Free Entropy, Free Fisher Information, and the Free Dimension¶

Job: develop Voiculescu's free entropy $\chi$ (microstates / non-microstates), the free Fisher information $\Phi$ via the conjugate variable, the free entropy dimension $\delta$ and its role in the free group factor isomorphism problem, and the free log-Sobolev / free Stam inequality — with the semicircular free entropy / the free CLT as the worked example.

Core¶

Free entropy $\chi$ (Voiculescu)¶

The free entropy $\chi(X_1,\dots,X_n)$ of an $n$-tuple of self-adjoint operators in a tracial von Neumann algebra $(M,\tau)$ — the noncommutative analog of the differential entropy / the Shannon entropy. The microstates definition (Voiculescu): $$ \chi(X)=\lim_{\varepsilon\downarrow0,\,R\uparrow\infty}\limsup_{N\to\infty}\frac{1}{N^2}\log\mathrm{Vol}\,\Gamma_R(X;N,\varepsilon)+\frac{n}{2}\log(2\pi e), $$ where $\Gamma_R(X;N,\varepsilon)$ is the set of $N\times N$ self-adjoint matrices $A_N=(A_1,\dots,A_n)$ with $\|A_i\|\le R$ and $|\frac1N\mathrm{Tr}P(A_N)-\tau(P(X))|<\varepsilon$ for all noncommutative polynomials $P$ of degree $\le N$ (the matricial microstates, the matrix models approximating the joint distribution of $X$). The $\frac{1}{N^2}$ scaling (not $1/N$) is the free / noncommutative normalization (the dimension of the matrix space is $N^2$).

Free entropy	Classical analog
$\chi(X)=\lim\frac1{N^2}\log\mathrm{Vol}\,\Gamma_R+\frac n2\log(2\pi e)$	the differential entropy $h(X)=-\int p\log p$
Microstates $\Gamma_R$ (matrix models)	the support of the distribution
Free entropy dimension $\delta(X)$	the Hausdorff / the Shannon dimension
$\delta(X)=\lim_{\varepsilon\to0}(\chi(X+\varepsilon S)-\chi(X))/\varepsilon\log\varepsilon\to n/2$	the dimension / the number of variables

Free Fisher information $\Phi$ (Voiculescu)¶

The free Fisher information $\Phi(X)$ — the noncommutative analog of the Fisher information, defined via the conjugate variable $D_X$ (the free difference quotient / the Voiculescu noncommutative derivative): the element $D_X\in W^*(X)$ satisfying $$ \tau(D_X\cdot P(X))=\tau\otimes\tau\bigl((\partial P)(X)\bigr) $$ for all polynomials $P$ (the integration-by-parts / the noncommutative Stein equation, $\partial$ the free difference quotient $\partial:\mathbb{C}\langle X\rangle\to\mathbb{C}\langle X\rangle\otimes\mathbb{C}\langle X\rangle^{\mathrm{op}}$ with $\partial X=1\otimes1$). The $\Phi(X)=\tau(D_X^*D_X)$ (the $L^2$-norm of the conjugate variable), the free analog of $\mathbb{E}[\nabla\log p]^2$.

Free Fisher information	Classical analog
Conjugate variable $D_X$	the score $\nabla\log p$
$\Phi(X)=\tau(D_X^*D_X)=\\|D_X\\|_2^2$	$I(X)=\mathbb{E}[\nabla\log p]^2$
Free log-Sobolev $\chi\le\frac n2\log(2\pi e\Phi^{-1})$	the Gross / the Stam inequality
Free Stam: $\chi(\sum X_i/\sqrt N)\le\chi(S)$	the Shannon–Stam entropy inequality

The free logarithmic Sobolev inequality (Voiculescu): $$ \chi(X)\le\frac{n}{2}\log\bigl(2\pi e\,\Phi(X)^{-1}\bigr), $$ the analog of the Gross logarithmic Sobolev (the free Stam inequality), with equality for the semicircular / the free Gaussian.

The free entropy and the isomorphism problem¶

The free group factor isomorphism problem (the Connes / Voiculescu problem: is $L(\mathbb{F}_n)\cong L(\mathbb{F}_m)$ for $n\ne m$?). The free entropy approach (Voiculescu): the free entropy dimension $\delta(X_1,\dots,X_n)$ of the generators of $L(\mathbb{F}_n)$ is $n$ (the full dimension), and if $L(\mathbb{F}_n)\cong L(\mathbb{F}_m)$ then $\delta=\delta$ (the free entropy dimension is an invariant), giving a potential obstruction.

