Abstract:
Despite the apparent similarity between shared randomness and shared entanglement in the context of Communication Complexity, our understanding of the latter is not as good as of the former.  In particular, there is no known ``entanglement analogue'' for the famous theorem by Newman, saying that the number of shared random bits required for solving any communication problem can be at most logarithmic in the input length ( i.e., using more than O(log(n)) shared random bits would not reduce the complexity of an optimal solution).

We prove that the same is not true for entanglement.  We establish a wide range of tight (up to a logarithmic factor) entanglement vs. communication tradeoffs for relational problems.
The "low-end" is: for any t>2, reducing shared entanglement from log^t(n) to o(log^{t-1}(n)) qubits can increase the communication required for solving a problem almost exponentially, from O(log^t(n)) to \omega(\sqrt n).
The "high-end" is: for any \eps>0, reducing shared entanglement from n^{1-\eps}\log(n) to o(n^{1-\eps}) can increase the required communication from O(n^{1-\eps}\log(n)) to \omega(n^{1-\eps/2}).
The upper bounds are demonstrated via protocols which are exact and work in the simultaneous message passing model, while the lower bounds hold for bounded-error protocols, even in the more powerful model of 1-way communication.  Our protocols use shared EPR pairs while the lower bounds apply to any sort of prior entanglement.