\\
\Gamma _{R}({Y_{1},\dots ,Y_{m}};k,l,\epsilon )
\end{array}} &  & \\
\log \lambda {\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k ,l ,\epsilon  )}+\frac{n}{2}\log k & \end{aligned}$$*

$$\chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})=\inf _{l,\epsilon }\chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}};l,\epsilon )$$

$$\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})=\sup _{R}\chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}).$$ The last quantity is called the relative free entropy of the $ n $-tuple $ ({X_{1},\dots ,X_{n}}) $ with respect to the $ m $-tuple $ ({Y_{1},\dots ,Y_{m}}) $.

If $ \omega  $ is a free ultrafilter on $ \mathbb N $, then one can also define $ \chi ^{\omega }({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}) $ exactly as in (\[ean:defofentropy\]), but replacing $ \limsup  $ by $ \lim _{k\to \omega } $.

\[rmq:relative\]It is not clear whether our definition of $ \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}) $ coninsides with that of Voiculescu (see [@dvv:entropy2]). His definition corresponds to defining $$\begin{aligned}
\chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}};l,k,\epsilon )= &  & \\
\limsup _{k}\log \frac{\lambda \Gamma _{R}({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}};k,l,\epsilon )}{\lambda \Gamma _{R}({Y_{1},\dots ,Y_{m}};k,l,\epsilon )}+\frac{n}{2}\log k. &  & \end{aligned}$$ The connection to our definition can be made as follows: let $$f({y_{1},\dots ,y_{m}})=\lambda {\Gamma _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}[{y_{1},\dots ,y_{m}}];k ,l ,\epsilon  )}$$ Then Voiculescu’s definition corresponds to taking as $ \chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}};l,k,\epsilon ) $ the average of $ 