# Quantum and classical bounds for two-state overlaps

## TL;DR

This paper investigates the constraints on unknown quantum state overlaps given some known pairwise overlaps, providing complete solutions for specific cases and applications in quantum coherence, dimension, and indistinguishability.

## Contribution

It offers a complete characterization of overlap bounds for three pure states with two known overlaps and extends to general cases, contrasting quantum and classical models.

## Key findings

- Complete bounds for three pure states with two known overlaps
- Method to derive bounds in general cases
- Applications in coherence, dimension, and indistinguishability

## Abstract

Suppose we have $N$ quantum systems in unknown states $\lvert\psi_i \rangle $, but know the value of some pairwise overlaps $\left| \langle \psi_k \lvert \psi_l \rangle \right|^2$. 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.