Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 BB'-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 𝒱𝒱\mathcal{V} \to \natural{\mathcal{V}} is given by the union over the reader monads

    B(𝒱)((𝒱)×B𝒱) *((𝒱)×B𝒱) * \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 : BIndifferent\;\;\; \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:


    D 0prog 1 B 1D 1\;\;\;\;\;\; D_0 \xrightarrow{ prog_1 } \bigcirc_{B_1} D_1 \;\; > discard measurements

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

    D 1prog 2 B 3D 3\;\;\;\;\;\; D_1 \xrightarrow{ prog_2 } \bigcirc_{B_3} D_3 \;\; > discard measurements


    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:


    LgclQBitteleport Bit×BitLgclQBit\;\;\;\;\;\; LgclQBit \xrightarrow{ teleport } \bigcirc_{Bit \times Bit} LgclQBit \;\; > discard measurements

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

    defining a map

    IndifferentLgclQBitIndifferentLgclQBitIndifferent \; 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 Mod 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 “\mathbb{C}-linear tangent \infty-topos” (H)Mod GrpdCh () 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