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A two-party quantum communication process with classical inputs and outcomescan be simulated by replacing the quantum channel with a classical one. Theminimal amount of classical communication required to reproduce the statisticsof the quantum process is called its communication complexity. In the case ofmany instances simulated in parallel, the minimal communication cost perinstance is called the asymptotic communication complexity. Previously, wereduced the computation of the asymptotic communication complexity to a convexminimization problem. In most cases, the objective function does not have anexplicit analytic form, as the function is defined as the maximum over aninfinite set of convex functions. Therefore, the overall problem takes the formof a minimax problem and cannot directly be solved by standard optimizationmethods. In this paper, we introduce a simple algorithm to compute theasymptotic communication complexity. For some special cases with an analyticobjective function one can employ available convex-optimization libraries. Inthe tested cases our method turned out to be notably faster. Finally, using ourmethod we obtain 1.238 bits as a lower bound on the asymptotic communicationcomplexity of a noiseless quantum channel with the capacity of 1 qubit. Thisimproves the previous bound of 1.208 bits.

Simple algorithm for computing the communication complexity of quantum communication processes