Two free entropies disagree even when both are finite

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Family 298 · Operator algebras
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

Entropy can describe how many microscopic configurations realize a macroscopic law. In free probability, the variables are operators that need not commute, and there are two major definitions of entropy.

One counts the volumes of large matrix tuples approximating the variables’ moments. The other uses an analytic measure of sensitivity to freely independent noise. A general theorem says the matrix-volume entropy is no larger than the analytic entropy. Are they equal whenever the matrix-volume entropy is finite?

Family 298 claims a negative answer: one bounded self-adjoint tuple has two finite entropies separated by at least one half in the stated normalization. The construction keeps a hidden block pattern visible in matrices even after enough noise has been added to make the analytic entropy almost Gaussian.

2. Background and essential definitions

A tracial von Neumann algebra $(M,\tau)$ provides a noncommutative probability space. Self-adjoint operators are real-valued random variables in this language, and $\tau(X_{i_1}\cdots X_{i_k})$ are their joint moments. The trace is faithful and normalized, so $\tau(1)=1$.

For a bounded self-adjoint tuple $X=(X_1,\ldots,X_n)$, a microstate is a tuple of self-adjoint $d$-by-$d$ matrices whose normalized trace moments approximate those of $X$. The set $\Gamma_R(X;m,d,\varepsilon)$ requires coordinate norms at most $R$ and accuracy $\varepsilon$ for every word of length at most $m$.

The original microstates entropy is

$$\chi_R(X)=\inf_{m,\varepsilon}\limsup_{d\to\infty} \left[d^{-2}\log\operatorname{Vol}\Gamma_R(X;m,d,\varepsilon) +\frac n2\log d\right],$$

followed by $\chi(X)=\sup_R\chi_R(X)$. Volume uses the Euclidean structure from the unnormalized Hilbert–Schmidt inner product. The normalization compensates for growth of matrix dimension. This is a differential-entropy-type quantity and can be negative.

The analytic entropy $\chi^*$ uses conjugate variables, free analogues of score functions, and their squared norms summed as free Fisher information $\Phi^*$. Its normalization is

$$\chi^*(X)=\frac n2\log(2\pi e) +\frac12\int_0^\infty \left[\frac n{1+s}-\Phi^*(X+\sqrt s\,S)\right]ds.$$

Here $S$ is a variance-one semicircular family free from $X$. Semicircular variables play the role of Gaussians in free probability; freeness replaces classical independence through vanishing traces of alternating centered products. A variance-one free semicircular $n$-tuple has entropy $\frac n2\log(2\pi e)$.

The two definitions ask different questions: how much matrix volume realizes the law, and how its analytic noise response integrates over time.

3. Development history and previous partial results

Voiculescu introduced microstates free entropy and later its nonmicrostates counterpart. They agree for one self-adjoint variable. Agreement for several variables became part of the broader program of unifying free-entropy theories.

Biane, Capitaine and Guionnet proved the general inequality $\chi\leq\chi^*$ using matrix Brownian-motion large deviations. Jekel and Pi later gave an elementary proof through classical entropy and Fisher information and extended the comparison to conditional free entropy.

Without a finiteness assumption, equality can fail because some noncommutative laws have no sufficiently good matrix approximations: their microstates entropy is minus infinity while analytic entropy is larger. That does not answer the sharper question posed explicitly in Guionnet’s formulation: does $\chi(X)>-\infty$ force equality?

Substantial positive results support equality under stronger regularity assumptions. Dabrowski proved it for suitable strictly convex potentials with Lipschitz derivatives and polynomial approximation properties. Jekel obtained matrix entropy and Fisher-information convergence for regular convex potentials with appropriate concentration and approximation control.

M1 targets the gap between these structured settings and mere finiteness. Its matrix argument also draws on older methods controlling microstate volumes through commutation constraints and projection geometry. The new claim is that residual simultaneous block structure can produce a strict entropy deficit even after smoothing ensures finite microstates entropy.

4. What the new manuscripts claim

M1 states that there is a finite integer $n\geq2$ and a bounded self-adjoint tuple $X$ in a von Neumann algebra with faithful normal tracial state such that

$$-\infty<\chi(X)\leq\chi^*(X)-\frac12<\infty.$$

Both values are finite, and the inequality concerns the same tuple. The definition of $\chi$ is the original operator-norm-cutoff version with a limsup over matrix sizes, not an alternative entropy defined by a preferred matrix ensemble or by a liminf.

The tuple has a large but fixed number of coordinates. The theorem does not establish the same phenomenon for every prescribed smaller number, such as exactly two variables.

