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Starting an entry (to rhyme on “quantum reader monad”) on the linear version of the costate comonad and how its coKleisli category on the tensor unit alone exhibits the structure of quantum observables.
Made some cosmetic rearrangements and did some polishing.
Have renamed the entry from “quantum costate comonad” to “quantum state monad”.
First, both fuse to a single Frobenius monad, so a priori either terminology is fine as the other. First I thought the “costate”-terminology would be more suggestive since the coKleisli morphisms are quantum observables.
But if we say “quantum state monad” then it becomes a theorem that
unitary quantum channels are quantum state transformations
which is too happy a terminological coincidence not to embrace.
added (here) the observation that quantum measurement channels are quantum state monad transformations.
Generally, unital quantum channels are quantum state monad transformations.
It seems that the space of quantum state monad transformations $\mathscr{H}State \to \mathscr{H}State$ cannot be much larger than that given by the unital quantum channels, but I don’t have a proof yet that they coincide.
added the observation (here) that
the action of those quantum state transformations which correspond to unitary quantum channels on the quantum state contextful scalars (the Kleisli maps $\mathscr{H} \otimes \mathscr{H}^\ast \to \mathbb{1}$) gives the Heisenberg-picture evolution of observables
added (still here) also the converse direction for the distributivity statement:
The QuantumState comomad distributes over the QuantumEnvironment monad
The QuantumEnvironment comonad distributes over the QuantumState monad.
All the many diagrams to be checked commute trivially (by ordinary distributivity of tensor over direct sum), though so far I don’t have an argument why they trivially commute trivially, so I just write the all out…
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