Symmetric Holevo information versus squared fidelity

For any type $I$ and $W:\mathbb Z_2\to\mathbb C^{I\times I}$, suppose each $W(b)$ is a density array in the following precise sense: it has an expansion $W(b)_{ij}=\sum'_r u_r(i)\overline{v_r(j)}$, with $u_r,v_r\in\ell^2(I;\mathbb C)$ and $\sum_r\|u_r\|\|v_r\|<\infty$; every finite quadratic form $\sum_{i,j\in s}\overline{z_i}W(b)_{ij}z_j$ is real and nonnegative; and its real diagonal is summable with sum one. For an array $X$, let $O_X$ be a chosen bounded operator with coefficients $X$, if one exists, and zero otherwise; let $R(T)$ be a chosen positive bounded square root of $T$, if one exists, and zero otherwise; and write $\operatorname{tr}_b(T)=\sum'_i(Te_i)_i$. Define

For arrays $X,Y$, define $D_2(X\Vert Y)$ as the supremum over all finite orthonormal families $v_0,\ldots,v_{m-1}$, including the empty family, of

where $A_{rs}=\langle v_r,O_Xv_s\rangle$, $C_{rs}=\langle v_r,O_Yv_s\rangle$, $f(t)=t\log t$, and matrix functions use continuous functional calculus. With $\overline W=(W(0)+W(1))/2$ and $\mathcal I(W)=(D_2(W(0)\Vert\overline W)+D_2(W(1)\Vert\overline W))/2$, the assertion is

where the bounds are embedded into $[0,\infty]$ and the real square root is zero on negative arguments. No countability assumption on $I$ is present; the density hypothesis cannot hold for empty $I$.