Manuscript Oai290-A@1 · Oai290-A.tex · proposition · 7c6e104a

Proposition: Exact duplicate noncommutative representations

Analytic examples and separation tests › Two commuting logical copies · lines 1836–1858 · proof included or linked

Oai290-A@1#prop:copies

Suppose unital representations of $B (C)$ on $A$ and $B$ have operators $X_{A}$ and $X_{B}$ satisfying

(X_{A} \otimes I_{B}) V = V X, (I_{A} \otimes X_{B}) V = V X

for every logical $X$ . Then all logical operators commute. Hence such two commuting exact copies do not exist for $d \geq 2$ .

Proof. For logical $X, Y$ , the two physical operators commute because they act on separate factors. Their intertwining identities give

V X Y = (X_{A} \otimes I) (I \otimes Y_{B}) V = (I \otimes Y_{B}) (X_{A} \otimes I) V = V Y X .

Injectivity of $V$ gives $X Y = Y X$ , contradicted by matrix units in $B (C)$ when $d \geq 2$ .

Displayed formulas in this passage

unlabeled · line 1840
$(X_{A} \otimes I_{B}) V = V X, (I_{A} \otimes X_{B}) V = V X$
unlabeled · line 1852
$V X Y = (X_{A} \otimes I) (I \otimes Y_{B}) V = (I \otimes Y_{B}) (X_{A} \otimes I) V = V Y X .$