In this manuscript, the authors present a novel technique for the study of steerability of quantum states. It is a question of both fundamental and practical importance to understand what are the entangled quantum states that are incapable of displaying Einstein-Podolski-Rosen (EPR) steering (and, consequently, Bell nonlocality), and not much is known in this sense, specially in more general scenarios where one does not assume that the parties realize projective value measuerements (PVMs). The main result in the manuscript is a method for certifying the steerability of quantum states in the more general scenario where Positive Operator-Valued Measures (POVMs) are allowed. As an application of the method, the authors show strong numerical evidence that the two-bit Werner states are unsteerable for POVMs for $p \leq 1/2$, thus showing that POVMs provide no advantage over PVMs for such states, in this context. They also apply the method to show new results concerning the class of T-states, or Bell-diagonal states.
Overall, the manuscript is very well written and organised. The introduction contextualises the problem very well, and the references are appropriate. All the definitions and main concepts are well presented and formalised, and the main problem is clearly stated. The results and examples, alongside with the supplemental material, are well presented as well.
The main drawback of the manuscript, in my opinion, is related to the range of the method. The strategy relies on the fact that a candidate LHS ensemble is provided, and then verifies the existence of responde functions that satisfy the unsteerability criterium given by eq. (1), of the manuscript. The more general problem of determining if a given quantum state is able to display EPR steering would have to consider, also, all possible choices of LHS ensembles. Thus, the strategy is mainly useful in cases where, for any reason, a good candidate LHS ensemble is known, as for the examples presented, and, hence, has its range limited. Also, it is worth noting that, even when a good LHS ensemble is provided, the implementation of the strategy may not be simple, since the computation of the gap function generally requires global optimisation. I do not recall any comments on the following, but I strongly believe that implementation would be unfeasible for higher dimensional systems due to the complexity of the optimisation problem.
In the following I address other questions and comments:
- In the abstract, I suggest exchanging the term "quantum steerability" to "Einstein-Podolski-Rosen (EPR) steering".
- In the last paragraph of the introduction, I suggest adding the word "general" before "POVMs provide an advantage...".
- There is a typo in eq. (1): the identity in system B, not A.
- What is the measure S in eq. (1)? I recommend it is changed to $du(P)$. (Same for the following equations.)
- There is a period missing in the caption of Fig. 1.
- In the second column of page 2, first paragraph, last sentence: it is claimed that all effects of non-special extremal POVMs are rank-1, which is not true. A counter-example, for $n=d=3$, is the POVM whose effects are $E_{1} = \ketbra{0}$, $E_{2} = \ketbra{1} + \ketbra{2}$, and $E_{3} = 0$.
- There is a typo below eq. (2): a sum is superindexed by $4$, when it should be $n$.
- There is a typo on footnote [28]: "Because $G_i \geq 1$..." should be "Because $G_i \geq 0...$"
- Above eq. (7): as noted above, $\mathcal{N}^n$ is not the convex closure of the non-special extreme points of $\mathcal{M}^n$, but rather a set which generates the latter by means of post-processing (coarse-graining). But, in terms of local models, the general strategy of focusing in $\mathcal{N}^n$ still works (the same point was made by J. Barrett in Ref. [29]).
- In the Conjecture 1, the side of (uns)steering is emphasised. However, due to their symmetry, the Werner states should be (uns)steerable from any side, not just Alice's. Also, I suggest exchanging "equivalent in steering" to "equivalent for steering".
- Below the conjecture, the authors write: "We first note that the optimal choice for the LHS ensemble is the uniform distribution[...]". Why is that so? This is an important claim, and the authors should either prove it or provide arguments and/or references to support it.
- In Fig. 3, it is not clear what is the meaning of the gap being zero for $p<1/2$ and ePOVMs. This point should be discussed with more detail, and be compared to the PVM case.
- There is a typo in the last paragraph of page 4: "Surprisingly \textbf{a} the numerical accuracy"...