Quantum and classical bounds for two-state overlaps
Ernesto F. Galvão, Daniel J. Brod
Abstract
Suppose we have $N$ quantum systems in unknown states $\left|{\ensuremath{\psi}}_{i}\right\ensuremath{\rangle}$, but we know the value of some pairwise overlaps ${\left|\ensuremath{\langle}{\ensuremath{\psi}}_{k}|{\ensuremath{\psi}}_{l}\ensuremath{\rangle}\right|}^{2}$. What can we say about the values of the unknown overlaps? We provide a complete answer to this problem for three pure states and two given overlaps and a way to obtain bounds for the general case. We discuss how the answer contrasts from that of a classical model featuring only coherence-free, diagonal states, and we describe three applications: basis-independent coherence witnesses, dimension witnesses, and characterization of multiphoton indistinguishability.