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We show that a qubit can be used to substitute for an arbitrarily largenumber of classical bits. We consider a physical system S interacting locallywith a classical field phi(x) as it travels directly from point A to point B.The field has the property that its integrated value is an integer multiple ofsome constant. The problem is to determine whether the integer is odd or even.This task can be performed perfectly if S is a qubit. On the otherhand, if S isa classical system then we show that it must carry an arbitrarily large amountof classical information. We identify the physical reason for such a hugequantum advantage, and show that it also implies a large difference between thesize of quantum and classical memories necessary for some computations. We alsopresent a simple proof that no finite amount of one-way classical communicationcan perfectly simulate the effect of quantum entanglement.Comment: 8 pages, LaTeX, no figures. v2: added result on entanglement  simulation with classical communication; v3: minor correction to main proof,  change of title, added referenc

Substituting a Qubit for an Arbitrarily Large Number of Classical Bits