Light hits a metal and knocks electrons out of it. This is the photoelectric effect, and it only works if the light has enough energy to overcome the barrier holding the electrons inside. Physicists have long been able to measure the wave properties of quantum objects through quantum coherence, that is, the ability of a state to be a superposition. But how much an object resembles a particle has so far not been measured.

The authors of the new work propose to define particleness precisely as the ability to overcome such a barrier. They build a simplified model: the quantum system has several energy levels, and the material from which electrons are knocked out has two levels with an energy gap between them. A state possesses particleness if its average energy is above this gap. The measure of particleness is the distance to states without particleness, those whose energy is insufficient to knock out an electron.

Numerical experiments showed that if you measure the particleness and coherence of the same state, their sum with a coefficient of 1.3 never exceeds 1.8. The more particleness, the less coherence, and vice versa. This is a new formulation of the old principle: an object cannot be both completely a wave and completely a particle. Such a measure could be useful in developing photodetectors and solar cells, where it is important how efficiently light knocks out electrons.