Unifying Classical Physics and Quantum Physics

I've written a quasi-popular level article that has been published on The Quantum Daily: Unifying Classical Physics and Quantum Physics. The principal audience, of people with an interest in quantum computing, will want more details, but I think it should be more-or-less readable on its own.
On Facebook, I've made the point that this is not as nuanced as An algebraic approach to Koopman classical mechanics, but of course you can take me to task for getting it wrong anyway.

The QCS axioms of Frauchiger&Renner, in an algebraic form

Piling on Frauchiger&Renner is a sport that I've never felt much inclination for, but a paper in SHPMP, "The Frauchiger-Renner argument: A new no-go result?", https://doi.org/10.1016/j.shpsb.2019.12.002, Fortin&Lombardi, states F&R's premises neatly enough that I feel like messing with the field:

(Q) Compliance with quantum theory: Quantum mechanics is universally valid, that is, it applies to systems of any complexity, including observers. Moreover, an agent knows or is certain that a given proposition is true whenever the Born rule assigns probability 1 to it.
(C) Self-consistency: Different agents' predictions are not contradictory.
(S) Single-world: From the viewpoint of an agent who carries out a particular measurement, this measurement has one single outcome.

If we take an algebraic approach, we have, instead,
(Q) There is a collection of measurements (modeled by a collection of operators) and recorded measurement results for each of those measurements (modeled by real non-negative normalized linear forms over the algebra generated by the collection of operators) [there are no "systems" and no "observers", only measurements, models of the measurements, and results and models of the results].
(C) If recorded measurement results match the modeled measurement results, we're good, otherwise change the recorded measurement results [just kidding! It's a lot safer to change either our model of the measurements or our model of the measurement results.]
(S) Journal editors are suspicious of recorded measurement results that change after they have accepted an article [it is well-known that journal editors are the closest anyone can come to being God, but there are also plenty of physicists ready to pillory a cheat.]

F&R's rules don't seem an interesting game. The above may seem too operational, but we can optionally introduce a very beyond-the-operational in-between rule:
(B) We can introduce a continuum of models of measurements and of their results that are, in various ways, in-between the measurements we actually performed or intend to perform, and which we might perform, perhaps even unexpectedly [that is, metaphysically, we can imagine a continuum of measurements and their results.]

"The Talk" — A Discussion

The Talk is a fun and very helpful read for anyone who wants to think about Quantum Mechanics, but for an analysis of just what Scott Aaronson and Zach Weinersmith do in it, I will here ignore the child's side of the dialogue and focus on what the grown up says that I think is most substantive:

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"[superposition] means a complex linear combination of a 0 state and a 1 state. You should think of it as a new ontological category: a way of combining things that doesn't really map onto any classical concept."

"Quantum Mechanics is just a certain generalization of probability."  ✔✔✔

"Quantum Mechanics has probability amplitudes, and they can be positive or negative or even complex."

"When you make a measurement, there's a rule for converting these amplitudes into ordinary probabilities. But when you're not looking, the amplitudes, well, sometimes they do something very special and private with each other. Something very... intimate."

Next panel, child: "Interference?"

"If an event could happen one way with a positive amplitude, and another way with a negative amplitude, the two amplitudes can cancel each other out. So the total amplitude is zero -"

"In quantum computing, the whole idea is just to choreograph a pattern of interference where the paths leading to each wrong answer interfere destructively and cancel out, while the paths leading to the right answer reinforce each other."  ✔

"the important thing for you to understand is that quantum computing isn't just a matter of trying all the answers in parallel."  ✔✔✔

 

"It's not the size that matters. It's the rotation through complex vector space."

 

"It's all just different consequences of one fact: classical events have probabilities, and quantum events have amplitudes. Remember that, and you'll do just fine."

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The paucity of approving ✔'s may look damning, if anyone cares about my approval, but there is a lot to like in the gaps. What I really like about "The Talk" is that events, which are what we record as our experimental raw data, are almost front and center. If only amplitudes and interference were not in front of them. Just if the grown up were to say "quantum computing is just about choreographing and analyzing noisy patterns of events," I would be much happier.

Starting with the last quote above, however, the concept of a "quantum event" as something distinct from a "classical event" is introduced without definition, so that putting so very much weight on "quantum events have amplitudes", where "amplitudes" are also barely defined except as somehow a precursor for probabilities, is a big jump. The word "event" is mentioned just in one earlier panel, without qualification as classical or quantum, a few panels later "answer" is substituted as, it seems, a synonym for "event", then suddenly the whole weight of how we should think about QM is loaded onto the distinction between "classical event" and "quantum event".

We certainly see and record "events" when we perform experiments that are characteristically quantum mechanical, which we could call "classical events", but there isn't a different kind of events that happen in experiments that we could call "quantum events". When, for example, we see spots or marks appear on a screen, all of them are events, even if each of them may play a different part in different patterns of many events, whether as a spot in a trajectory-like pattern or as a spot in a wave-like pattern. More generally and abstractly, we construct hardware that examines the many electric currents on the wires that come out of an experiment for patterns that may be very complicated, then that hardware records what type of event happened and when it did, often millions per second, without human seeing or intervention.

One significant failure, I think, is in not introducing the idea of a state as something [in fact: a normalized, positive, complex linear map] that we can think of as giving us a probability distribution for every measurement we think we could perform. Given that thing, a state, there is a mathematical way to construct a Hilbert space of vectors and to derive the existence of pure states and of transformations, and then we have Quantum Mechanics. A vector is best thought of as not a state, but instead as a way to construct a pure state. Because so much is made of superposition, we should be clear that convex linear combinations of states are called mixtures, whereas sums of vectors are called superpositions, but if we think in terms of states, these are two different ways of constructing new states, resulting in different patterns of probability distributions for each measurement. I want to minimize the mention of amplitudes, which, as a way of discussing states, assumes that one kind of measurement is special (most often the measurement of position is chosen), but this is largely against a sometime principle of QM that takes all measurements to be equal.

Quantum mechanics is indeed "a certain generalization of probability", but the difference does map onto classical concepts. Noncommutativity, which is arguably the single generalization that takes us from classical to quantum, is an entirely usual property of classical symmetries of classical mechanics. The representation theory of symmetry groups is as classical as it is quantum. Fourier analysis, rooted in mathematical concepts that are as classical as any mathematics can be, is intimately associated with the sometimes supposed to be only-in-quantum Heisenberg group. I can't put this in simple language that Zach Weinersmith could use, but I'm sure it can be done and it will be done by someone.

Despite all this, there is no question that "The Talk" is a significant step forward in the communication of quantum mechanics and quantum computing. In a number of places it is delightfully deflationary —quantum mechanics almost becomes second nature— but on a close reading there is too much missing. Although my discussion here is far from suiting all tastes, and it only hints towards something that could be as accessible but even more deflationary, "The Talk II" will hopefully make fewer jumps.