e $ if and only if $ {X_{1},\dots ,X_{n}}$ are a free semicircular family, which is free from $ B $. The same statement holds for $ \chi ^{\omega } $.

The condition that $ \chi ({X_{1},\dots ,X_{n}}|B) $ achieves a local algebraic maximum is weaker than the condition that it achieves its maximum, so if the maximum is achieved, the local algebraic maximum is achieved, and Proposition \[prop: maximality\] applies. Conversely, it was shown in Corollary \[corr: max\] that the given number is indeed a maximum.

We end with the following theorem, whose proof is identical to that of [@dvv:entropy4 Proposition 4.3].

If $ \chi (X|B)=\chi (X)\neq -\infty  $, then $ X $ is free from $ B $. The same statement holds for $ \chi ^{\omega } $.

$ \chi ({X_{1},\dots ,X_{n}}|B\otimes M_{N})\protect $. 
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Let $ X_{ij}^{k} $, $ i,j=1,\dots ,N $, $ k=1,\dots ,n $ be non-commutative random variables, such that $ X_{ij}=X_{ji}^{*} $, and let $ B $ be a unital subalgebra of $ (M,\tau ) $ . Then the joint $ * $-distribution of the family $ \{X_{ij}^{k}\}\cup B $ completely determines and is completely determined by, the joint distribution of the matrices $$Z_{k}=\left( 