Quantum fingerprinting

Classical fingerprinting associates with each string a shorter string (its fingerprint), such that any two distinct strings can be distinguished with small error by comparing their fingerprints alone. The fingerprints cannot be made exponentially smaller than the original strings unless the parties preparing the fingerprints have access to correlated random sources. We show that fingerprints consisting of quantum information can be made exponentially smaller than the original strings without any correlations or entanglement between the parties. This implies an exponential quantum/classical gap for the equality problem in the simultaneous message passing model of communication complexity.