Multi-User Distillation of Common Randomness and Entanglement From Quantum States

We construct new protocols for the tasks of converting noisy multipartite quantum correlations into noiseless classical and quantum ones using local operations and classical communication (LOCC). The former task is known as common randomness (CR) distillation, and it requires offsetting the amount of classical communication against the randomness created. We obtain a new lower bound on the “distillable common randomness,” an operational measure of the total genuine (classical) correlations in a quantum state. Our proof relies on a generalization of communication for omniscience (CO) [Csiszár and Narayan, IEEE Trans. Inf. Theory 50:3047-3061, 2004], and our contribution here is a novel simultaneous decoder for the compression of correlated classical sources by random binning with quantum side information at the decoder. For the latter task, we derive two lower bounds on the rate at which Greenberger-Horne-Zeilinger (GHZ) states can be asymptotically distilled from any given pure state under LOCC. Our approach consists in “making coherent” the proposed CR distillation protocols and recycling of resources [Devetak
et al.
IEEE Trans. Inf. Theory 54(10):4587-4618, 2008]. The first lower bound is identical to a recent result by Vrana and Christandl [IEEE Trans. Inf. Theory 65(9):5945-5958, 2019], which is based on a combinatorial method to achieve the same rate. Our second lower bound generalises and improves upon this result, and unifies a number of other known lower bounds on GHZ distillation.