Free Fisher information of a self-adjoint tuple 
in a von Neumann algebra with a faithful normal
tracial state 
is
 |
where vectors 
satisfy
 |
for every noncommutative polynomial 
. Here 
, and 
is the derivation determined
by 
and the product rule. These vectors lie in the 
completion
of the algebra generated by the tuple. When they do not
exist, 
.
Integrating this quantity along addition of freely
independent semicircular noise defines
nonmicrostates free entropy.
