Infinite algebraic state and finite-region restrictions

Let $\mathcal{A}$ be the unital quasi-local spin C*-algebra of Norm-completed quasi-local quantum spin algebra, whose C*-structure uses Countable Choice as in Quasi-local algebra completion and locality. An infinite algebraic state is a bounded complex-linear functional $\omega :\mathcal{A}\to \mathbb{C}$ satisfying $\omega (I)=1$ and $\omega (A^{\ast }A)\ge 0$ for every $A\in \mathcal{A}$.

For each finite region $\mathrm{\Lambda }\subseteq {\mathbb{Z}}^d$, the restriction $\omega {\mathrm{\mid }}_{{\mathcal{A}}_{\mathrm{\Lambda }}}$ is a finite-matrix state. By Finite matrix C-star algebra and density-state duality, every positive normalized functional on a finite matrix algebra is represented by a unique density matrix. Define its local density matrix ${\rho }_{\mathrm{\Lambda }}$ by

$\omega (A)={\mathrm{Tr}}_{\mathrm{\Lambda }}({\rho }_{\mathrm{\Lambda }}A)(A\in {\mathcal{A}}_{\mathrm{\Lambda }}),$

where ${\rho }_{\mathrm{\Lambda }}={\rho }_{\mathrm{\Lambda }}^{\ast }\ge 0$ and ${\mathrm{Tr}}_{\mathrm{\Lambda }}{\rho }_{\mathrm{\Lambda }}=1$. For $\mathrm{\Lambda }\subseteq {\mathrm{\Lambda }}^{\mathrm{\prime }}$, put $\mathrm{\Delta }={\mathrm{\Lambda }}^{\mathrm{\prime }}\setminus \mathrm{\Lambda }$ and use the configuration-basis factorization ${\mathcal{H}}_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}\cong {\mathcal{H}}_{\mathrm{\Lambda }}\otimes {\mathcal{H}}_{\mathrm{\Delta }}$ of Local spin matrix algebras and their factor inclusions. If $T\in {\mathcal{A}}_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}$, define its partial trace over $\mathrm{\Delta }$, ${\mathrm{Tr}}_{\mathrm{\Delta }}(T)\in {\mathcal{A}}_{\mathrm{\Lambda }}$, by its matrix coefficients

$⟨{\mathrm{Tr}}_{\mathrm{\Delta }}(T)e_s,e_{s^{\mathrm{\prime }}}⟩:={\sum }_{t\in S_{\mathrm{\Delta }}}⟨T(e_s\otimes e_t),e_{s^{\mathrm{\prime }}}\otimes e_t⟩(s,s^{\mathrm{\prime }}\in S_{\mathrm{\Lambda }}),$

where $e_s$ and $e_t$ are the configuration basis vectors of ${\mathcal{H}}_{\mathrm{\Lambda }}$ and ${\mathcal{H}}_{\mathrm{\Delta }}$. The finite sums depend linearly on $T$, so they define a linear map ${\mathcal{A}}_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}\to {\mathcal{A}}_{\mathrm{\Lambda }}$ with a unique output matrix for each $T$. Expanding both configuration-basis traces gives its pairing identity

${\mathrm{Tr}}_{\mathrm{\Lambda }}\text{\hspace{0.17em}⁣}(A{\mathrm{Tr}}_{\mathrm{\Delta }}(T))={\mathrm{Tr}}_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}\text{\hspace{0.17em}⁣}((A\otimes I_{\mathrm{\Delta }})T)(A\in {\mathcal{A}}_{\mathrm{\Lambda }}),$

with the common value given by

${\sum }_{s,s^{\mathrm{\prime }}\in S_{\mathrm{\Lambda }}}{\sum }_{t\in S_{\mathrm{\Delta }}}⟨A{e}_{s^{\mathrm{\prime }}},e_s⟩⟨T(e_s\otimes e_t),e_{s^{\mathrm{\prime }}}\otimes e_t⟩\mathrm{.}$

The pairing identity determines this matrix uniquely: if $D\in {\mathcal{A}}_{\mathrm{\Lambda }}$ pairs to zero with every $A$, taking the configuration-basis matrix units $A=E_{ji}$ gives ${\mathrm{Tr}}_{\mathrm{\Lambda }}(E_{ji}D)=⟨D{e}_j,e_i⟩=0$ for every $i,j$, so $D=0$. The local densities are consistent when ${\mathrm{Tr}}_{{\mathrm{\Lambda }}^{\mathrm{\prime }}\setminus \mathrm{\Lambda }}({\rho }_{{\mathrm{\Lambda }}^{\mathrm{\prime }}})={\rho }_{\mathrm{\Lambda }}$. No density operator on a globally chosen Hilbert space is part of this definition.

This agrees with the canonical state convention of States and positive functionals on a C star algebra: the finite restrictions have norm one by Finite matrix C-star algebra and density-state duality, giving $\mathrm{\mid }\omega (A)\mathrm{\mid }\le \mathrm{\parallel }A\mathrm{\parallel }$ on the dense local algebra and hence on its completion; evaluation at the norm-one identity gives $\mathrm{\parallel }\omega \mathrm{\parallel }=1$. Conversely a canonical state has local restrictions with density matrices of trace one, since the finite norm formula and continuity on the dense local algebra imply $\mathrm{\parallel }\omega \mathrm{\parallel }=\omega (I)$. No global density representation is asserted.