Generalized unitary induced channels are equal in the commuting and tensor models if and only if Tsirelson's conjecture holds, so the models differ.
Boca, Free products of completely positive positive maps and spectral sets, J
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Unitary induced channels and Tsirelson's problem
Generalized unitary induced channels are equal in the commuting and tensor models if and only if Tsirelson's conjecture holds, so the models differ.