bibkey: johnston2010generalized authors: Nathaniel Johnston and David W. Kribs year: 2011 title: Generalized Multiplicative Domains and Quantum Error Correction doi: 10.1090/S0002-9939-2010-10556-7 url: https://arxiv.org/abs/1004.5112v1 claim: A unital completely positive map has a multiplicative domain characterized by its two square equalities; its use for Schur channels retains the unital and trace-preserving hypotheses. strata_touched: [] license: citation-only triage: anchor

Multiplicative-domain criterion consumed by calibrated five-label blocks

Primary text: arXiv:1004.5112v1, section 3, equation (3), Theorem 3; the preprint is from 2010. Published article: Proceedings of the American Mathematical Society 139, 627–639 (2011). The consumer is Hidden Relations, C-29.5.

For a unital completely positive map $\Phi$, the cited classical criterion is

$$\mathrm{MD}(\Phi )=\{A:\Phi (A{)}^{†}\Phi (A)=\Phi (A^{†}A),\Phi (A)\Phi (A{)}^{†}=\Phi (A{A}^{†})\}.$$

On this algebra $\Phi$ respects multiplication and adjoints. The paper recalls this criterion as Theorem 3; its generalized-domain theorems have additional hypotheses. In particular, Theorems 5 and 7 require a unital CPTP map. Theorem 11 instead treats correctable subsystems of a specified code space using an actual representation. These statements are not an unconditional convergence theorem for repeated channels.

For the consumer’s correlation-matrix Schur map $\Phi (A)=\Gamma \circ A$, complete positivity and unit diagonal imply unitality and trace preservation. The two Schwarz defects are positive, and each has trace

$$\sum _{s,t}(1-|{\Gamma }_{st}{|}^2)|A_{st}{|}^2.$$

Because a positive matrix of trace zero is zero, the cited criterion says that the multiplicative domain consists exactly of entries with $|{\Gamma }_{st}|=1$. Common unit records make that relation an equivalence relation. In each such class the channel acts by a diagonal unitary; its inverse is an actual correction only when the phase calibration and gate are supplied. Raw fixed points instead require ${\Gamma }_{st}=1$ on each nonzero matrix entry.

The five-label consumer applies this distinction to an actual square with an $\{O,T\}$ block and three singleton classes. It derives the repeated Schur formula and its phase-corrected limit from matrix-unit eigenvalues and fresh independent environments. Those conclusions are not attributed to this paper’s general theorem. Positive class mass is required before normalizing a conditional block, and class weights remain part of the state.