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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2022
    • (edited Oct 27th 2022)

    starting something.

    I claim that in terms of quantum circuits via dependent linear types, the principle of deferred measurement is immediately formalized and proven by the Kleisli equivalence:

    Namely a quantum circuit involving measurement in the BB-basis anywhere is a Kleisli morphism Circ: B Circ : \mathscr{H} \coloneqq \Box_B \mathscr{H}_\bullet \longrightarrow \mathscr{H}_\bullet for the linear necessity-comonad, and the Kleisli equivalence says that this equals a coherent (non-measurement) quantum circuit δ Circ:\delta^\Box \circ \Box Circ \colon \mathscr{H} \to \mathscr{H} postcomposed with the \Box-counit: But the latter is the measurement gate.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2022
    • (edited Oct 27th 2022)

    I have added a diagram (here) showing how the deferred measurement principle looks just like the Kleisli equivalence for the necessity comonad on dependent linear types in quantum modal logic.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2022

    added pointer to

    • Sam Staton, Algebraic Effects, Linearity, and Quantum Programming Languages, POPL ’15: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2015) [doi:10.1145/2676726.2676999, pdf]

    where the statement is Axiom B (p. 6 of 12)

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2023
    • (edited Aug 7th 2023)

    started (here) a section “Deferred measurement and Interpretations”, so far with a full quote of the fable which is Everett (1957)’s key argument for rejecting the Copenhagen interpretation, followed by a brief comment that the deferred measurement principle may be understood as resolving the apparent paradox (which is really the paradox of Schrödinger’s cat, in different words).

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2023

    uploaded a mildly improved version of the diagram (here)

    which proves the deferred measurement principle in quantum modal logic

    diff, v9, current

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 8th 2023

    Interesting! Is there any way to use the Kleisli formulation to help people past what they take to be paradoxical in Section 3? Not sure what that would be. Maybe some new ordinary language rendition of the type theory/category theory.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 8th 2023
    • (edited Aug 8th 2023)

    I was thinking about spelling out the formalization of the “paradox” further, but need to be looking into other tasks with higher priority.

    Briefly, you need to identify the cat/observer1 with the controlled quantum gate (which sets the cat’s quantum state in dependence of the observed qbit). The cat/observer1-perspective is that on the left hand of the Kleisli equivalence, the perspective of the experimentor who opens the cat’s box (observer2) is that on the right. On the right, the cat/observer1 remains in superposition and is only collapsed by observer2.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 8th 2023

    have spelled out the example (here) of (deferred) measurement on a CNOT-gate

    diff, v10, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2023
    • (edited Sep 23rd 2023)

    added a couple of lead-in paragraphs to the section “Interpretation of Deferred measurement” bringing out two points/claims more explicitly:


    The folklore of quantum physics knows paradoxical-sounding stories under the title of

    The author of these paragraphs asserts that:

    1. These are all the same story, recast with different actors: Schrödinger’s cat plays the same role as Everett’s observer A and the same role as Wigner’s friend. The point in any case is that this first observer makes a quantum measurement and (only) ofterwards is himself observed by a second observer.

    2. This is just what is formalized by the set-up of the deferred measurement principle:

      1. The first observer (called “cat” or “A” or “friend”) is the controlled quantum gate denoted “GG” above,

      2. the quantum system observed by the first observer is QW\mathrm{Q}W above,

      3. the state space of the first observer is \mathscr{H} (before) and \mathscr{H}' (after the observation).

      4. The second observer inspecting the scene at the end is the right hand side of the above setup, where the measurement is made at the end of the circuit execution. Before it is made, the first observer may have been in a superposition (in \mathscr{H}').

      5. But the deferred measurement principle says the outcome is indistinguishable from the situation where the first observer already collapses the original state in QW\mathrm{Q}W.


    diff, v12, current