Two notions of free entropy differ even when both are finite

1 · The essentials

There are two ways to assign an entropy—a numerical measure—to certain collections of mathematical objects. One uses approximations by finite matrices, while the other uses a quantity called free Fisher information. The question was whether these two recipes always give the same answer when both answers are finite. The manuscript constructs a collection for which they do not. Both values are finite, but the second exceeds the first by at least one half. The example uses a large, fixed number of objects, each bounded in size. The main result has a Lean formalization.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine two measuring procedures intended to describe the same feature of a collection of items. Here both procedures return ordinary finite readings, yet their readings differ by a definite amount; the analogy does not explain how either entropy is calculated.

Why it matters

The result answers negatively the finite-entropy equality question attributed to Voiculescu in the repository summary. It establishes that the matrix-approximation and free-Fisher-information definitions can differ without either entropy becoming infinite.

2 · Going further

The precise problem concerns a bounded self-adjoint tuple $X$ in a von Neumann algebra equipped with a faithful normal tracial state. The two quantities are microstates free entropy $χ (X)$ and nonmicrostates free entropy $χ^{*} (X)$ . The repository summary states the separation as $- \infty < χ (X) < χ^{*} (X) < \infty$ .

The main manuscript gives the more quantitative bound $χ (X) \leq χ^{*} (X) - \frac{1}{2}$ , with both entropies finite. It specifies that $χ$ is the original microstates entropy, using an operator-norm cutoff and a limsup over matrix sizes. Thus the convention for the matrix-approximation definition is part of the result, not an unspecified choice.

The construction uses a large but fixed number of variables. Beyond this construction-based description, the supplied abstract does not explain the mechanism of the argument. Only the main manuscript is supplied, so no relationship to companion manuscripts can be described from this source. The main result has a Lean formalization, although the supplied material does not detail its coverage of subsidiary claims.

Useful notions

Finite matrices and matrix approximations
Bounded self-adjoint operators
Tracial von Neumann algebras
Microstates and nonmicrostates free entropy
Limits superior

What was known before / what is new

Before

The source identifies two definitions of free entropy: one based on matrix approximations and the other on free Fisher information. It describes their equality when both are finite as a question posed by Voiculescu. It supplies no further account of earlier results.

New

The manuscript constructs a bounded self-adjoint tuple for which both free entropies are finite but unequal. The stated gap is at least $\frac{1}{2}$ , with microstates entropy the smaller quantity. This gives a negative answer to the finite-entropy equality question.

Limits and assumptions

The example lies in a von Neumann algebra with a faithful normal tracial state and uses a large but fixed number of bounded self-adjoint variables. The result concerns the specified original microstates definition, including its operator-norm cutoff and limsup over matrix sizes. It is an existence result, not a claim that the two entropies differ for every tuple. The main result has a Lean formalization; the supplied material does not specify the detailed scope of that formalization. It also does not state which related questions remain open.

Mini glossary

Microstates free entropy: The matrix-approximation version of free entropy, denoted $χ$ , used here with an operator-norm cutoff and a limsup over matrix sizes.
Nonmicrostates free entropy: The version of free entropy based on free Fisher information, denoted $χ^{*}$ .
Tuple: The collection of variables, denoted $X$ , to which the two entropy definitions are applied.
Finite-entropy separation: A strict difference between the two entropy values in an example where both values are finite.

Keywords : Free entropy · Microstates · Operator algebras

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/298.md

Manuscripts in this family

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A finite-entropy separation of microstates and nonmicrostates free entropy

25/09/2026

To be verified

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 $- \infty < χ (X) \leq χ^{*} (X) - \frac{1}{2} < \infty$ . 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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