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Entanglement assistance is known to reduce the quantum communicationcomplexity of evaluating functions with distributed inputs. But does the typeof entanglement matter, or are EPR pairs always sufficient? This is a naturalquestion because in several other settings maximally entangled states are knownto be less useful as a resource than some partially entangled state. Theseinclude non-local games, tasks with quantum communication between players andreferee, and simulating bipartite unitaries or communication channels. Bycontrast, we prove that the bounded-error entanglement-assisted quantumcommunication complexity of a function cannot be improved by more than aconstant factor by replacing maximally entangled states with arbitraryentangled states. In particular, we show that every quantum communicationprotocol using $Q$ qubits of communication and arbitrary shared entanglementcan be $\epsilon$-approximated by a protocol using$O(Q/\epsilon+\log(1/\epsilon)/\epsilon)$ qubits of communication and only EPRpairs as shared entanglement.  Our second result concerns an old question in quantum information theory: Howmuch quantum communication is required to approximately convert one purebipartite entangled state into another? We show that the communication cost ofconverting between two bipartite quantum states is upper bounded, up to aconstant multiplicative factor, by a natural and efficiently computablequantity which we call the $\ell_{\infty}$-Earth Mover's Distance (EMD) betweenthose two states. Furthermore, we prove a complementary lower bound on the costof state conversion by the $\epsilon$-smoothed $\ell_{\infty}$-EMD, which is anatural smoothing of the $\ell_{\infty}$-EMD that we will define via aconnection with optimal transport theory.

Universality of EPR pairs in Entanglement-Assisted Communication Complexity, and the Communication Cost of State Conversion