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now creating this entry.
The technical material under “Details” (here) is copied over from what I had written at reader monad – Examples – quantum reader monad. (There may still be room left to adjust the wording in order to reflect that this material moved to a new entry.)
To this I have now added an Idea-section (here) which highlights the relation to (equivalence with) Bob Coecke’s “classical structures” (which term I made redirect to here now)
With the quantum reader monad $\bigcirc_B$ modelling dependency on quantum $B$-measurement, we next want to be able to discard (forget) given $B$-measurement results while possibly proceeding with making any $B'$-measurements, and so on.
I am thinking of using a modality – maybe to be called “indifferently” and maybe to be denoted by a dashed circle –
which is the monad on the full category of bundles of linear types (e.g. VectBund over Set) that on a bundle $\mathcal{V} \to \natural{\mathcal{V}}$ is given by the union over the reader monads
$\bigcirc_B (\mathcal{V}) \;\coloneqq\; \big( \natural(\mathcal{V}) \times B \to \natural{\mathcal{V}}\big)_\ast \big( \natural(\mathcal{V}) \times B \to \natural{\mathcal{V}}\big)^\ast$with respect to all finite types:
$Indifferent (-) \;\; \coloneqq \;\;\; \underset{B : \FinTypes}{\coprod} \bigcirc_{B} E (-) \,.$The monad product is meant to be given by commuting $\bigcirc_B$ through the coproduct (which works since the $B$ are assumed finite) and then using the natural isomorphisms (of underlying functors)
$\bigcirc_B \bigcirc_{B'} \to \bigcirc_{B \times B'} \,,$and the monad unit includes into the component of $\bigcirc_\ast \;\simeq\; id$.
Moreover, the underlying functor of this monad receives canonical natural injections from $\bigcirc_B$. If we sugar all these transformations to
discard measurements
$\;\;\; \colon \;\; \bigcirc_B \to Indifferent$
then we can code a general quantum protocol consisting of a sequence of circuits followed by measurements, via do
-notation for the indifference monad like this:
do
$\;\;\;\;\;\; D_0 \xrightarrow{ prog_1 } \bigcirc_{B_1} D_1 \;\;$ > discard measurements
$\;\;\;\;\;\; D_1 \xrightarrow{ prog_2 } \bigcirc_{B_2} D_2 \;\;$ > discard measurements
$\;\;\;\;\;\; D_1 \xrightarrow{ prog_2 } \bigcirc_{B_3} D_3 \;\;$ > discard measurements
etc.
For example, a teleportation protocol for a noisy teleportation channel (we can do all this with mixed states, but let me not make that notationally explicit) followed by error correction would be given by the follwing indifference-do
-notation:
do
$\;\;\;\;\;\; LgclQBit \xrightarrow{ teleport } \bigcirc_{Bit \times Bit} LgclQBit \;\;$ > discard measurements
$\;\;\;\;\;\; LgclQBit \xrightarrow{ correct } \bigcirc_{Syndrome} LgclQBit \;\;$ > discard measurements
defining a map
$Indifferent \; LgclQBit \to Indifferent \; LgclQBit$
below the paragraph on the Coecke-Pavlovic incarnation of the quantum reader monad, I added the line:
In this guise, the quantum reader monad is reflected by the “green spiders” in the ZX-calculus.
Is it expected that the naive analogue to linear n-categories correctly captures the concept of (higher?) measurement in Quantum Field Theory? Also, since the symmetries of the chosen B act on the image, would it be more desirable to talk about soft quotients [B,X]//Aut(B) (in whatever is the correct implementation for vector spaces)?
Under caveat that there is currently not much to go by regarding “expectations”, I do think the answer is certainly: yes.
The general picture is:
traditional classically interacting quantum information theory (such as captured by the language Quipper) has its categorical semantics in the category $\mathbb{C}Mod_{Set}$ (as expanded on in the entry quantum circuits via dependent linear types)
that is just a small sub-category of the “$\mathbb{C}$-linear tangent $\infty$-topos” $(H \mathbb{C})Mod_{\infty Grpd} \,\simeq\, Ch_\bullet(\mathbb{C})_{\infty Grpd}$ which ought to interpret all of linear homotopy type theory (I am busy building the model, here, as we speak) which may be regarded as a universal quantum programming language (exposition here)
including “topological effects” which is essentially the way that physics (physicists) currently get(s) closest to seeing “higher homotopy quantum information” structure (as argued at some length here)
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