Free entropy is an entropy for tuples of noncommuting self-adjoint variables, with microstates and nonmicrostates definitions. For a bounded tuple 
in a von
Neumann algebra with a faithful normal tracial state 
, the microstates version measures volumes
of matrix approximations to its moments. Let 
consist of self-adjoint 
matrix
tuples whose coordinate operator norms are at most
and whose normalized trace moments of
all words of length at most 
differ from those of 
by less than 
. Then
![chi_R(X)=inf_(m,epsilon>0)limsup_(d->infty)[d^(-2)lnVolGamma_R(X;m,d,epsilon)+n/2lnd],](/images/equations/FreeEntropy/NumberedEquation1.svg) |
where 
is a positive integer and volume
uses the real Euclidean structure from the unnormalized Hilbert-Schmidt inner
product. The microstates free entropy is 
. Both
the operator-norm cutoff and the upper limit over
matrix sizes are part of this convention.
The nonmicrostates version is
![chi^*(X)=n/2ln(2pie)+1/2int_0^infty[n/(1+s)-Phi^*(X+sqrt(s)S)]ds,](/images/equations/FreeEntropy/NumberedEquation2.svg) |
where 
is free Fisher information and 
is a variance-one semicircular
family freely independent of 
. The free
entropy equality problem asks whether these two quantities agree when the microstates
entropy is finite.
