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REVIEW 4 major objections 4 minor 73 references

Block entanglement bounds distribution of regionally localized entanglement

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read For a wide class of multi-qubit pure states, the total entanglement that can be localized on all two-qubit regions sharing a common hub is bounded above and below by the block entanglement between that hub and the rest of the system.

desk verdict Defines a promising new entanglement quantity, but the stated GHZ bound likely needs a normalization factor and the core proofs are not in the extract. read the letter →

arxiv 2508.05431 v1 pith:6D6RNDQ2 submitted 2025-08-07 quant-ph

Jithin G. Krishnan , Aditi Sen De , Amit Kumar Pal This is my paper · ORCID
classification quant-ph MSC 81P4081P45 PACS 03.67.Mn03.65.Ud
keywords regionallylocalizedentanglementlocalizableblockquantumnetworksGHZstateWDickestatesphase-flipnoise
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces 'regionally localized entanglement' (RLE): the average entanglement concentrated on a two-qubit region of a multi-qubit state by measuring the other qubits, with all regions sharing a common 'hub' qubit. It proves that for a broad class of pure states — permutation-symmetric states such as GHZ, W, Dicke states and their superpositions, and single-excitation generalized W states — the total RLE over all regions around a hub is bounded above and below by the 'block localized entanglement' between the hub and the rest. This matters for quantum networks, where idle nodes are often eliminated by measurements; the bounds set fundamental limits on how much entanglement can remain among active links after pruning. Numerical tests indicate the bounds survive Haar-random states and local phase-flip noise except for very small system sizes.

What carries the argument

The key objects are three entanglement quantifiers defined via measurement optimization: regionally localized entanglement (RLE), block localized entanglement (BLE), and block entanglement (BE). The proof machinery is a chain of inequalities that links the optimal projective measurements for each two-qubit region to the optimal measurement for the hub-rest bipartition, exploiting symmetry properties (permutation symmetry or fixed magnetization) of the state family to evaluate the sums in closed form.

What would settle it

Numerically search over a large ensemble of, say, 6-qubit Haar-random pure states for a case where the total regionally localized entanglement for a hub exceeds the block localized entanglement upper bound; a single such instance would invalidate any claim that the bounds are universal beyond the studied state families.

Watch

Extended reading notes

Core claim

The paper defines RLE for a hub qubit as the sum over all other qubits of the maximum average entanglement (measured by a pure-state entanglement measure) achievable on each two-qubit pair after projective measurements on the remaining qubits. It also defines block localized entanglement (BLE) as the maximum average entanglement between the hub and the whole remaining block after measurements on the complementary set, and block entanglement (BE) as the bipartite entanglement of the reduced hub-rest state. The central claim is that for the paradigmatic pure states listed, total RLE is sandwiched between an upper bound set by BLE (and often BE) and a lower bound set by BLE, with the precise fo

Load-bearing premise

The proof assumes that the optimal localizing measurements for the different two-qubit regions can be treated independently and that their combined effect is bounded by the hub-rest block measurement; if these measurements interfere, the bounds could fail.

Editorial extensions

If this is right

  • For a quantum network with a hub node, the bounds tell operators the maximum entanglement that can be concentrated on the hub's links after discarding idle nodes, and the minimum needed to certify a given amount of link entanglement.
  • The closed-form bounds for GHZ and W states allow quick estimates of achievable link entanglement without performing the full optimization.
  • The persistence of the bounds under local phase-flip noise for large system sizes suggests they are robust enough for noisy intermediate-scale networks.
  • The differing bounds for single-excitation generalized W states show that breaking permutation symmetry changes the allowed distribution of localized entanglement, so symmetry must be accounted for in network design.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because localizable entanglement optimization is over all measurement bases, the bounds could potentially be tightened or extended to other entanglement measures (e.g., negativity or concurrence) beyond the pure-state measures used here.
  • The paper's proof assumes the optimal localizing measurements for different regions can be treated independently; if they interfere, there may exist states outside the proven classes where the total RLE violates the upper bound, suggesting a direction for counterexample search.
  • The numerical result that violations appear only for small systems under phase-flip noise hints that finite-size effects are the main danger to the bounds, so future work could map the exact system-size threshold as a function of noise strength.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript introduces 'regionally localized entanglement' (RLE), defined as the average entanglement concentrated on a two-qubit region after projective measurements on the remaining qubits, and its 'total' over all two-qubit regions sharing a common hub. It claims that for permutation-symmetric states (GHZ, generalized GHZ, W, Dicke states and their superpositions) and for states in a specific magnetization sector (generalized W states), the total RLE is bounded above and below by the block-localized entanglement (BLE) or the block entanglement (BE) between the hub and the rest. It further claims numerical verification for Haar-random states and robustness under Markovian and non-Markovian phase-flip noise except at small system sizes. The supplied text contains the abstract, introduction, the beginning of Sec. II (through Eq. (2)), and the references; the analytical derivations, numerical results, and concluding sections are not included.

