Light strikes a metal plate, and electrons fly out. This is the photoelectric effect, and it works only if the light has enough energy to cross the threshold that holds electrons inside. Until now, physicists could rigorously describe the wave side of the quantum world through quantum coherence, that is, the ability of a state to be a superposition. But particle-like behavior remained without a precise measure: we would say 'light behaves like a particle,' but we couldn't say how much.

The authors of the new work propose to define particleness precisely as the ability to cross such a threshold. They build a simplified model of the photoelectric effect: the incoming quantum system is described by a set of energy levels, while the material from which the electron is knocked out is just two levels with an energy gap between them. A state possesses particleness if its average energy lies above this gap. From there, everything follows the theory of quantum coherence: 'free states' without particleness and 'free operations' that do not create it are defined. The measure of particleness is the distance to the set of free states, just as in the geometric measure of entanglement.

Numerical experiments with thousands of random three-level states revealed an intriguing relation. If you measure the particleness and coherence of the same state, their sum with a coefficient of 1.3 never exceeds 1.8. For some pure states, this bound is reached, while mixed states lie below. This is a new formulation of an old principle: the more strongly an object behaves as a particle, the weaker its wave-like features, and vice versa.

Previously, wave-particle duality was discussed in words or in specific experiments with interferometers. Now both sides have a common language of resource theories: wave properties have coherence, particle-like properties have particleness. This is not just a beautiful symmetry. Such a measure could be useful in analyzing photodetectors and solar cells, where it matters how efficiently light knocks out electrons.

The threshold that started this story turned out to be not only a physical phenomenon but also a convenient metaphor for all of quantum mechanics. Wave and particle are no longer two different entities but two measures of the same state, and there is a precise numerical relation between them. The next step is to test this theory on real photons or trapped ions and see how particleness behaves in larger systems.