Classicality with(out) decoherence: Concepts, relation to Markovianity, and a random matrix theory approach

This paper develops the formalism of decoherent histories in an interesting and careful manner, emphasizing the emergence of probabilities that satisfy the Kolmogorov consistency conditions as the key to classical behavior, with "classical" defined as meaning that the effects of quantum interference are negligibly small. Numerics with a simple random-matrix model serve to demonstrate the abstract results.

This paper is a study of the emergence of classicality in Hamiltonian quantum systems through the behaviour of coarse-grained observables. The author first analyses a number of different concepts in the literature, and derives relations between them. Then he derives the conditions under which his favoured concept of classicality, the validity of the Kolmogorov condition for observations, can be derived. This is done by considering when a classical-like observables can be found that is suitably coarse-grained and slow, for a Hamiltonian drawn at random according to random matrix theory. Next, he studies this numerically, in detail, for quite large systems, and finds quite good agreement with his theory.

The paper is generally written with great care and honesty, which is appreciated. In most parts, it certainly builds a lot on work by previous authors. But it seems to me that it contains substantial conceptual, analytical, and numerical advances, and so deserves publication in SciPost, once the comments below are addressed.

1. “Emergence of classicality” could mean many things to many people. In the introduction, one pair of approaches that the author mentions because it is going to be relevant in the paper (as indeed it is) is decoherent histories / consistent histories. In my understanding these were proposed as an interpretation of quantum mechanics, a solution to the quantum measurement problem. By contrast, the viewpoint adopted by the author in this paper seems very much an operational one: to study “an experimentally well-defined procedure to access the time evolution of a quantum system”. That is, he assumes the existence of measurements as a primitive, and thus is not addressing the measurement problem at all. That is fine in itself, but it would help the reader greatly to make this clear very early in the paper. That is, the author should make clear the scope of answer he is considering to the question he raises in paragraph 1, “what needs to be proved?” Or, if this is not what the author meant by that question, then it seems that there is an even more basic question that the author needs to ask, and answer for the purpose of this paper.

2. For the relations in Fig. 2, are all the (non-double-headed) arrows one-way? That is, in all cases do there exist counter-examples to show that the concepts are not equivalent? If so, then the author should spell-out (where not already done) the required counter-examples. If the author cannot find a counter-example, then it would be useful to introduce a distinction between arrows known to be one-way, and arrows which could possibly be two-way.

3. In the first sentence of the second-last paragraph on page 7, it would help to be explicit that what is sufficient is Markovianity *on top of* Eq.(3). Assuming that my understanding is correct.

4. In the first paragraph on page 12, dim H_S << dim H_B may be a sufficient condition for coarseness. But I don’t see that it follows from the previously discussed sufficient condition, which would translate to merely requiring 1 << dim H_B.

5. In the final paragraph on Sec. 2, why is it “unfortunate” that certain assumptions are questionable? I’m sure this is clear to the author, but there are a lot of concepts and assumptions for the reader to keep track of so it would help to spell out the argument a bit more.

6. Similarly, in the remarks following Eq. (28) it would help to remind the reader of the meaning of the terms therein, and why it is “certainly negligibly small”.

7. Above Eq.(34), why call this operator an “energy imbalance”? If I understood correctly, there is as much energy spread within each band as there is from one band to the next.

8. From Fig. 3 on, the figure captions are extremely brief. I don’t know if SciPost has a policy on this, but in other high-quality journals it is required to put enough information in the caption so that a reader can look at the figure and caption and understand what is being plotted. Here that information is buried in the text, often across several paragraphs. I suggest the author to make proper captions and remove the explicit descriptions (of colours and line-styles etc.) from the main text. The main text can just say what is plotted. The caption says how. Quantities plotted should either be given a description or an equation reference in the caption.

9. In the second-last paragraph of page 22, it would help to remind the reader what q_{t,s}(x_0) is.

10. This statement in the conclusion: "In fact, by looking at Eq. (18) it even becomes clear that we have derived the stronger decoherent histories condition” should maybe be discussed at that point in the main text.

There are also some minor things that need fixing:

1. The sentence crossing from page 1 to page 2 is not grammatically well formed.
2. In the second-last sentence on page 11, “classical” should be “classically”.
3. On page 16, “fascillitated” should be “facilitated”.
4. On page 18, “hugh” should be “huge”
5. First sentence, last paragraph of page 19, “plays” would be better than “played”.