Accessible output algebras

For ${\mathcal{A}}_L=\mathbb{C}I$, this is the record criterion and $d_{{\mathcal{O}}_L}={\delta }_{\mathrm{rec}}$. For the full residual algebra, it compares complete instrument outputs.

Let $\Lambda :L({\mathcal{H}}_{\mathrm{in}})\to L({\mathcal{H}}_{\mathrm{out}})$ be a channel and let $\mathcal{A},\text{cB}$ be unital ${}^{\ast }$-subalgebras. Equality of input states on $\mathcal{A}$ implies equality of their outputs on $\text{cB}$ for every pair of states if and only if

For inputs restricted to a classical block-diagonal state space, replace ${\Lambda }^{\ast }(\text{cB})$ by ${\mathcal{P}}_{\mathrm{cl}}({\Lambda }^{\ast }(\text{cB}))$, where ${\mathcal{P}}_{\mathrm{cl}}$ is its block pinching and $\mathcal{A}$ lies in that block algebra.