Boltzmann counting in Hilbert space

• URL: http://arxiv.org/abs/2608.20136v1
• Date: Thu, 20 Aug 2026 15:01:51 GMT
• Title: Boltzmann counting in Hilbert space
• Abstract summary: We introduce a geometric entropy for quantum preparations, defined as the logarithm of the Hilbert-space volume of pure states compatible with a given set of constraints.<n>We analyze three classes of constraints: restriction to a subspace, fixed expectation values, and coarse-grained subsystem descriptions.<n>The resulting framework provides a geometric measure of quantum ignorance at the level of the preparation.
• Score: 39.146761527401424
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce a geometric entropy for quantum preparations, defined as the logarithm of the Hilbert-space volume of pure states compatible with a given set of constraints. This construction extends Boltzmann's counting perspective to the quantum setting, where compatible states need not be orthogonal and the relevant notion of "number of states" is naturally replaced by a volume in state space. We analyze three classes of constraints: restriction to a subspace, fixed expectation values, and coarse-grained subsystem descriptions. For representative examples, including subspace projection, spin expectation values, partial trace, and an imperfect detector map, we obtain explicit scaling laws and closed-form expressions for the associated volumes. The resulting framework provides a geometric measure of quantum ignorance at the level of the preparation and complements entropy notions based on density matrices and coarse graining.

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