Quantum Information Complexity

We define a new notion of information cost for quantum protocols, and a corresponding notion of quantum information complexity for bipartite quantum tasks. These are the fully quantum generalizations of the analogous quantities for bipartite classical tasks that have found many applications recently, in particular for proving communication complexity lower bounds and direct sum theorems. Finding such a quantum generalization of information complexity was one of the open problems recently raised by Braverman (STOC'12). Previous attempts have been made to define such a quantity for quantum protocols, with particular applications in mind; our notion differs from these in many respects. First, it directly provides a lower bound on the quantum communication cost, independent of the number of rounds of the underlying protocol. Secondly, we provide an operational interpretation for quantum information complexity: we show that it is exactly equal to the amortized quantum communication complexity of a bipartite task on a given input. This generalizes a result of Braverman and Rao (FOCS'11) to quantum protocols. Along the way to prove this result, we even strengthens the classical result in a bounded round scenario, and also prove important structural properties of quantum information cost and complexity. We prove that using this definition leads to the first general direct sum theorem for bounded round quantum communication complexity. Previous direct sum results in quantum communication complexity either held for some particular classes of functions, or were general but only held for single-round protocols. We also discuss potential applications of the new quantities to obtain lower bounds on quantum communication complexity.