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Maximum entanglement of mixed symmetric states under unitary transformations
by Eduardo SerranoEnsástiga, John Martin
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  John Martin · Eduardo SerranoEnsástiga 
Submission information  

Preprint Link:  https://arxiv.org/abs/2112.05102v3 (pdf) 
Date accepted:  20230811 
Date submitted:  20230712 16:26 
Submitted by:  SerranoEnsástiga, Eduardo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study the maximum entanglement that can be produced by a global unitary transformation for systems of two and three qubits constrained to the fully symmetric states. This restriction to the symmetric subspace appears naturally in the context of bosonic or collective spin systems. We also study the symmetric states that remain separable after any global unitary transformation, called symmetric absolutely separable states (SAS), or absolutely classical for spin states. The results for the twoqubit system are deduced analytically. 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. As an application of our results, we also analyse the temperature dependence of the maximum entanglement that can be obtained from the thermal state of a spin1 system with a spinsqueezing Hamiltonian. For the symmetric threequbit case, our results are mostly numerical, and we conjecture a 3parameter family of states that achieves the maximum negativity in the unitary orbit of any mixed state. In addition, we derive upper bounds, apparently tight, on the radii of balls containing only/all SAS states.
Author comments upon resubmission
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 pointbypoint.
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 SerranoEnsástiga and John Martin
List of changes
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 threequbit results in the abstract and the introduction, the authors could mention upfront that only numerical results and conjectures are obtained in the threequbit 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 nonzero
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 4dimensional 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 3qubit 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 nonsymmetric setting in 2002. However, the authors
only computed these radii in the twoqubit 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 crystalclear that we studied only these radii in the two and threequbit 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 twoqubit 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.
Published as SciPost Phys. 15, 120 (2023)