Polar coding: ensemble block error below the symmetric information rate

Let $I$ be any type and let $W_0,W_1$ be density coefficient arrays on $\ell^2(I;\mathbb C)$: each admits a countable expansion $X(i,j)=\sum_r u_r(i)\overline{v_r(j)}$ with $\sum_r\|u_r\|\|v_r\|<\infty$, has real nonnegative finite quadratic forms, and has summable real diagonal of sum $1$. For any array $X$, write $O(X)$ for a chosen bounded operator with those coordinate coefficients, or $0$ if none exists; write $R(T)$ for a chosen positive bounded square root of $T$, or $0$ if none exists; and write $\operatorname{tr}_bT=\sum_j(Te_j)_j$, using the total infinite sum. Define $F(V)=[\operatorname{Re}\operatorname{tr}_bR(R(O(V_0))O(V_1)R(O(V_0)))]^2$. For arrays $X,Y$, define $D_2(X\mid Y)$ as the supremum over every integer $d\ge0$ and every orthonormal family $v_0,\ldots,v_{d-1}$ of the following quantity: put $X_v=(\langle v_r,O(X)v_s\rangle)_{r,s}$, $Y_v=(\langle v_r,O(Y)v_s\rangle)_{r,s}$; if $\ker Y_v\subseteq\ker X_v$, take

and otherwise take $+\infty$, where the matrix functions are the continuous functional calculus and $[a]_+=\max(a,0)$. Put $J(V)=\tfrac12[D_2(V_0\mid(V_0+V_1)/2)+D_2(V_1\mid(V_0+V_1)/2)]$, with values in $[0,\infty]$. For each $n\ge0$, put $N=2^n$, let $G_n(r,c)=\prod_{k<n}\kappa(\operatorname{bit}_k(\operatorname{rev}_n r),\operatorname{bit}_k(c))$, where $\kappa(a,0)=1,\ \kappa(a,1)=a$, and form $V_u(y,z)=\prod_{c<N}W_{\sum_r u_rG_n(r,c)}(y_c,z_c)$, with all bit arithmetic in $\mathbb F_2$. For $i<N$, $h\in\mathbb F_2^i$, and $b\in\mathbb F_2$, let $T_{i,h,b}$ be the uniform average of $V_u$ obtained by fixing $u_{<i}=h,u_i=b$ and filling later coordinates from a uniformly chosen full $N$-bit word; define $W_{n,i}^b((h,y),(h',z))=\mathbf1_{h=h'}2^{-i}T_{i,h,b}(y,z)$. To define the error for a set $A\subseteq\{0,\ldots,N-1\}$, project onto the closed span of negative-real-eigenvalue eigenvectors of $R(O(T_{i,h,0}))-R(O(T_{i,h,1}))$, use its complement for bit $0$ and the projection itself for bit $1$, and multiply the selected projections in increasing $i$, with later factors on the left, only at indices in $A$; call the product $K_u$. Set $s(u)=\operatorname{Re}\operatorname{tr}_b(K_u^*K_uO(V_u))$, $e_f=1-2^{-|A|}\sum_{m\in\mathbb F_2^A}s(m\cup f)$ for $f\in\mathbb F_2^{A^c}$, and $\bar e_n(A)=2^{-|A^c|}\sum_f e_f$. For all real $R,\beta$ satisfying $R\ge0$, $[R]_+<J(W)$, and $\beta<\tfrac12$, and every sequence of sets $A_n\subseteq\{0,\ldots,2^n-1\}$ such that $|A_n|=\lfloor2^nR\rfloor_{\mathbb N}$ and $F(W_{n,i})\le F(W_{n,j})$ whenever $i\in A_n,j\notin A_n$, the assertion is

Precisely, for every $a>0$, eventually $|\bar e_n(A_n)|\le a\,2^{-\frac12(2^n)^\beta}$. There is no lower bound on $\beta$; negative $\beta$ and $\beta=0$ are included. The sets must satisfy the stated cardinality and ordering for every $n$, including $n=0$; if no such sequence exists, the implication has no instances.