Quantum and classical bounds for two-state overlaps

Suppose we have $N$ quantum systems in unknown states $\lvertψ_i \rangle$, but know the value of some pairwise overlaps $\left| \langle ψ_k \lvert ψ_l \rangle \right|^2$. 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 describe three applications: basis-independent coherence witnesses, dimension witnesses, and characterisation of multi-photon indistinguishability.

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