Maximum entanglement of mixed symmetric states under unitary transformations

Dear Editor,
We thank the referees for their valuable reports and their time reading and reviewing our
manuscript. We also appreciate their recommendations for publication in Sci. Post. We have
added all their requested changes in the new version of the manuscript. We comment below their
observations point-by-point.

We hope we have clarified all concerns and adequately addressed all questions. We believe this
reviewed version of our manuscript is in a suitable form for publication.

Yours sincerely,
Eduardo Serrano-Ensástiga and John Martin

Answers to Report 1

Weakness 1. The numerical experiments (i.e., Section 4) are less interesting and convincing than the results that came earlier in the paper. Maybe when mentioning the three-qubit results in the abstract and the introduction, the authors could mention up-front that only numerical results and conjectures are obtained in the three-qubit case.

We are following the referee’s recommendation. The new version includes changes to highlight
this comment in the abstract and introduction (page 4, before Section 2).

1. Typo near the bottom of page 5: ” if it has more than one zero eigenvalues” should
be ” if it has more than one zero eigenvalue”

We have corrected this typo.

2. Page 6, right before the start of Section 3: the authors talk about the maximally
mixed symmetric state on 3 qubits. This state is indeed rank 4, so has 4 non-zero
eigenvalues. The authors say that 2 of its 6 eigenvalues are 0... what are those 2?
I could understand the statement ”4 of its 8 eigenvalues are 0” since the symmetric
2 subspace is a 4-dimensional subspace of (C^2)⊗3 = C^8... but I can’t figure out where
6 dimensions are coming from. Maybe some clarification would help.

In this part of the text, we are considering the symmetric 3-qubit state as an 2 × 3 system,
ρ ∈ B(H_1^{∨3} ) ⊂ B(H_1 ⊗ H_1^{∨2} ). Consequently, the density operator has 6 eigenvalues. For clarity of
the reader, we have added additional text in the discussion.

3. In Equation (33), it took me too long to figure out what ”s” is (I see now that it is
a spin system parameter). Maybe remind the reader of what ”s” is here, or replace
it by N/2?

We have rephrased this paragraph and Eqs. (33) and (34) in terms of the variable N.

4. In the abstract, the authors say ”In particular, we determine the maximal radius
of a ball of SAS states around the maximally mixed state in the symmetric sector, and
the minimal radius of a ball that contains the set of SAS states.” I read this sentence
as saying that the authors computed these radii in all dimensions; something that
Gurvits and Barnum did in the non-symmetric setting in 2002. However, the authors
only computed these radii in the two-qubit case. I think that the abstract should be
updated to make this clearer. I suggest a similar change on Page 4, right before the
start of Section 2.

We agree with the observation of the referee. We have updated the text before Section 2 and
the abstract to make crystal-clear that we studied only these radii in the two- and three-qubit case.

Answers to Report 2

1. Sometimes in the paper, (for instance in the abstract) the authors use the
expression ‘permutation symmetry’ when talking about mixed states acting on the
on the symmetric subspace of the entire Hilbert space. I would modify this expression
somehow because in the literature permutationally invariant states are
those that are invariant under a permutation of any pair of subsystems (see e.g.
https://arxiv.org/pdf/1302.4100.pdf), and, while for pure states they are the same
as those belonging to the symmetric subspace, for mixed states permutationally invariant
states form a superset of the symmetric ones. For instance, a projection onto the two-qubit singlet state is permutationally invariant in the above sense.

We thank the referee for this accurate observation. We have made the necessary changes to
avoid the term ”permutation invariant state”. More specifically, we have slightly edited the text
in the abstract and on pages 3 (last paragraph), 4 (second paragraph), and 6 (first sentence).

2. Perhaps the authors could include also [K. Eckert et al., Ann. Phys. 299, 88
(2002)] in the list of references, where the observation that partial transposition is an
iff criterion for separability was made for the first time.

We thank the referee for mentioning the reference. We have added it, as well as the Reference
to Peres. The new citations are [12] and [14] on the manuscript.