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.

• Received 9 April 2002

DOI:https://doi.org/10.1103/PhysRevLett.90.087902