Boundary energy bounds thermal mutual information

Assume the Axiom of Choice. Let ${\mathcal{H}}_A,{\mathcal{H}}_B$ be nonzero finite-dimensional complex inner-product spaces. Let $H_A=H_A^{\ast }$ act on ${\mathcal{H}}_A$, let $H_B=H_B^{\ast }$ act on ${\mathcal{H}}_B$, and let $H_{\mathrm{\partial }}=H_{\mathrm{\partial }}^{\ast }$ act on ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$. Set

$H:=H_A\otimes I_B+I_A\otimes H_B+H_{\mathrm{\partial }}\mathrm{.}$

For $\beta >0$, define $Z_{\beta }:=\mathrm{Tr}(e^{-\beta H})$ and the finite Gibbs density ${\rho }_{\beta }:=e^{-\beta H}\mathrm{/}{Z}_{\beta }$. Then

$I(A:B{)}_{{\rho }_{\beta }}\le 2\beta \mathrm{\parallel }{H}_{\mathrm{\partial }}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}},$

where $I(A:B)$ uses dimensionless entropy in nats and $\mathrm{\parallel }K{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}:={\sup }_{\mathrm{\parallel }v\mathrm{\parallel }=1}\mathrm{\parallel }Kv\mathrm{\parallel }$ is the operator norm.

For the optional lattice-family conclusion, fix $m\ge 1$. For each member $s$ of a specified family, take a finite site set ${\mathrm{\Lambda }}_s\subseteq {\mathbb{Z}}^m$, a partition ${\mathrm{\Lambda }}_s=A_s\bigsqcup B_s$, and nonzero finite-dimensional complex inner-product spaces ${\mathcal{K}}_{s,x}$ for $x\in {\mathrm{\Lambda }}_s$. Use the lexicographic order on ${\mathbb{Z}}^m$ in ${\mathcal{H}}_{s,X}:={⨂}_{x\in X}{\mathcal{K}}_{s,x}$, with the empty tensor product defined as $\mathbb{C}$, and identify ${\mathcal{H}}_{s,{\mathrm{\Lambda }}_s}$ with ${\mathcal{H}}_{s,A_s}\otimes {\mathcal{H}}_{s,B_s}$ in the displayed factor order. Let $J_s$ be a finite set of interaction labels with nonempty supports $X_{s,j}\subseteq {\mathrm{\Lambda }}_s$ and self-adjoint terms $h_{s,j}$ on ${\mathcal{H}}_{s,X_{s,j}}$. For $Y\in \{{\mathrm{\Lambda }}_s,A_s,B_s\}$ containing $X_{s,j}$, write ${\widehat{h}}_{s,j}^Y$ for the extension of $h_{s,j}$ to ${\mathcal{H}}_{s,Y}$ by the identity on every factor in $Y\setminus X_{s,j}$. The crossing-label set is

$J_{\mathrm{\partial },s}:=\{j\in J_s:X_{s,j}\cap A_s\mathrm{\ne }\mathrm{\varnothing }\text{ and }{X}_{s,j}\cap B_s\mathrm{\ne }\mathrm{\varnothing }\},\mathrm{\mid }\mathrm{\partial }{A}_s\mathrm{\mid }:=\mathrm{\mid }{J}_{\mathrm{\partial },s}\mathrm{\mid }\mathrm{.}$

Let $J_{A,s}:=\{j\in J_s:X_{s,j}\subseteq A_s\}$ and $J_{B,s}:=\{j\in J_s:X_{s,j}\subseteq B_s\}$. Set $H_{A_s}:={\sum }_{j\in J_{A,s}}{\widehat{h}}_{s,j}^{A_s}$, $H_{B_s}:={\sum }_{j\in J_{B,s}}{\widehat{h}}_{s,j}^{B_s}$, and $H_{\mathrm{\partial },s}:={\sum }_{j\in J_{\mathrm{\partial },s}}{\widehat{h}}_{s,j}^{{\mathrm{\Lambda }}_s}$. Under the stated identification these give the displayed Hamiltonian decomposition. For each $s$, assume ${\beta }_s>0$ and define ${\rho }_{{\beta }_s}$ by the Gibbs formula above. If one specified $C\ge 0$ satisfies $\mathrm{\parallel }{H}_{\mathrm{\partial },s}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le C\mathrm{\mid }\mathrm{\partial }{A}_s\mathrm{\mid }$ for every family member, the bound gives $I(A_s:B_s{)}_{{\rho }_{{\beta }_s}}\le 2{\beta }_{s}C\mathrm{\mid }\mathrm{\partial }{A}_s\mathrm{\mid }$ memberwise. No finite-range condition or additional geometric scaling assumption is used. When $H_{\mathrm{\partial },s}$ and each interaction term are measured in joules, ${\beta }_s$ has units J${}^{-1}$ and $C$ has units J, so the bound is dimensionless.

Dimensionless entropy, mutual and conditional information

Finite Gibbs free-energy and entropy optimization

The Axiom of Choice