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nLab > Latest Changes: quantum reader monad

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 13th 2022
    • (edited Nov 13th 2022)

    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)

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2022

    With the quantum reader monad ◯ B\bigcirc_B modelling dependency on quantum BB-measurement, we next want to be able to discard (forget) given BB-measurement results while possibly proceeding with making any B′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 𝒱→♮𝒱\mathcal{V} \to \natural{\mathcal{V}} is given by the union over the reader monads

    ◯ B(𝒱)≔(♮(𝒱)×B→♮𝒱) *(♮(𝒱)×B→♮𝒱) * \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(−)≔∐B:FinTypes◯ BE(−). Indifferent (-) \;\; \coloneqq \;\;\; \underset{B : \FinTypes}{\coprod} \bigcirc_{B} E (-) \,.

    The monad product is meant to be given by commuting ◯ B\bigcirc_B through the coproduct (which works since the BB are assumed finite) and then using the natural isomorphisms (of underlying functors)

    ◯ B◯ B′→◯ B×B′, \bigcirc_B \bigcirc_{B'} \to \bigcirc_{B \times B'} \,,

    and the monad unit includes into the component of ◯ *≃id\bigcirc_\ast \;\simeq\; id.

    Moreover, the underlying functor of this monad receives canonical natural injections from ◯ B\bigcirc_B. If we sugar all these transformations to

    discard measurements :◯ B→Indifferent\;\;\; \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→prog 1◯ B 1D 1\;\;\;\;\;\; D_0 \xrightarrow{ prog_1 } \bigcirc_{B_1} D_1 \;\; > discard measurements

    D 1→prog 2◯ B 2D 2\;\;\;\;\;\; D_1 \xrightarrow{ prog_2 } \bigcirc_{B_2} D_2 \;\; > discard measurements

    D 1→prog 2◯ B 3D 3\;\;\;\;\;\; 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→teleport◯ Bit×BitLgclQBit\;\;\;\;\;\; LgclQBit \xrightarrow{ teleport } \bigcirc_{Bit \times Bit} LgclQBit \;\; > discard measurements

    LgclQBit→correct◯ SyndromeLgclQBit\;\;\;\;\;\; LgclQBit \xrightarrow{ correct } \bigcirc_{Syndrome} LgclQBit \;\; > discard measurements

    defining a map

    IndifferentLgclQBit→IndifferentLgclQBitIndifferent \; LgclQBit \to Indifferent \; LgclQBit

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 7th 2023

    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.

    diff, v5, current

    • CommentRowNumber4.
    • CommentAuthorperezl.alonso
    • CommentTimeApr 27th 2023

    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)?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2023

    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 ℂMod Set\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 “ℂ\mathbb{C}-linear tangent ∞\infty-topos” (Hℂ)Mod ∞Grpd≃Ch •(ℂ) ∞Grpd(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)

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 1st 2023

    added observation and proof (here) that the quantum reader monad is a monoidal monad for the evident lax monoidal structure.

    diff, v12, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2023

    improved rendering (here)

    diff, v13, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2023

    Added a remark (here) on how the monoidal monad structure on ◯ W\bigcirc_W generalizes the quantum measurement typing from pure to mixed states by implementing the decoherence process which discard the off-diagonal terms in the density-matrix.

    diff, v14, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 30th 2023

    For the record, I have spelled out (now here) also the comomonoidal comonad structure on the quantum coreader, even though it’s formally dual to the monoidal monad structure

    diff, v17, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 16th 2023
    • (edited Nov 16th 2023)

    added pointer to today’s replacement

    • Stefano Gogioso, Finite-dimensional Quantum Observables are the Special Symmetric Dagger-Frobenius Algebras of CP Maps, EPTCS 394 (2023) 432-441 [arXiv:2110.07074]

    diff, v18, current

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