Finite local interfaces

All statements concern finite time windows and finite experiments. At each site and tick, reserve a finite interface large enough for that experiment. An interface may contain quantum registers, classical records, occupation and placement labels, and controller memory.

A local observable algebra has the form

where $L_{x,s}$ is a finite classical label set and each ${\mathcal{H}}_{x,s,\lambda }$ is finite dimensional. Different occupation patterns may have different quantum dimensions. Inactive reserved slots are not accessible surviving quantum systems.

For a finite region $I$, write

A state is represented by positive blocks

The trace is the ordinary sum of matrix traces.

On a countable substrate we require a compatible family of states on finite local algebras. No density operator on an unspecified infinite tensor product is required. All calculations below reduce to finite regions.