A drop of cream falls into a cup of coffee, and within a minute, the entire drink turns beige. This mixing process can be quantified: how much "stirring" is needed to transform one distribution of particles into another? In mathematics, this approach is called optimal transport, and the measure of effort is the Wasserstein distance.
The authors applied this idea to quantum systems. They took the simplest example — a single quantum harmonic oscillator (a pendulum from the micro-world)—and tracked how its state changes as it exchanges energy with its surroundings. If you measure the distance between the oscillator's states using optimal transport, the very history of its changes arranges itself into a geometry resembling spacetime around a black hole. Where information about the system is irretrievably lost, an analog of the event horizon emerges — a surface from which not even light can return.
The same recipe was tested on a more complex model — a subsystem of the SYK model, which in holographic theory describes a two-dimensional black hole. The resulting geography exactly matched the known anti-de Sitter space, confirming the conjecture. Interestingly, the distance measure used turned out to be related to the so-called Krylov complexity — a quantity that estimates how "entangled" a quantum system becomes over time.
The Wasserstein distance is named after the Soviet mathematician Leonid Vaserstein, but the method of optimal transport itself dates back to the work of French engineer Gaspard Monge, who in the 18th century pondered how to move earth most efficiently when building fortifications.
In Greg Egan's novel "Permutation City," virtual universes arise from cellular automata — here, spacetime is born from the play of probabilities in quantum systems.