Many-body Bell inequalities for bosonic qubits

The main aim of this work is to construct Bell or Bell-type inequalities for bosonic many-qubit states. First, the Author considers a simple Bell inequality, constructed for instance in Ref. [24] for a pair of binary observables per party, and derives a symmetric version of it in which all observers measure the same observables. Second, the Author considers a situation in which a cloud of particles is divided into two groups and, building on the above symmetric inequality, he derives another inequality, symmetrized within each group, whose aim is to detect nonlocal correlations between these groups. A highly interesting feature of these constructions is that the obtained inequalities can be rewritten in terms of expectation values of some collective quantities and their powers and thus in principle could be used to detect Bell correlations within current experiments; an example of such an experiment was recently performed in Ref. [40]. Finally, the obtained inequalities are tested on two particular physical systems, for instance the BEC condesate in a double-well potential.

While I understand the general aim of this work and find it certainly interesting, I have certain doubts about the approach used here to achieve this goal:

1. While the inequality (3) seems to be derived for arbitrary pairs of binary observables measured by the parties, which might also differ between different parties, in the ‘quantum version’ of it, stated in (4), the Author seems to assume that all observers measure the same particular observables which are the Pauli matrices sigma_x and sigma_y. This is not a serious problem because at the end of the day one always needs to choose a particular Bell operator to test whether a given state is non-local or not. Here, however, the passage from (3) to (4) is made without basically any explanation, and it would make the paper easier to follow if the Author elaborated on this step. It should be noticed here that (3) is a Bell inequality in which also in the quantum regime every party can in principle measure any pair of observables.

2. A slightly more serious problem is with the inequality (13). To derive it the Author starts from (10) which is an expectation value of a particular quantum operator being simply a symmetrized version of (4). This is certainly not a proper way to derive Bell inequalities. And, even if this procedure could lead to a valid Bell inequality because it relies on (3), in my opinion this part of the paper is not properly phrased. In particular, I would first introduce a symmetrized version of the Bell expression (3), that is, one which is invariant under permutation of any pair of parties; let me add here that the fact that a Bell expression is symmetric does not impose any constraints on the observables that the parties measure. Now, by fixing all the observables to be sigma_x and sigma_y, one would recover the expectation value in (10) or (13).

Another possibility is that the Author simply derives a permutationally-invariant version of the particular Bell operator given in (4) in order to rewrite it in terms of powers of collective observables. But then, I am not sure whether it’s entirely correct to name (13) Bell inequality. If it’s the case, I suggest to add a few more lines to elaborate on what exactly is being done there.

3. A similar problem is with (21) or (23). Here, the Author again considers an expectation value of a particular quantum operator being a tensor product of two sequences of the rising operators of different lengths. Then, in Eq. (21) he attempts to put an upper bound on it for LHV models by using Eq. (6), which, I believe, is the maximal expectation value of a tensor product of such a sequence of the rising operators. However, I don’t think that this procedure leads to a proper Bell inequality because to obtain a Bell inequality one needs to optimize over all possible LHV models. For instance, assuming that each party decides to measure two same observables sigma_x (which is a valid possibility in a Bell scenario), the expectation value of (6) will be larger than ¼. It seems that what the Author derives is rather an entanglement witness. I thus suggest to modify this part of the paper accordingly.

Concluding, in my opinion the paper presents a very interesting line of research that merits publication, however, prior to that I suggest to revise the manuscript significantly, in particular taking into account the above comments.

Other comments:

1. ‘a simplest case’ → ‘the simplest case’.
2. Eq. (25), ‘n.m=0’ → ‘n,m=0’.
3. The Author uses ‘nonlocality bound’ to name the maximal value of (3) or other expressions over the LHV models. Usually one uses the ‘LHV bound’ or ‘local realistic bound’ for that purpose.

1- This is a study about Bell correlation witnesses in bosonic systems, showcasing interesting tools for the study of high-order correlator functions.
2- The paper tackles some hard questions such as the ones of Bell correlations across two space-like separated regions (double-well BEC), giving elegant analytical results.
3- The paper proposes interesting physically relevant situations in which bosonic systems could potentially reveal Bell correlations, and discusses the challenges towards their eventual detection in an experiment.

1- The work's motivation is the study of Bell's nonlocality in bosonic qubits, highly emphasized through examples of Bose-Einstein condensates, where individual addressing is manifestly not possible. In my view, the manuscript glosses over the issue of space-like separation too much, so that it may be confusing to the reader.
2- Another item that is confusing is that the paper uses interchangeably the domain of quantum physics, with qubits and bosons and operators acting on Hilbert or Fock spaces with that of Bell inequalities, which talk about correlations and probability distributions. This is a key difference in talking about entanglement vs Bell nonlocality and it should be made clear in the paper.
3- The study is restricted to the so-called Bell correlation witnesses (strictly speaking, there is no single Bell inequality with a classical bound, inputs and outputs in a black box scenario, etc.). Although duly acknowledged, some of the examples are almost impossible to implement experimentally, due to the non-robustness of particle losses.
4- One of the highlights is that stronger Bell correlations reside in fixed N sectors, but that is not very surprising. For instance, in [40] it was shown how the operators in the Bell correlation witness commute with the particle number operator.

My recommendation for the paper is a major revision. The idea and the motivation are good and the math seems technically correct. However, the weaknesses I highlighted should be either resolved or explicitly acknowledged in the paper.

Given the originality and the advancement of the current state of knowledge, I think this work should be eventually publishable in SciPost Physics Core, and does not transcend its criteria significantly enough to be more appropriate for a more selective journal.

1- Before Eq. 2, lambda is a "hidden variable", but Bell's theorem is about local hidden variables. The issue of locality is key here, especially when dealing with indistinguishable particles. How does Bell's theorem formulate in this case? What are the measurement choices in the Bell experiment, since there appears only to be the \sigma_plus observable for each particle. That connection should be made explicit, beyond what Fig. 1 hints at.
2- k \in [1,m] is perhaps better to say k \in \{1,\ldots,m\}
3- The Cauchy-Schwarz inequality in integral form is used repeatedly in the paper. To make it more self-contained I would suggest to state it in general form the first time it is used.
4- In the 3.1 example, it would be nice if more details about how the ground state for N=100 was found were indicated.
5- In Figure 1, what would be the physical mechanism by which one would space-like separate the particles in a BEC without changing their (relevant) quantum degrees of freedom?

References:
R1- In the introduction, two more loophole-free Bell tests were performed in 2015 almost simultaneously with the one cited (and another in 2017) by independent groups.
R2- In the discussion about Eq. 18 the problem about nonlocality depth is highlighted. There exist works that already address this problem in the context of device-independent witnesses of entanglement depth that should be referenced as well.
R3- The triplet of experiments on split BEC https://www.science.org/doi/abs/10.1126/science.aat4590 seems also a good motivation for this paper worthy of mentioning.
R4- The entanglement witness bound 4^{-m} in Section 4 needs proper referencing.