Universal limitations on quantum key distribution over a network

The possibility to achieve secure communication among trusted parties by means of the quantum entanglement is intriguing both from a fundamental and an application purpose. In this work, we show that any state (after distillation) from which a quantum secret key can be obtained by local measurements has to be genuinely multipartite entangled. We introduce the most general form of memoryless network quantum channel: quantum multiplex channels. We define and determine asymptotic and non-asymptotic LOCC assisted conference key agreement capacities for quantum multiplex channels and provide various strong and weak converse bounds in terms of the divergence based entanglement measures of the quantum multiplex channels. The structure of our protocol manifested by an adaptive strategy of secret key and entanglement (GHZ state) distillation over an arbitrary multiplex quantum channel is generic. In particular, it provides a universal framework to study the performance of quantum key repeaters and - for the first time - of the MDI-QKD setups of channels. For teleportation-covariant multiplex quantum channels, which are channels with certain symmetries, we get upper bounds on the secret key agreement capacities in terms of the entanglement measures of their Choi states. For some network prototypes of practical relevance, we evaluate upper bounds on the conference key agreement capacities and MDI-QKD capacities. Upper bounds on the LOCC-assisted conference key agreement rates are also upper bounds on the distillation rates of GHZ states, a class of genuinely entangled pure states. We also obtain bounds on the rates at which conference key and GHZ states can be distilled from a finite number of copies of an arbitrary multipartite quantum state. Using our bounds, in particular cases, we are able to determine the capacities for quantum key distribution channels and rates of GHZ-state distillation.