Free entropy dimension	Value
$\delta(S_1,\dots,S_n)$ (free semicircular $n$-tuple, generators of $L(\mathbb{F}_n)$)	$n$ (the full dimension, Voiculescu)
$\delta(X)$ for $X$ the circular element	$1$ (the $L(\mathbb{F}_1)=L(\mathbb{Z})$ the hyperfinite II$_1$, the Dykema–Radulescu)
$\delta$ as an isomorphism invariant	if $L(\mathbb{F}_n)\cong L(\mathbb{F}_m)$ then $\delta=\delta$, giving $n=m$ as the partial obstruction

The Connes embedding (every II$_1$ factor embeds into $R^\omega$ the ultrapower of the hyperfinite — the $L(\mathbb{F}_n)$ embed, the microstates / the free entropy). The current status: the free entropy dimension is the partial obstruction (the $\delta=n$ for the generators), but the full isomorphism problem remains open (the $\delta$ is not known to be a complete invariant).

The non-microstates approach (Biane–Capitaine–Guionnet)¶

The non-microstates free entropy $\chi^*$ (defined without the microstates, via the free Gibbs–Bogoliubov / the variational formula): $$ \chi^*(X)=\sup_{\omega\text{ free Gibbs state}}\bigl{\tau(\omega(X))-\Phi(\omega)\bigr}+\frac n2\log(2\pi e), $$ an alternative to the Voiculescu microstates entropy, defined via the free Fisher information / the variational principle.

Microstates vs non-microstates
$\chi$ (microstates, Voiculescu)	the volume of the matrix microstates
$\chi^*$ (non-microstates, BCG)	the variational / the free Fisher formula
Equivalence	$\chi=\chi^*$ in the full / semicircular case (Biane–Capitaine–Guionnet theorem); the general equivalence is the conjecture / partial
Free pressure	the noncommutative pressure / the Donsker–Varadhan variational principle

The equivalence (the two definitions agree in the full / semicircular case, the Biane–Capitaine–Guionnet theorem — the general equivalence is not known).

See it¶

Free entropy of the semicircular $n$-tuple and the free CLT. The $n$-tuple of free semicircular elements $S_1,\dots,S_n$ (the generators of $L(\mathbb{F}_n)$, the free analog of the Gaussian) has free entropy $$ \chi(S_1,\dots,S_n)=\frac{n}{2}\log(2\pi e\sigma^2) $$ (the Gaussian free entropy, the maximum for a given variance $\sigma^2$, the free Stam equality — the semicircular is the free Gaussian / the maximum entropy distribution). The free Fisher information $\Phi(S)=1/\sigma^2$ (the inverse variance, the free analog of the Gaussian Fisher). The free CLT: the sum of free i.i.d. centered variables $X_i$ (variance $\sigma^2$) $\to S$ the semicircular, and the free entropy increases (the free Stam inequality $\chi(\sum X_i/\sqrt N)\le\chi(S)$, the free analog of the Shannon–Stam entropy inequality, the free CLT as the entropy maximization). The key point: the free entropy / Fisher information is the noncommutative analog of the differential entropy / Fisher, with the semicircular as the free Gaussian (the maximum entropy, the minimum Fisher), the free CLT as the entropy maximization.

Check¶

Verify that the free entropy dimension $\delta(S_1,\dots,S_n)=n$ for the free semicircular $n$-tuple, and explain why this is the partial obstruction to the isomorphism problem.

The free semicircular $n$-tuple $(S_1,\dots,S_n)$ generates $L(\mathbb{F}_n)$, and the microstates $\Gamma_R(S;N,\varepsilon)$ have volume $\sim\exp(N^2\cdot\frac n2\log(2\pi e\sigma^2))$ (the Gaussian volume, the $n$ free semicirculars as $n$ independent Gaussians on the $N^2$-dim matrix space). The entropy $\chi(S)=\frac n2\log(2\pi e\sigma^2)$, and the entropy dimension $\delta(S)=\lim_{\varepsilon\to0}(\chi(S+\varepsilon S')-\chi(S))/(\varepsilon\log\varepsilon)=n$ (the perturbation by $\varepsilon$ free semicirculars increases the entropy by $n\cdot\frac12\log\varepsilon$, so $\delta=n$). The partial obstruction: if $L(\mathbb{F}_n)\cong L(\mathbb{F}_m)$, then the generators have the same $\delta$ (the entropy dimension is an algebraic invariant), so $n=m$ — the free entropy dimension distinguishes $L(\mathbb{F}_n)$ from $L(\mathbb{F}_m)$ for $n\ne m$ (a partial obstruction; the full isomorphism problem remains open since $\delta$ is not known to be a complete invariant).

