Manuscript XIX-E@1 · XIX-E.tex · proposition · 702f146e

Proposition: Superoperator commutation is insufficient

Operational support does not determine spacetime or evolution › Commuting channels do not furnish an observable algebra · lines 797–819 · proof included or linked

XIX-E@1#prop:channel-algebra

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A unital completely positive map need not be multiplicative. Commuting channel superoperators, without additional regional algebra hypotheses, do not imply commutation of their images as observables.

Proof. On $M_{2} (C)$ , let

D (A) = P_{0} A P_{0} + P_{1} A P_{1} .

For the Pauli matrix $X$ ,

D (X) = 0, D (X^{2}) = D (I) = I \neq = D (X)^{2} .

For the second assertion, two identity superoperators commute. Their images of the Pauli matrices $X$ and $Z$ do not commute: $[X, Z] \neq = 0$ .

Displayed formulas in this passage

unlabeled · line 807
$D (A) = P_{0} A P_{0} + P_{1} A P_{1} .$
unlabeled · line 811
$D (X) = 0, D (X^{2}) = D (I) = I \neq = D (X)^{2} .$

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section: Commuting channels do not furnish an observable algebra