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Operator algebras · Result 298

Two notions of free entropy differ even when both are finite

1 paperLean

Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy \(-\infty\lt \chi\lt \chi^*\lt \infty\). This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.

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First page of A finite-entropy separation of microstates and nonmicrostates free entropy

A finite-entropy separation of microstates and nonmicrostates free entropy

OpenAI · September 25, 2026

We answer the finite-entropy equality question for microstates and nonmicrostates free entropy negatively. We construct a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state such that \(-\infty\lt \chi(X)\leq\chi^*(X)-\tfrac12\lt \infty\). Here χ is the original microstates entropy with an operator-norm cutoff and a limsup over matrix sizes. The counterexample uses a large but fixed number of variables.

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@misc{OAI:A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026,
  author = {{OpenAI}},
  title = {{A finite-entropy separation of microstates and nonmicrostates free entropy}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf}{OAI:A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization disproves equality of microstates and nonmicrostates free entropy even when both quantities are finite. It constructs a bounded self-adjoint tuple \(X\) in a von Neumann algebra with a faithful normal tracial state such that \(-\infty<\chi(X)\le\chi^*(X)-1/2<\infty\). Here \(\chi\) is the microstates entropy with an operator-norm cutoff and a limsup over matrix sizes, and \(\chi^*\) is the nonmicrostates entropy defined using free semicircular noise. The example has a fixed finite number of variables.

Comparator links

ResultComparator statement
Finite separation of microstates and nonmicrostates free entropyFiniteEntropySeparation.lean

Scope note on GitHub →