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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 $B$-basis anywhere is a Kleisli morphism $Circ : \mathscr{H} \coloneqq \Box_B \mathscr{H}_\bullet \longrightarrow \mathscr{H}_\bullet$ for the linear necessity-comonad, and the Kleisli equivalence says that this equals a coherent (non-measurement) quantum circuit $\delta^\Box \circ \Box Circ \colon \mathscr{H} \to \mathscr{H}$ postcomposed with the $\Box$-counit: But the latter is the measurement gate.
added pointer to
where the statement is Axiom B (p. 6 of 12)
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).
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.
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.
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
Schrödinger’s cat (1935)
Everett’s observers (1957)
Wigner’s friend (1961)
The author of these paragraphs asserts that:
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.
This is just what is formalized by the set-up of the deferred measurement principle:
The first observer (called “cat” or “A” or “friend”) is the controlled quantum gate denoted “$G$” above,
the quantum system observed by the first observer is $\mathrm{Q}W$ above,
the state space of the first observer is $\mathscr{H}$ (before) and $\mathscr{H}'$ (after the observation).
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 $\mathscr{H}'$).
But the deferred measurement principle says the outcome is indistinguishable from the situation where the first observer already collapses the original state in $\mathrm{Q}W$.
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