Substituting a qubit for an arbitrarily large number of classical bits

We show that a qubit can be used to substitute for a classical analog system requiring an arbitrarily large number of classical bits to represent digitally. Let a physical system S interact locally with a classical field varphi(x) as S travels directly from point A to point B. Our task is to use S to answer a simple yes/no question about varphi(x). If S is a qubit, the task can be done perfectly. We show that any classical system S must encode an arbitrarily large number of classical bits to solve the same task. This result implies a large quantum advantage in the memory size necessary for some computations. We also show that no finite amount of one-way classical communication can perfectly simulate the effect of quantum entanglement.