute many identical copies of n-qubit bipartite pure states, using computationally efficient local operations and classical communication. We demonstrate that computational entanglement measures diverge considerably from their information-theoretic counterparts. Whereas the information-theoretic distillable entanglement is determined by the von Neumann entropy of the reduced state, we show that the min-entropy governs the computationally efficient setting. On the other hand, computationally efficient entanglement dilution requires maximal consumption of entangled bits, even for nearly unentangled states. Furthermore, in the worst-case scenario, even when an efficient description of the state exists and is fully known, one gains no advantage over state-agnostic protocols. Our findings establish sample-complexity bounds for measuring and testing the von Neumann entropy, fundamental limitations on efficient state compression and efficient local tomography protocols.