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…

]]>better rendering of the distributivity conditions

]]>added the description (here) of the QuantumStateReader “double Kleisli category” induced by the distributivity of QuantumState over QuantumReader

]]>added statement and proof (here) that the QuantumState comonad distributes over the QuantumReader monad

(which comes down to nothing but the distributivity of the tensor product over the direct sum)

]]>Of course I was wrong in #6 to say that quantum measurement channels give QuantumState (co)monad transformations. (They do preserve the (co)unit but not the (co)join.)

But partial trace quantum channels do give comonadic QuantumState transformations, have now added this here.

]]>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 (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.

]]>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 the observation (here) that unitary quantum channels are quantum store comonad transformations.

]]>added a remark (here) that on a compact closed category the $\mathscr{H}$-store comonad is left adjoint equivalent to the $\mathscr{H}^\ast$-state monad, and as such a Frobenius monad.

]]>have written out statement and proof (here) characterizing the general Kleisli composition of the linear store comonad on fin-dim vector spaces as the direct composition of (the linear duals of) the corresponding “superoperators”

]]>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.

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