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The question is as stated in the title, very simple. I just need a definite answer on this since I am working with a diagrammatic method of quantum mechanics where this fact might come in very handy.

Maybe I should take this to the math stackexchange?

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closed as unclear what you're asking by G. Smith, Thomas Fritsch, Kyle Kanos, Jon Custer, ZeroTheHero Sep 11 at 21:19

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ You need to define what you mean by “positive map”. If there are two outputs, it isn’t obvious what “positive” means. $\endgroup$ – G. Smith Sep 10 at 19:15
  • $\begingroup$ @G.Smith why should the number of inputs or outputs matter here? A positive map is a map $\mathcal E$ such that $\mathcal E(\rho)\ge0$ for all states $\rho$ (see e.g. here). Or is the OP referring to some other definition? $\endgroup$ – glS Sep 11 at 9:33