Certified transfer gap gives exponential MPS clustering

Define the linear functional $\phi (X):=\mathrm{tr}(\mathrm{\ell }X)$ and the rank-one map $P(X):=\rho \phi (X)$. Since $\mathrm{tr}\rho =1$, $\rho \mathrm{\ne }0$; the pairing normalization $\phi (\rho )=1$ gives $P^2=P$. The fixed-point equations give $\mathcal{E}P=P$ and $P\mathcal{E}=P$, since $\phi (\mathcal{E}(X))=\mathrm{tr}({\mathcal{E}}^{\ast }(\mathrm{\ell })X)=\phi (X)$. Hence $M_D(\mathbb{C})=\mathbb{C}\rho \oplus \ker \phi$ is an $\mathcal{E}$-invariant direct sum, $\mathcal{E}$ is the identity on $\mathbb{C}\rho$, and $P$ is zero on $\ker \phi$. If the restriction of $\mathcal{E}$ to $\ker \phi$ had eigenvalue $1$, it would have a nonzero eigenvector $X$ there. Since $\phi (X)=0$ and $\phi (\rho )=1$, $X$ and $\rho$ would be linearly independent eigenvectors of $\mathcal{E}$ for $1$, so $\dim E_1(\mathcal{E})\ge 2$. But [F7]–[F8] identify the Statement's simple-root hypothesis with algebraic multiplicity one, and [F9] bounds this geometric multiplicity by the algebraic multiplicity. This contradiction shows that the restriction has no eigenvalue $1$. Therefore ${\mathcal{E}}^g-P$ is zero on $\mathbb{C}\rho$ and equals $(\mathcal{E}{\mathrm{\mid }}_{\ker \phi }{)}^g$ on the invariant complement.

Use $\mathrm{\parallel }X{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}:={\sup }_{\mathrm{\parallel }v\mathrm{\parallel }=1}\mathrm{\parallel }Xv\mathrm{\parallel }$ on $M_D(\mathbb{C})$ and the induced norm $\mathrm{\parallel }\mathrm{\Phi }\mathrm{\parallel }:={\sup }_{\mathrm{\parallel }X{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}=1}\mathrm{\parallel }\mathrm{\Phi }(X){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ on its linear maps. If $\ker \phi =\{0\}$, then $\mathcal{E}=P=I$ and ${\mathcal{E}}^g-P=0$ for every $g\ge 0$; for every allowed $r$, take $K_r=0$. Otherwise the restriction is a finite-dimensional complex endomorphism of positive dimension $m$. Its characteristic polynomial is monic of degree $m$ by [F7]. By [F4] it has a root $\mu$. Write this characteristic polynomial as $p(x)={\sum }_{k=0}^m{c}_k{x}^k$. Since $p(\mu )=0$, the identities $x^k-{\mu }^k=(x-\mu ){\sum }_{t=0}^{k-1}{x}^{k-1-t}{\mu }^t$ give $p(x)=(x-\mu )h(x)$ with $\mathrm{deg}h=m-1$. Induction on degree therefore shows that this characteristic polynomial splits into linear factors over $\mathbb{C}$. Now [F5] gives a similarity to finitely many Jordan blocks $J_s(\lambda )=\lambda I+N$ with $N^s=0$. Every such $\lambda$ belongs to ${\sigma }_{\mathbb{C}}(\mathcal{E})\setminus \{1\}$, since an eigenvector of the restriction is also an eigenvector of $\mathcal{E}$ and step 1.1 excludes $1$ from the restricted spectrum; hence $\mathrm{\mid }\lambda \mathrm{\mid }\le q$. If $q=0$, all Jordan powers of the restriction vanish for sufficiently large $g$, so ${\mathcal{E}}^g-P=0$ for those $g$. Since every allowed $r>q=0$ is positive, the finitely many remaining quantities $\mathrm{\parallel }{\mathcal{E}}^g-P\mathrm{\parallel }{r}^{-g}$ have a finite maximum; take this maximum as $K_r$ to obtain the required bound for all $g$. If $q>0$, fix $r>q$. For a block with $\lambda \mathrm{\ne }0$ and $g\ge s$, the binomial expansion gives $J_s(\lambda {)}^g={\sum }_{j=0}^{s-1}\left(\genfrac{}{}{0px}{}{g}{j}\right){\lambda }^{g-j}{N}^j$. For every fixed $j$, $g^j(\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{/}r{)}^g$ is bounded because the ratio of consecutive terms tends to $\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{/}r<1$; hence each matrix entry is bounded by a constant times $r^g$. Blocks with $\lambda =0$ vanish for $g\ge s$, and all finitely many smaller $g$ are absorbed into the constant. In the maximum-coordinate norm of the Jordan basis, the norm of the block-diagonal power is bounded by a constant times $r^g$, since there are finitely many blocks and each has finitely many entries; the zero action of ${\mathcal{E}}^g-P$ on $\mathbb{C}\rho$ adds only a one-dimensional zero block. The fixed change-of-basis map and its inverse have finite induced norms, so submultiplicativity transfers this bound to the stated superoperator norm: $\mathrm{\parallel }{\mathcal{E}}^g-P\mathrm{\parallel }\le K_r{r}^g$ for a finite $K_r$ and every $g\ge 0$.

By [F1]–[F2], the transfer contraction for the two blocks with a gap of $g$ identities is $\mathrm{tr}(\mathrm{\ell }{\mathcal{E}}_{O_A}{\mathcal{E}}^g{\mathcal{E}}_{O_B}(\rho ))$. The product of the separate expectations equals $\mathrm{tr}(\mathrm{\ell }{\mathcal{E}}_{O_A}P{\mathcal{E}}_{O_B}(\rho ))$, since $P(X)=\rho \mathrm{tr}(\mathrm{\ell }X)$. Subtracting gives $C_g=\mathrm{tr}(\mathrm{\ell }{\mathcal{E}}_{O_A}({\mathcal{E}}^g-P){\mathcal{E}}_{O_B}(\rho ))$.

Diagonalize $\mathrm{\ell }\ge 0$ using [F6] as $\mathrm{\ell }={\sum }_j{t}_j\mathrm{\mid }{u}_j⟩⟨u_j\mathrm{\mid }$ with $t_j>0$. For every matrix $X$, $\mathrm{\mid }\mathrm{tr}(\mathrm{\ell }X)\mathrm{\mid }=\mathrm{\mid }{\sum }_j{t}_j⟨X{u}_j,u_j⟩\mathrm{\mid }\le \mathrm{tr}(\mathrm{\ell })\mathrm{\parallel }X{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$. The block insertion maps are finite sums of matrix multiplications, so their induced norms in step 2.1 are finite; submultiplicativity gives $\mathrm{\mid }{C}_g\mathrm{\mid }\le \mathrm{tr}(\mathrm{\ell })\mathrm{\parallel }{\mathcal{E}}_{O_A}\mathrm{\parallel }{K}_r\mathrm{\parallel }{\mathcal{E}}_{O_B}(\rho ){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}{r}^g$. All factors before $r^g$ are finite and independent of $g$, so their product is an admissible $C$. This proves the claimed exponential clustering bound, including Jordan blocks and the case $q=0$.