Communication cost of classically simulating a quantum channel with subsequent rank-1 projective measurement

A process of preparation, transmission and subsequent projective measurement of a qubit can be simulated by a classical model with only two bits of communication and some amount of shared randomness. However no model for n qubits with a finite amount of classical communication is known at present. A lower bound for the communication cost can provide useful hints for a generalization. It is known for example that the amount of communication must be greater than c 2n, where c ≃ 0.01. The proof uses a quite elaborate theorem of communication complexity. Using a mathematical conjecture known as the “double cap conjecture”, we strengthen this result by presenting a geometrical and extremely simple derivation of the lower bound 2n − 1. Only rank-1 projective measurements are involved in the derivation.