Abstract:
Alice and Bob want to know if two strings of length $n$ are almost equal. That is, do they differ on at most $a$ bits? Let $0\le a\le n-1$. We show that any deterministic protocol, as well as any error-free quantum protocol ($C^*$ version), for this problem requires at least $n-2$ bits of communication. We show the same bounds for the problem of determining if two strings differ in exactly $a$ bits. We also prove a lower bound of $n/2-1$ for error-free $Q^*$ quantum protocols. Our results are obtained by lower-bounding the ranks of the appropriate matrices.