Lemma 5: mutual-information mixing for a monoorthogonal decomposition

Let $e$ consist of nonnegative summable real weights $w_k$ with $\sum_k w_k=1$ and kernels $X_k$ indexed by pairs of natural numbers. For each $k$ with $w_k>0$, require that $X_k$ admit a representation $X_k(x,y)=\sum_r u_r(x)\overline{v_r(y)}$ by square-summable vectors with $\sum_r\|u_r\|\|v_r\|<\infty$, have real nonnegative finite quadratic forms, and have a summable real diagonal of sum $1$; zero-weight components are unrestricted. Put $X(x,y)=\sum_k w_kX_k(x,y)$. For $C\subseteq\{0,1\}$, define the reduced kernel by

where $a,b:C\to\mathbb N$ and $a\sqcup z$ combines the coordinates on $C$ and its complement. For any kernel $Y$, let $H(Y)$ be the infimum of $\sum_r[-a_r\log a_r]_+$ over all entrywise expansions $Y=\sum_r a_rq_rq_r^*$ with nonnegative summable weights of sum $1$ and square-summable vectors whose positive-weight members are unit and mutually orthogonal; zero-weight vectors are unrestricted, and an empty infimum is $+\infty$. Write $S_Y(C)=\operatorname{toReal}H(Y_C)$, where $\operatorname{toReal}(+\infty)=0$, and define $J_Y(A,B)=S_Y(A)+S_Y(B)-S_Y(A\cup B)$. Assume that, for distinct $k,l$ with $w_k,w_l>0$, the reductions on coordinate $0$ obey

and assume $H(X_C)\ne+\infty$ for every $C\subseteq\{0,1\}$, including the empty set. For kernels $Y,Z$ on the same label set, define $D(Y\|Z)$ as the supremum over all finite orthonormal families $v$, including the empty family, of the quantity obtained from compressions $Y^v_{ab}=\sum_i\overline{v_a(i)}\sum_jY(i,j)v_b(j)$ and $Z^v$: it is $+\infty$ if $\ker Z^v\not\subseteq\ker Y^v$, and otherwise is $[\operatorname{Re}\operatorname{tr}(f(Y^v)-Y^v g(Z^v)-Y^v+Z^v)]_+$, where $f(t)=t\log t$, $g(t)=\log t$, and matrix functions use continuous functional calculus. Then

Here $[t]_+=\max(t,0)$ takes values in $[0,\infty]$, and $0\cdot(+\infty)=0$. Individual component marginal entropies are not assumed finite; their conversion to real numbers uses the stated convention.