The construction begins with

$$C=(B_1,\ldots,B_p,Y),$$

where the $B_i$ are free variance-one semicircular variables and $Y$, in a separate tensor factor, is uniformly distributed on $p$ equally spaced points in $[-1,1]$. Thus $Y$ commutes with every $B_i$. With $S$ free from $C$, the tuple is

$$X=C+\sqrt h\,S$$

for a suitably small positive $h$, after choosing one sufficiently large finite $p$. An additional noise time $t=p^{3/4}$ is used in the comparison argument. All matrix-size limits are then taken with $p$ and $h$ fixed.

The result is distinct from the entropy-dimension consequences in families 287 and 296. Those compare different generating tuples of one algebra; M1 compares two entropy definitions on a single tuple.

5. Why the advance matters

Importance: 5/5. If correct, the theorem separates the two principal free-entropy theories even in the finite regime where a simple failure of matrix approximability cannot explain the difference.

This would change the role of regularity assumptions in equality theorems. Convexity, concentration and approximation hypotheses would be substantive reasons for agreement, rather than technical routes toward an equality implied by finite entropy alone.

The construction also identifies a mechanism. Analytic noise estimates can become nearly Gaussian while a matrix ensemble retains collective geometry restricting its volume. Two variables can have the same broad variance scale without erasing all joint block information. The discrepancy is structural, not just a mismatch in additive normalization constants.

6. The main idea

Build a shared block signal

Because $Y$ commutes with all the $B_i$, its spectral projections divide the original tuple into simultaneous blocks. In a matrix approximation, the last coordinate organizes subspaces on which much of the energy of the first coordinates is concentrated.

Adding the small free noise $\sqrt h\,S$ makes the microstates entropy finite while leaving enough of this pattern detectable. M1 then compares what happens after a larger positive noise time.

Make the analytic deficit small

Free cumulants organize joint moments into contributions associated with noncrossing partitions. A semicircular family has only second-order cumulants; higher cumulants measure departures from semicircular behavior.

M1 bounds a Fisher-information deficit by squared sums of higher cumulant arrays, using semicircular Wick-polynomial orthogonality. The signal’s cumulants are sparse enough that additional noise makes $\chi^*$ close to the entropy of a semicircular family with matching coordinate variances.

This is an analytic comparison with a Gaussian-type reference, not a claim that the entire tuple becomes exactly semicircular.

Keep a rare matrix event visible

For the matrix side, the source chooses increasingly accurate cutoff microstate sets for $X$, takes uniform laws on them, and evolves those laws by independent Gaussian unitary ensemble noise. GUE matrices are the standard self-adjoint Gaussian matrix model whose large-size law is semicircular.

The evolved selected ensembles retain a simultaneous block-energy event with probability bounded away from zero. For independent GUE reference matrices with the same limiting variances, that event has probability at most $e^{-cd^2}$ for some fixed $c>0$.

Relative entropy cannot increase when observations are simplified to an event indicator. Comparing these two event probabilities therefore produces a deficit of order $d^2$, which remains nonzero after the entropy normalization by $d^2$.

Balance the two scales and retain the same ensembles

The analytic deficit becomes small when $t/\sqrt p$ grows, while the matrix block estimate survives when $t\log p/p$ becomes small. The choice $t=p^{3/4}$ satisfies both as $p$ grows.

A finite-time classical-to-free Fisher comparison transfers the positive-time matrix deficit back to the original tuple. M1 keeps the same selected terminal ensembles throughout this step. Their normalized entropy limsup need not equal the microstates entropy of the limiting law, so replacing one by the other would lose precisely the distinction being exploited.

After selecting sufficiently large finite $p$ and then small $h$, the estimates yield the stated fixed gap.

7. Scope, limitations, and open questions

The theorem concerns the original $\chi$ and $\chi^*$ with the displayed conventions. It does not assert an exact formula for either entropy of the constructed tuple. The lower bound on their gap is quantitative, but the minimal number of variables realizing a finite gap is not determined.

The parameter choices are made before matrix size tends to infinity. The result is not merely a sequence of tuples with increasing coordinate count whose limiting entropies disagree. Likewise, the selected noisy matrix laws are analytical tools, not a redefinition of microstates entropy.

The known one-variable equality and regular convex equality theorems remain outside the counterexample’s scope. The paper does not claim that every finite-entropy tuple has a gap, nor that the two theories lose their general comparison inequality.

Further questions include identifying broader equality classes, reducing the coordinate count of counterexamples, and understanding which matrix structures analytic Fisher information can fail to capture. These go beyond the stated finite-entropy separation.

8. Sources and further reading

• Jekel and Pi, An Elementary Proof of the Inequality chi ≤ chi* for Conditional Free Entropy, for the comparison used in the scale-transfer argument.
• Dabrowski, A Laplace Principle for Hermitian Brownian Motion and Free Entropy I, for a substantial convex equality class.
• M1 fixes both entropy normalizations and explains why the terminal matrix ensembles must be retained through its finite-time comparison.

Source manuscripts

• M1. OpenAI, A finite-entropy separation of microstates and nonmicrostates free entropy (September 25, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.