Average ensemble error and existence of a good frozen vector

Let $I$ be any type, let $\mathcal H_I=\ell^2(I;\mathbb C)$, and let $W$ assign a complex coefficient array $W_b$ on $I\times I$ to each bit $b\in\mathbb F_2$. Assume each $W_b$ is a density array in the following precise sense: it has an expansion $W_b(i,j)=\sum_{r=0}^{\infty}u_r(i)\overline{v_r(j)}$ with $u_r,v_r\in\mathcal H_I$ and $\sum_r\|u_r\|\|v_r\|<\infty$; every quadratic form restricted to a finite subset of $I$ is real and nonnegative; and its real diagonal is summable with sum $1$. Fix any $n\in\mathbb N$, put $N=2^n$, and let $A\subseteq\{0,\ldots,N-1\}$. Define $G_n(r,c)=\prod_{k=0}^{n-1}\kappa(\operatorname{bit}_k(\operatorname{rev}_n r),\operatorname{bit}_k(c))$ over $\mathbb F_2$, where $\kappa(a,0)=1$, $\kappa(a,1)=a$, and $\operatorname{rev}_n$ reverses the $n$ binary digits of an index. For $u\in\mathbb F_2^N$, set $x_c(u)=\sum_r u_rG_n(r,c)$ and $V_u(y,z)=\prod_{c=0}^{N-1}W_{x_c(u)}(y_c,z_c)$, for $y,z\in I^N$. For an index $i$, prefix $h\in\mathbb F_2^i$, and bit $b$, let $T_{i,h,b}=2^{-N}\sum_{v\in\mathbb F_2^N}V_{\operatorname{fill}(h,b,v)}$, where the filled word equals $h_j$ for $j<i$, equals $b$ at $i$, and equals $v_j$ for $j>i$. The split array $W_{n,i}$, indexed by $\mathbb F_2^i\times I^N$, is $W_{n,i}^b((h,y),(h',z))=\mathbf1_{h=h'}2^{-i}T_{i,h,b}(y,z)$. For any coefficient array $X$, let $O(X)$ be a chosen bounded operator having coefficients $X$ in the coordinate basis if one exists, and $0$ otherwise; let $R(T)$ be a chosen positive bounded operator satisfying $R(T)^2=T$ if one exists, and $0$ otherwise; and put $\operatorname{tr}_bT=\sum_j(Te_j)_j$. For a binary array $V$, define $F(V)=[\operatorname{Re}\operatorname{tr}_bR(R(O(V_0))O(V_1)R(O(V_0)))]^2$. For decoding, let $P_{i,h}$ be the orthogonal projection onto the closed span of all eigenvectors with negative real eigenvalue of $R(O(T_{i,h,0}))-R(O(T_{i,h,1}))$, and set $\Pi_{i,h,0}=1-P_{i,h}$, $\Pi_{i,h,1}=P_{i,h}$. Given $u$, form $K_u$ by processing $i=0,\ldots,N-1$, left multiplying by $\Pi_{i,u_{<i},u_i}$ when $i\in A$ and by the identity otherwise, and set $s(u)=\operatorname{Re}\operatorname{tr}_b(K_u^*K_uO(V_u))$. For frozen bits $f\in\mathbb F_2^{A^c}$, define $e_f=1-2^{-|A|}\sum_{m\in\mathbb F_2^A}s(m\cup f)$, and define $\bar e=2^{-|A^c|}\sum_f e_f$. Then

This includes $n=0$, when $N=1$, and $A=\varnothing$ or $A=\{0,\ldots,N-1\}$; an assignment on an empty set has one possible value. The basis traces use the total infinite-sum operation, which is $0$ for a nonsummable family.