Statistical signatures of integrable and non-integrable quantum hamiltonians

Integrability has long served as a cornerstone concept in classical mechanics, where it possesses a precise and unambiguous definition. Extending this notion to the quantum domain, however, remains a far more subtle and elusive problem. In particular, deciding whether a given quantum Hamiltonian – viewed simply as a matrix of its elements – does or does not define an integrable system is far from obvious. Yet this question is crucial: it bears directly on non-equilibrium dynamics, spectral correlations, the behaviour of cor- relation functions, and other fundamental properties of many-body quantum systems. In this work, we develop a statistical framework for addressing quantum integrability from a purely probabilistic standpoint. Our approach begins with the observation that a necessary signature of integrability is the finite probability of encountering vanishing energy gaps in the spectrum. On this basis, we formulate a twofold protocol capable of distinguishing be- tween integrable and non-integrable Hamiltonians. The first step consists of a systematic Monte Carlo decimation of the spectrum, designed to reveal the emergence (or absence) of Poissonian level spacing statistics. The iterative decimation compresses the Hilbert space exponentially, and its termination point determines whether the spectrum is governed by a mixed distribution or approaches the Poisson limit. In the second step, the distinction is instead obtained by analysing the distributions of k-step gaps which can help in discrim- inating between Poisson and mixed statistics. This procedure applies to Hamiltonians of arbitrary finite size, regardless of whether their structure involves a finite number of blocks or an exponentially fragmented Hilbert space. As a concrete benchmark, we implement the protocol on a class of quantum Hamiltonians constructed from the permutation group SN , thereby demonstrating both its effectiveness and its broad applicability.