Conditional nonmicrostates entropy and a duplicated-generator obstruction

Definitions and mathematical statement

The source asks for a non-microstates analogue of microstates free entropy in the presence, $\chi(X:Y)$. Let $X=(X_1,\ldots,X_m)$ and $Y=(Y_1,\ldots,Y_r)$ be selfadjoint tuples in a tracial von Neumann probability space $(\mathcal M,\tau)$, and put $B=W^*(Y)$. The aid identifies Voiculescu's algebra-conditioned non-microstates entropy $\chi^*(X:B)$ as a canonical candidate, while distinguishing it from an asserted equality with projected microstates entropy in the presence.

For a formal noncommutative polynomial $P$ in $X$ with coefficients in $B$, let $\partial_jX_k=\delta_{jk}1\otimes1$, $\partial_jb=0$ for $b\in B$, and extend by the Leibniz rule. Relative conjugate variables $J_j$ satisfy \[ \tau(J_jP(X))=(\tau\otimes\tau)((\partial_jP)(X)) \] for all such polynomials, interpreted as identities after evaluation. Relative free Fisher information is \[ \Phi^*(X:B)=\sum_{j=1}^m\|J_j\|_2^2, \] with value $+\infty$ when the conjugate system does not exist. If $S$ is a standard semicircular $m$-tuple free from $W^*(B,X)$, the entropy convention is \[ \chi^*(X:B)=\frac m2\log(2\pi e) +\frac12\int_0^\infty\left[ \frac m{1+t}-\Phi^*(X+\sqrt tS:B)\right]\,{\,\mathrm{d}} t, \] with its usual extended-real interpretation. The supplied properties are invariance under replacement of $Y$ by generators of the same algebra, decrease when $B$ is enlarged, equality with $\chi^*(X)$ when $B$ is free from $X$, and invariance under $X_j\mapsto X_j+b_j$ for selfadjoint $b_j\in B$.

For a centered semicircular family $T$ free from $B$, with covariance matrix $C$, and $X_j=T_j+b_j$, the aid reports \[ \chi^*(X:B)=\frac m2\log(2\pi e)+\frac12\log\det C \quad(C>0), \] and $\chi^*(X:B)=-\infty$ when $C$ is singular positive semidefinite. In the nonsingular case the associated perturbed Fisher information is $\operatorname{Tr}(C+tI)^{-1}$.

A duplicated-generator test rules out joint-minus-marginal entropy as a general definition. Let $S,Z$ be free standard scalar semicirculars and let $c=\tfrac12\log(2\pi e)$. With $Y=(Z)$, \[ \chi^*(S,Z)-\chi^*(Z)=2c-c=c=\chi^*(S:W^*(Z)). \] But $Y'=(Z,Z)$ generates the same $B$, while \[ \chi^*(Z,Z)=-\infty,\qquad \chi^*(S,Z,Z)=-\infty. \] Their formal difference is undefined, although the intrinsic conditioned entropy remains $c$.

Microstates entropy in the presence uses projection of joint matrix-microstate sets onto the $X$ coordinates, allowing compatible $Y$ microstates; it is not defined by subtracting two joint entropies. The algebra-conditioned Fisher-information construction is likewise not established here to coincide with that projected notion in general. The aid supplies a canonical conditional construction and an obstruction to a naive alternative, not a general identification theorem. The appended "Focus Group on Operator Theory (day 3)" is the next section heading.

Short English statement

Use relative free Fisher information to define a generator-invariant non-microstates entropy conditioned on $W^*(Y)$. The aid's duplicated-semicircular-generator example shows why joint entropy minus marginal entropy is not a valid general definition, while a full identification with projected microstates entropy remains separate.

Sources

• American Institute of Mathematics, Free Analysis problem list, 24 August 2006, page 3 as identified by supplied verification, https://aimath.org/WWN/freeanalysis/freeanalysis.pdf.
• Local UnsolvedMath registry, permanent ID 20002673, source code catalogue entry, canonical location aim-probability-notes.json, notes[114], record dated 14 August 2026. The verification notes uncertain conditioning-tuple typography and identifies the next section heading; the original literature field is empty.
• The supplied assessment attributes the relative non-microstates construction and projected microstates notion to Voiculescu without a complete bibliographic reference. Research attempt 1: partial result, problem status at run partially solved, rigor labelled proved, novelty confidence low. Its translated-semicircular calculation and duplicated-generator obstruction are machine-generated research aid, not peer reviewed; independently verify before citation.