Conservation and ordering of symmetric information

Let $I$ be any type and $W_0,W_1$ complex arrays on $I\times I$. Assume each array admits a nuclear expansion $X(i,j)=\sum_r u_r(i)\overline{v_r(j)}$ with $u_r,v_r\in\ell^2(I;\mathbb C)$ and $\sum_r\|u_r\|\|v_r\|<\infty$, every finite quadratic form of $X$ is real and nonnegative, and its real diagonal is summable with sum $1$. For any array $X$, let $O(X)$ be a chosen bounded operator with coordinate coefficients $X$, or $0$ if no such operator exists. Define $D_2(X\mid Y)\in[0,\infty]$ by taking the supremum over every $d\in\mathbb N$ and every orthonormal $d$-tuple $v$ in the relevant $\ell^2$ space: with $X_v=(\langle v_r,O(X)v_s\rangle)$ and $Y_v=(\langle v_r,O(Y)v_s\rangle)$, its contribution is $+\infty$ if $\ker Y_v\not\subseteq\ker X_v$, and otherwise is

where matrix functions use the continuous functional calculus and the real logarithm has its total value $\log0=0$. Define $J(V)=\tfrac12[D_2(V_0\mid(V_0+V_1)/2)+D_2(V_1\mid(V_0+V_1)/2)]$. The binary arrays $W^-$ on $I\times I$ and $W^+$ on $\mathbb F_2\times(I\times I)$ are given by $W_b^-((a_1,a_2),(c_1,c_2))=\tfrac12\sum_{d\in\mathbb F_2}W_{b+d}(a_1,c_1)W_d(a_2,c_2)$ and $W_d^+((b,(a_1,a_2)),(b',(c_1,c_2)))=\mathbf1_{b=b'}\tfrac12W_{b+d}(a_1,c_1)W_d(a_2,c_2)$, with addition modulo $2$. Then

These are equalities and inequalities in the extended nonnegative reals; no finiteness hypothesis is imposed on these quantities, and the defining supremum includes the empty orthonormal family $d=0$.