Quantum conditional query complexity

We define and study a new type of quantum oracle, the quantum conditional oracle, which provides oracle access to the conditional probabilities associated with an underlying distribution. Amongst other properties, we (a) obtain speed-ups over the best known quantum algorithms for identity testing, equivalence testing and uniformity testing of probability distributions; (b) study the power of these oracles for testing properties of boolean functions, and obtain an algorithm for checking whether an n-input m-output boolean function is balanced or ε-far from balanced; and (c) give a sub-linear algorithm, requiring tildeO(n3/4/ε) queries, for testing whether an n-dimensional quantum state is maximally mixed or not.