spin system. Since the Hilbert space for multiple particles is given by the tensor product of the Hilbert spaces of the individual particles, its dimension grows exponentially with the number of particles. This exponential behavior is therefore the key obstruction to classical modeling of quantum systems. The observation is as old as quantum theory itself, and physicists ever since have tried to find ways around it. In that spirit, our paper presents an effective method to extract those physical features that “only depend on the one-body eigenvalues” associated to randomly-chosen quantum states of any fixed number of particles. Our methods are rooted in geometry, and so further a nascent dialog between algebraic and symplectic geometry on the one hand, and the theory of quantum computation and quantum information on the other, with new results for each subject. We therefore begin with some context, and make an effort in the body of the paper to build a correspondence in terminologies.

Typically, to address the aforementioned exponential complexity, physicists make use of a very simple yet powerful observation: Important properties such as energy and entropy often do not depend on the whole wavefunction but rather on only a small part, namely the 