A mathematical framework proposes measuring quantum uncertainty by asking how much of a system’s state space remains compatible with the way it was prepared. In an arXiv version 1 preprint dated 20 August 2026, the authors define a geometric entropy as the logarithm of that compatible volume.
The approach applies Boltzmann-style counting to quantum preparations by assigning a measure to the pure states that remain possible. The authors present it as a preparation-level measure of quantum ignorance that could complement density-matrix and coarse-grained entropy descriptions.
A volume that needs a measuring rule
The analysis covers restrictions to subspaces, fixed expectation values and coarse-grained descriptions of a subsystem after the rest has been traced out. It studies finite-dimensional Hilbert spaces using Haar-distributed pure states.
Exact equality constraints create a technical problem because the set of states satisfying them exactly has zero Haar volume. To make such calculations possible, the paper gives the constraint manifold a small nonzero thickness, a procedure known as regularization.
What the calculations find
In a subspace-projection example, the regularized volume of compatible states rapidly approaches zero as codimension—the number of excluded directions—increases.
For fixed spin expectation values, the exact volume depends only on the length of the Bloch vector used to represent those values. The expression is piecewise polynomial and becomes close to a Gaussian at moderately large dimension.
Testing states that are only partly known
A more involved calculation examines partial-trace coarse graining, in which an effective state describes part of a larger system. The compatible volume scales with the determinant of that effective state and is largest for maximally mixed states.
The authors checked that prediction with rejection sampling of random local qutrit states. For each effective state, the simulation sampled 100,000 microscopic states using a constraint tolerance of ε = 0.2, and the numerical trends followed the predicted constant, linear and quadratic determinant patterns across the tested environment dimensions.
The partial-trace construction is also formally extended to mixed effective states by introducing an auxiliary system that purifies them. For an imperfect-detector map, the final volume is expressed through the effective state’s Bloch-vector coordinates; the derivation uses Laplace or Fourier transforms followed by inverse transforms.
A proposal, not a finished entropy theory
The work is a mathematical construction supported by derivations, model quantum-state calculations and selected simulated rejection-sampling checks. It does not establish that the proposed quantity is a universally valid thermodynamic entropy.
Questions about additivity, subadditivity, continuity and behavior under composition or coarse graining remain open. The paper also leaves unresolved whether compatible-set volume can be defined independently of the particular constraint equations.
Several results carry narrower qualifications: the Gaussian behavior of the spin vector is suggested rather than fully established; the partial-trace derivation assumes nonzero eigenvalues and treats zero eigenvalues through a limiting procedure; and the numerical check uses a fitted constant and selected simulated examples without formal inferential uncertainty.
The preprint therefore offers a geometric language for describing what remains unknown about a quantum preparation, while leaving its broader physical and thermodynamic status open to further analysis.
Paper data and sources
Original title: Boltzmann counting in Hilbert space
Authors: Raúl O. Vallejos, Isadora Veeren, Frederico Brito, Fernando de Melo
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text