Proposition: One-shot environment ledger

Let

where $J$ is the unnormalized Choi matrix. Among dilations with orthogonal pointer sectors for the reported outcomes $h$, the minimum environment dimension is $R_g$.

A common pointer reserve of

qubits suffices. If matter is encoded in $s$ qubits, a unitary extension of (eq:dilation) uses those $s$ matter qubits and $e$ fresh pointer qubits. Full fine readout produces at most $e$ classical bits per step.

Proof. A Choi matrix of rank $r_{h|g}$ has a decomposition into precisely that many rank-one positive terms, giving a Kraus family of that size. Conversely, a family with $q$ operators gives a Choi sum of $q$ rank-one terms, so $q\ge r_{h|g}$.

An environment sector of dimension $q_h$ gives at most $q_h$ Kraus operators when expanded in a sector basis. Thus $q_h\ge r_{h|g}$, and summing sectors gives the lower bound $R_g$. Minimal Kraus families in (eq:dilation) attain it.

Encode the pointer in $e$ qubits. The isometry on the subspace with pointer input $|0{⟩}^{\otimes e}$ extends to a unitary by completing orthonormal bases. Destructive readout consumes the pointer and returns a finite classical fine index.