Access is not automatically a spatial tensor factor

In a finite physical output space, declare a unital observable algebra $\mathcal{A}\subseteq \mathcal{B}({\mathcal{H}}_{\mathrm{phys}})$. For a tensor factor this can be $\mathcal{B}(R)\otimes I_B$. For a general finite algebra, a decomposition is

Its restricted state can be described by weights $p_{\alpha }$ and normalized accessible matrices ${\rho }_{\alpha }$. With the explicitly chosen ordinary matrix trace on each accessible block, one may define

This entropy convention excludes the multiplicity factors $m_{\alpha }$; it is not automatically the entropy of an ambient density operator.

Cheung's En05-C compares state restrictions with Heisenberg containment without assuming a spatial complement CheungEn05C. A continuum gravitational algebra requires its own representation, trace or entropy prescription, and comparison theorem.