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

Preprint Link:  https://arxiv.org/abs/2112.05102v2 (pdf) 
Date submitted:  20230417 11:52 
Submitted by:  SerranoEnsástiga, Eduardo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We study the maximum entanglement that can be produced by a global unitary transformation for systems of two and three qubits when state permutation invariance is imposed. This constraint of permutation symmetry 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. 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, 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, which our numerical results suggest are tight, on the radii of balls containing only/all SAS states.
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Reports on this Submission
Anonymous Report 2 on 202374 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.05102v2, delivered 20230704, doi: 10.21468/SciPost.Report.7446
Report
This work is concerned with the characterization of entanglement in two or threequbit states that live (fully) symmetric subspace of the underlying multipartite Hilbert space. The main reason to consider these particular situations is that in these systems partial transposition is an iff criterion for separability/entanglement. First, the authors study the twoqubit case and provide the following results:
1. Derivation of the maximal value of negativity that can be achieved from a given symmetric mixed state by acting on it with an arbitrary global unitary operation that acts only on the symmetric subspace (Theorem 1). A direct corollary to Theorem 1 is a necessary and sufficient criterion for a state to be symmetric absolutely separable (SAS), where the latter stands for a state that is separable and remains separable under the action of any symmetric global unitary operation.
2. Building on the above results, the authors then determine the minimal and maximal radii of the balls containing the set of symmetric absolutely separable twoqubit states.
3. Then, the authors aim to determine the maximal amount of entanglement that can be obtained from a thermal state of a certain Hamiltonian by applying symmetric global unitary operations. In particular, values of the parameters that the Hamiltonians depends on are found for which the thermal states are symmetric absolutely separable.
The above results are then generalized to the symmetric threequbit states for which partial transposition is also an iff criterion for separability. Since the threequbit state is already too difficult to study analytically the authors employ some numerical methods. Based on the numerical search the authors pose a few conjectures such as that to obtain the maximal value of negativity over all operations from SU(4) it is enough to restrict to a certain fourparameter class of them. The authors also provide a sufficient condition for a symmetric threequbit state to be SAS, formulated in terms of its eigenvalues.
I find this paper a solid piece of work. It asks interesting questions regarding characterization of entanglement in composite quantum systems. I particularly like the idea of studying the notion of absolutely separable state within quantum systems for which partial transposition detects all entangled states. The paper is also clearly written. I therefore recommend publication in Sci. Post.
Comments:
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.
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.
Anonymous Report 1 on 202362 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.05102v2, delivered 20230602, doi: 10.21468/SciPost.Report.7295
Strengths
1. The problems considered by the paper are completely natural, and I'm honestly surprised they weren't considered earlier. This work can be seen as a symmetric (bosonic) version of the paper "F. Verstraete, K. Audenaert, and B.D. Moor, Maximally entangled mixed states of two qubits, Phys. Rev. A, vol. 64, p. 012316, 2001" which I'm surprised wasn't written 10 or 15 years ago.
2. The results obtained by the authors (in particular, Theorem 1 and Equation (13)) are deep. Neither the results nor their proofs are obvious, but they are simpletostate and make use of.
Weaknesses
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.
Report
I believe that this paper will have high impact and will be of broad interest to the quantum information theory community. It clearly meets all of the journal's "general acceptance" criteria, and I believe that I meets "Expectations" #1 and #3 as well: Theorem 1 and Equation (13) are deep theoretical discoveries that are likely to launch new research projects.
Requested changes
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"
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 subspace is a 4dimensional subspace of (C^2)^{\otimes 3} = C^8... but I can't figure out where 6 dimensions are coming from. Maybe some clarification would help.
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?
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.