Show that the free Stam inequality $\chi(\sum X_i/\sqrt N)\le\chi(S)$ (the free CLT as the entropy maximization) follows from the free log-Sobolev.

The free log-Sobolev: $\chi(X)\le\frac n2\log(2\pi e\Phi(X)^{-1})$. For the sum $Y_N=\sum X_i/\sqrt N$ of free i.i.d. centered variables (variance $\sigma^2$), the free Fisher $\Phi(Y_N)\to1/\sigma^2=\Phi(S)$ (the free Fisher information converges to the semicircular Fisher, by the free CLT / the free Fisher convergence). The free log-Sobolev gives $\chi(Y_N)\le\frac n2\log(2\pi e\sigma^2)=\chi(S)$ (the entropy of the partial sum is bounded by the entropy of the limit, the semicircular). The free CLT says $Y_N\to S$ in distribution, and the entropy increases to the limit $\chi(S)$ (the Stam equality at the limit). So the free Stam inequality $\chi(Y_N)\le\chi(S)$ is the entropy increase / the entropy maximization, the free CLT as the approach to the maximum entropy (the semicircular).

Pitfalls¶

The free entropy dimension $\delta$ is the partial obstruction (not the full invariant — the isomorphism problem remains open; the $\delta=n$ distinguishes $L(\mathbb{F}_n)$ but is not known to be a complete invariant).
The microstates free entropy $\chi$ and the non-microstates $\chi^*$ agree in the full / semicircular case (the Biane–Capitaine–Guionnet theorem), but the general equivalence is not known (not a formal consequence).
The free Fisher / the conjugate variable requires the $L^2$-smoothness (the free Gibbs state, the existence of the conjugate variable $D_X\in L^2(M,\tau)$), not every state has the conjugate variable (the non-smooth / the singular states).
The free group factor isomorphism is open — the free entropy is the best current obstruction, but no full invariant is known (the $\delta$ is partial).
The Connes embedding (the microstates / the embedding into $R^\omega$) is the input to the microstates free entropy — without the Connes embedding, the microstates entropy is not defined (the non-microstates $\chi^*$ is the alternative).
The free log-Sobolev / free Stam is the analog of the classical log-Sobolev / Stam, but the constants and the normalization differ (the $n/2$ vs the $n$, the $2\pi e$ the Gaussian normalization).

Refs¶

Voiculescu, "The analogues of entropy and of Fisher's information in free probability theory I–VI" (Invent. / CMP 1993–1999)
Biane–Capitaine–Guionnet, "Free stochastic calculus and free entropy" (1999); the non-microstates free entropy
Dykema, "On the free entropy dimension of some $L(\mathbb{F}_n)$-generators" (1998); Radulescu, "The fundamental group of $L(\mathbb{F}_\infty)$" (1994)
Connes, "Classification of injective factors" (the embedding into $R^\omega$); Voiculescu, the free group factor isomorphism problem

Next¶

FP index · Foundations · Operator-Algebraic Freeness · Free Stochastic Calculus · Subordination & Harmonic Analysis · NCG foundations · Operator algebras

Free Fisher information	Classical analog
Conjugate variable \(D_X\)	the score \(\nabla\log p\)
\(\Phi(X)=\tau(D_X^*D_X)=\\|D_X\\|_2^2\)	\(I(X)=\mathbb{E}[\nabla\log p]^2\)
Free log-Sobolev \(\chi\le\frac n2\log(2\pi e\Phi^{-1})\)	the Gross / the Stam inequality
Free Stam: \(\chi(\sum X_i/\sqrt N)\le\chi(S)\)	the Shannon–Stam entropy inequality

Free entropy dimension	Value
\(\delta(S_1,\dots,S_n)\) (free semicircular \(n\)-tuple, generators of \(L(\mathbb{F}_n)\))	\(n\) (the full dimension, Voiculescu)
\(\delta(X)\) for \(X\) the circular element	\(1\) (the \(L(\mathbb{F}_1)=L(\mathbb{Z})\) the hyperfinite II\(_1\), the Dykema–Radulescu)
\(\delta\) as an isomorphism invariant	if \(L(\mathbb{F}_n)\cong L(\mathbb{F}_m)\) then \(\delta=\delta\), giving \(n=m\) as the partial obstruction