Our first result concerns an old question in quantum information theory: how much quantum communication is required to approximately convert one pure bipartite entangled state into another? We give a simple and efficiently computable bound in terms of the earth mover or Wasserstein distance. We show that the communication cost of converting between two pure states is bounded (up to a constant factor) by the Absolute Earth Mover distance between the distributions given by the negative logarithm of the Schmidt coefficients of each state. Here the Absolute Earth Mover distance may be informally described as the minimum, over all transports between the two distributions, of the maximum distance that any amount of mass must be moved in that transport. We also provide an analogous lower bound, showing that, for two different bipartite shared states, the amount of communication required to transform one into the other with error epsilon via a quantum communication protocol is lower bounded up to a constant by the epsilon-Smoothed Absolute Earth Mover's Distance between the two states.
Using the first result above we consider the nature of entanglement-assistance in quantum communication protocols. Maximally entangled states are known to be less useful than partially entangled states such as embezzling states for tasks that involve quantum communication between players and referee, for nonlocal games, and for simulating bipartite unitaries or communication channels. By contrast, we prove that the bounded-error one-way or two-way quantum entanglement-assisted communication complexity of a partial or total function cannot be improved by more than a constant factor by replacing maximally entangled states with arbitrary entangled states. In particular, we show that every quantum communication protocol using Q qubits of communication and arbitrary shared entanglement can be epsilon-approximated by a protocol using O(Q/epsilon) qubits of communication and only EPR pairs as shared entanglement. Note, in particular, that this conclusion is opposite of the common wisdom in the study of non-local games, where it has been shown, for example, that the I3322 inequality has a non-local strategy using a non-maximally entangled state, which surpasses the winning probability achievable by any strategy using a maximally entangled state of any dimension.
This leaves open the question of how much maximally entangled states can reduce the quantum communication complexity of functions.
Joint work with Aram Harrow.