Mixed subsystem and algebra Choi targets

For $H=C_L\otimes S$, with dimensions $\ell ,s$ and $d=\ell s$,

The reference includes both $L^{\prime }$ and $S^{\prime }$. Testing only $L^{\prime }$ with a fixed spectator prepares a different, weaker task.

For the algebra target, with $c_{\alpha }=\dim C_{\alpha }$ and $s_{\alpha }=\dim S_{\alpha }$,

where each block pairs its output sector with the matching input reference sector. Equations (eq:subchoi) and (eq:algchoi) follow by expanding the normalized maximally entangled input in matrix units.

A single overlap with these mixed states does not permit the pure-target proof of Corollary cor:pure. The general trace-norm certificate below avoids that mistake.