Remark: Partners and extra commuting copies

The normal form (eq:normalform) represents the recovered logical algebra on the accessible support as $T(\mathcal{B}(C)\otimes I_K)T^{†}$. Its commutant on that support is $T(I_C\otimes \mathcal{B}(K))T^{†}$, as follows by commuting with all logical matrix units. The syndrome algebra is not a second recoverable copy of the same unknown logical state. An abstract algebra of the same dimension may exist there without carrying that input.

This is the distinction relevant to the supplied partner comparisons in Hawk03-A CheungHawk03A. No independent-factor entropy bound is used to estimate complete leakage or identify its maximum.