Quantum hypothesis testing in many-body systems

Review "Quantum hypothesis testing in many-body systems"
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Techniques and concepts from quantum information theory are receiving increasing interest in the field of many-body physics and quantum gravity over the last years. In this submission the authors discuss the task of quantum hypothesis testing for one-parameter families of quantum states, assuming that the parameter (e.g., temperature) is perturbatively close. Their main results
are a new bound between the relative entropy variance and the relative entropy in this perturbative limit (which is important in this setting) and evaluating the respective quantities in some exemplary models of qubits, free fermionic (Gaussian) many-particle systems and conformal field theory. In the latter two examples they apply the results to reduced density matrices of the many-body
system in question. While these calculations are interesting from a technical point of view, a clear discussion of what one actually learns from these computations and how to interpret the results physically is lacking.

Besides deriving their results, the authors give a relatively thorough and nice review of quantum hypothesis testing. This review may in particular be useful for researchers outside the field of quantum information theory as it collects various results (but in itself does not yield new results). It does, however, have the down-side of making the paper rather long. To put it a bit strongly, at times one has the feeling that the authors want to report all that they have learned about hypothesis testing and quantum information theory, but it's less clear what the overall aim is.

The discussion mentions some natural open problems, which I agree should be studied, such as whether in many-body systems one can study, instead of taking many independent copies of the system, many disconnected subsystems of the single large many-body system. (Given the title of the work, it was my hope that some of them would have been tackled in this work.)
The authors also mention the connection of their work to research in quantum gravity which indeed seems like an interesting avenue for future work.

Overall I think the submission fulfills the "General acceptance criteria", but does not provide a breakthrough result. One may argue, though, that it
"opens a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work;"
or
"provides a novel and synergetic link between different research areas."
Hence it may be suitable for publication in scipost physics.

Remarks
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- The authors study parametrically close many-body states. However, there is a subtle issue of (non-)commuting limits here, namely taking the limit λ->0 and taking the thermodynamic limit. For example, even if two many-body systems have arbitrarily close temperatures (independent of system size) then in the thermodynamic limit their global states will generically be perfectly distinguishable in the single shot by a global energy measurement (their trace-distance will approach 1).
I think it would be important to discuss this issue in this paper.

- There are two different kind of expansions in this paper: One with respect to the parameter value, one with respect to the number of copies one takes in the hypothesis test. The expressions "first order" or "second order" are used differently in the two settings. It may therefore be useful to somehow distinguish these (for example by saying "leading order in n" vs. "first order in λ" or so).

- In secs. 3.2.1--3.2.3 the authors demonstrate that their, rigorously proven, theorem holds in particular simple examples. It's not clear to me what the use of these sections is, since the examples don't seem to deliver additional insight.

- As someone not very well versed in CFT techniques, I found the discussion of the CFT computations extremely dense and hard to follow (especially in contrast to the very explicit, detailed and pedagogical calculations in the rest of the paper). It would be very useful to be more explicit in the calculations, in particular if the aim of the paper is (as it seems to be) to bring together different communities.

Smaller points, typos etc.
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- In the introduction it says that subsystem reduced density matrices of fermion chains are determined by two-point functions. This is only true for Gaussian states (free fermions) as studied later in the paper and should be specified.

- After eq. (2.6) it says that "[...] -log Q_s(ρ,σ) are the relative Rényi entropies [...]".
The quantity -log Q_s(ρ,σ) is only proportional to the relative Rényi entropy of order s?!

- At the bottom of page 5 it says that Q satisfies the data-processing inequality. Given the pedagogical aim of the paper, it may be useful to state what the data-processing inequality is.

- Footnote 9 on p. 12: ΔK is not yet defined.

- What the authors call "capacity of entanglement" is known under various names in information theory, such as variance of surprisal, varentropy etc.

- After (3.49) the authors say that if "β_2 -> ∞, ρ_2 reduces to the ground state, and the relative entropy variance vanishes (along with C(β_2)->0)." This is only true if the ground state of the Hamiltonian is non-degenerate.

- In sec 4.2 shouldn't the acceptance condition be | |E| - |\tilde E| |>= \mathcal E? I.e. with absolute value sign? Similarly for (4.23) etc.?

- After (4.33) it should probably read "completely positive" instead of "positive".

- In (6.28) and alter ΔA is not defined.

- Why is the sandwiched Rényi relative entropy suddenly mentioned after (6.33)?

- In sec. 6.3.3. it says that the XY spin chain can be mapped to a free fermion chain in the thermodynamic limit. This is also possible for finite chains (care must be taken with regard to the boundary condition) and indeed done in (6.85).

- I didn't understand what is meant with the last sentence before Sec. 7: "It is interesting that the breaking of translation invariance leads to non-commutativity from the perspective of two fermions"

- In (7.3), L_0 and c hasn't been defined.

- In (7.9,7.10) are referred to as giving the "entanglement spectrum", but they give states
(the eigenstates not the eigenvalues of the modular Hamiltonians).

- While maybe not directly relevant for the questions studied here (and hence does not need to be cited here), the authors may be interested in the recent preprint arxiv:2009.08391, which discusses mathematical properties of the relative entropy variance from an information theoretic point of view.