f $ over $ \Gamma _{R}({Y_{1},\dots ,Y_{m}};l,k,\epsilon ) $. It follows that the quantity obtained in our definition is bigger than that of Voiculescu. We mention that it is possible to define, in the spirit of [@dvv:entropy2] the relative entropy as $$\chi '({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})=\chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}})-\chi ({Y_{1},\dots ,Y_{m}}:{X_{1},\dots ,X_{n}}).$$ Such a definition corresponds to defining $ \chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}};l,k,\epsilon ) $ as the average of $ f $ over $ \Gamma _{R}({Y_{1},\dots ,Y_{m}}:{X_{1},\dots ,X_{n}}) $. We don’t know whether $ \chi ' $ coincides with Voiculescu’s or our definition of $ \chi , $ and whether Voiculescu’s and our definitions are the same or different. Note, however, that we always have:

Let $ {X_{1},\dots ,X_{n}}$, $ {Y_{1},\dots ,Y_{m}}$ be non-commutative random variables. Then we have $$\begin{aligned}
\chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}})-\chi ({Y_{1},\dots ,Y_{m}})\leq  &  & \\
\chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}})-\chi ({Y_{1},\dots ,Y_{m}}:{X_{1},\dots ,X_{n}})\leq  &  & \\
\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}). &  & \end{aligned}$$ In particular, if $ \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})=-\infty  $, then $ \chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}})=-\infty  $.

In the case that $ \chi ({Y_{1},\dots ,Y_{m}}) $ is finite, the inequalities follow from the discussion in Remark \[rmq:relative\]. If $$\chi ({Y_{1},\dots ,Y_{m}})=-\infty ,$$ then $$\chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots 