Quantum and classical bounds for two-state overlaps

Abstract

Suppose we have $N$ quantum systems in unknown states $|{\psi }_i⟩$, but know the value of some pairwise overlaps ${\left|⟨{\psi }_k|{\psi }_l⟩\right|}^2$. 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 describe three applications: basis-independent coherence witnesses, dimension witnesses, and characterisation of multi-photon indistinguishability.

8 pages (+8 page Appendix), 6 figures. v3: Expanded content with more examples, figures and applications