Bounded channel-tree root-fidelity and information martingales

Let $I$ be any type, let $(\Omega,\mathcal M,\mu)$ be any probability space, and let $W_b$, $b\in\mathbb F_2$, be density coefficient arrays: each admits $W_b(i,j)=\sum_r u_r(i)\overline{v_r(j)}$ with $u_r,v_r\in\ell^2(I;\mathbb C)$, $\sum_r\|u_r\|\|v_r\|<\infty$, real nonnegative finite quadratic forms, and summable real diagonal of sum $1$. Let $B_k:\Omega\to\{0,1\}$ be measurable for every $k\ge0$, and assume that for every $n\ge0$ and every $h\in\{0,1\}^n$,

For an array $X$, let $O(X)$ be a chosen bounded operator having coefficients $X$, or $0$ if none exists; let $R(T)$ be a chosen positive bounded square root, or $0$ if none exists; and put $\operatorname{tr}_bT=\sum_j(Te_j)_j$, the total infinite sum. Define $F(V)=[\operatorname{Re}\operatorname{tr}_bR(R(O(V_0))O(V_1)R(O(V_0)))]^2$. Define $D_2(X\mid Y)$ as the supremum over all finite orthonormal families $v$, including the empty family, of $+\infty$ if $\ker Y_v\not\subseteq\ker X_v$, and otherwise of $\max(0,\operatorname{Re}\operatorname{Tr}(f(X_v)-X_v\log Y_v-X_v+Y_v)/\log2)$, where $X_v=(\langle v_r,O(X)v_s\rangle)$, $Y_v=(\langle v_r,O(Y)v_s\rangle)$, $f(t)=t\log t$, and matrix functions use continuous functional calculus; set $J(V)=\tfrac12[D_2(V_0\mid(V_0+V_1)/2)+D_2(V_1\mid(V_0+V_1)/2)]$. For each $n$, put $N=2^n$, encode $u\in\mathbb F_2^N$ by $x_c=\sum_r u_rG_n(r,c)$, where $G_n(r,c)=\prod_{k<n}\kappa(\operatorname{bit}_k(\operatorname{rev}_n r),\operatorname{bit}_k(c))$, $\kappa(a,0)=1,\ \kappa(a,1)=a$, and $\operatorname{rev}_n$ reverses $n$ bits. Write $V_u(y,z)=\prod_{c<N}W_{x_c}(y_c,z_c)$; let $T_{n,i,h,b}$ be its uniform average over full $N$-bit words used to fill coordinates after a fixed prefix $h$ and bit $b$ at $i$; and set $W_{n,i}^b((h,y),(h',z))=\mathbf1_{h=h'}2^{-i}T_{n,i,h,b}(y,z)$. Let

where the conversion to a real number sends $+\infty$ to $0$. Let $\mathcal F_n$ be the sigma-algebra generated by $B_0,\ldots,B_{n-1}$, so $\mathcal F_0$ is the bottom sigma-algebra. Then $J(W_{n,i_n(\omega)})\ne+\infty$ for every $n,\omega$, and $0\le Z_n(\omega)\le1$ and $0\le J_n(\omega)\le1$ for every $n,\omega$; every $J_n$ is integrable; $Z_n$ is a supermartingale with respect to $(\mathcal F_n)$ and $\mu$; and $J_n$ is a martingale with respect to the same filtration and measure. Thus the processes are adapted and integrable, with $\mathbb E[Z_m\mid\mathcal F_n]\le Z_n$ almost everywhere and $\mathbb E[J_m\mid\mathcal F_n]=J_n$ almost everywhere whenever $n\le m$; the displayed bounds and finiteness conclusion are pointwise, including at $n=0$.