Universality of EPR pairs in entanglement-assisted communication complexity, and the communication cost of state conversion

In this work we consider the role of entanglement assistance in quantum communication protocols, focusing, in particular, on whether the type of shared entangled state can affect the quantum communication complexity of a function. This question is interesting because in some other settings in quantum information, such as non-local games, or tasks that involve quantum communication between players and referee, or simulating bipartite unitaries or communication channels, maximally entangled states are known to be less useful as a resource than some partially entangled states. By contrast, we prove that the bounded-error entanglement-assisted quantum 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-approximated by a protocol using O(Q/+log(1/)/) qubits of communication and only EPR pairs as shared entanglement. 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 [17]. We leave open the question of how much the use of a shared maximally entangled state can reduce the quantum communication complexity of a function. Our second 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 simple and efficiently computable upper and lower bounds. Given two bipartite states |χi and |υi, we define a natural quantity, d∞(|χi, |υi), which we call the `∞ Earth Mover’s distance, and we show that the communication cost of converting between |χi and |υi is upper bounded by a constant multiple of d∞(|χi, |υi). Here d∞(|χi, |υi) may be informally described as the minimum over all transports between the log of the Schmidt coefficients of |χi and those of |υi, of the maximum distance that any amount of mass must be moved in that transport. A precise definition is given in the introduction. Furthermore, we prove a complementary lower bound on the cost of state conversion by the-Smoothed `∞-Earth Mover’s Distance, which is a natural smoothing of the `∞-Earth Mover’s Distance that we will define via a connection with optimal transport theory