Proposition: Accessible-current criterion

Let $\Phi ,\Psi$ be channels with the same input and output interfaces, and let $j=j^{†}$ be a fixed output observable. On the full input state space,

Moreover,

Agreement on every observable in a unital $\ast$-algebra ${\mathcal{A}}_{\mathrm{src}}$ is equivalent to equality of the adjoints on that algebra.

If inputs are restricted to the complete classical block-diagonal state space, replace adjoint equality by

and replace the norm in (eq:current-error) by the norm of this pinched operator. Here ${\mathcal{P}}_{\mathrm{cl}}$ is the input block pinching.

Proof. Put $\Delta ={\Phi }^{\ast }(j)-{\Psi }^{\ast }(j)$, which is Hermitian. The expectation difference is $\mathrm{Tr}(\varrho \Delta )$. Pure states in the eigenspaces of $\Delta$ show that all these expectations vanish precisely when $\Delta =0$, and that the largest absolute expectation is $\Vert \Delta {\Vert }_{\mathrm{op}}$. Every algebra element is a complex linear combination of Hermitian elements, proving the algebra statement.

For block-diagonal $\varrho$,

All density operators within every classical block are admitted. Spectral testing within those blocks proves both restricted statements.