Communication Complexity of Common Randomness Generation with Isotropic States

This paper addresses the problem of generating a common random string with min-entropy k using an unlimited supply of noisy EPR pairs or quantum isotropic states, with minimal communication between Alice and Bob. The paper considers two communication models -- one-way classical communication and one-way quantum communication, and derives upper bounds on the optimal common randomness rate for both models. We show that in the case of classical communication, quantum isotropic states have no advantage over noisy classical correlation, and that the optimal common randomness rate can be achieved by a classical strategy, in which Alice and Bob share classical $\rho$-correlated random variables. In the case of quantum communication, we demonstrate that the common randomness rate can be increased by using superdense coding on quantum isotropic states. Our main result is an upper bound on the optimal common randomness rate achievable by using one-way quantum communication. We also provide an application of this result, which yields upper bounds on the classical capacity of the noiseless quantum channel assisted by noisy entanglement.