Significance. If the claimed inequalities are correct, they would provide a practical, measurement-based method to bound how much entanglement can be concentrated on the links incident to a hub in a quantum network, complementing monogamy relations. The quantities are defined independently of the bounds and the claimed inequalities are falsifiable and free of fitting parameters. However, because the core proofs and numerical evidence are absent from the submitted text, the significance is conditional. The manuscript does not include code, data, or detailed simulation protocols, so the numerical claims cannot be independently checked as submitted.

major comments (4)
  1. [Secs. III–V] The supplied manuscript stops after Eq. (2) of Sec. II; Secs. III–V, which should contain the theorem statements, analytical proofs, and numerical simulations, are absent. The central claim is therefore unverified. Please provide the complete derivations and the exact statement of each bound, including directions, constants, and whether 'total' is a sum or an average over the two-qubit regions.
  2. [Abstract / Sec. I] The scaling of the claimed upper bound is ambiguous and, on the natural reading, appears problematic. For an N-qubit GHZ state (with S0 empty), the localizable entanglement on each hub-other pair is 1 ebit after measuring the remaining qubits in the X basis, so the sum over the N-1 regions is N-1. The block-localized entanglement between the hub and the rest is also 1 ebit. Hence any literal bound 'total RLE ≤ BLE' is false for N>2. The paper must state whether the bound contains a factor (N-1) or whether 'total' is normalized as an average.
  3. [Sec. II] The RLE for each region j is defined by optimizing measurements on S0 ∪ S\{hub,j} independently; these measurement bases generally differ with j. A bound on their sum via the BLE, which optimizes only measurements on S0, requires a relation between these optimizations that is not evident from Eqs. (1)–(2). The missing argument in Sec. III is load-bearing and must be supplied explicitly.
  4. [Sec. IV] The numerical claims in the abstract (Haar-random states; Markovian and non-Markovian phase-flip channels; small-system violations) are not accompanied in the supplied text by any simulation protocol, channel parameters, system sizes, sample counts, or figures/tables. These claims are part of the central robustness statement and need to be fully documented before the paper can be assessed.
minor comments (4)
  1. [Sec. II, Eqs. (1)–(2)] The projection basis {|b_k>} is not specified. If it is an arbitrary local basis on S0, state this; if the bounds require a particular basis, define it.
  2. [Abstract / Sec. I] The phrase 'average entanglement concentrated over a two-qubit region' conflates two averages: one over measurement outcomes and one over the choice of region. Please define RLE and total RLE with explicit formulas in Sec. II.
  3. [Fig. 1] The caption of Fig. 1 appears garbled in the posted version; please ensure that the definitions of S0, S, the hub, and the sets S0 ∪ S\{i,j} are legible and consistent with the main text.
  4. [Sec. I] The term 'specific magnetization sector' is used but not defined in the supplied text; please provide the definition and the precise family of states considered.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity identified; RLE and BLE are independent definitions and the derived bounds are not re-descriptions of the inputs.

full rationale

The paper introduces RLE as the average entanglement concentrated over two-qubit regions sharing a common hub, and BLE as the localized entanglement between the hub and the rest after measurements on the S0 block. These are distinct quantities defined through different measurement protocols. The claimed bounds are stated as theorems over specific state classes, not as fits to data; the numerical simulations are tests of the inequalities, not inputs to them. The visible text (Sec. II, Eqs. (1)-(2)) defines the post-measurement ensemble and gives no indication that the bound is imposed by construction: no parameter is adjusted to force the inequality, and no prior result by the same authors is the sole load-bearing justification. For pure states, the standard entropy-concavity inequality (average entanglement of assistance cannot exceed the local entropy of the hub) already yields an upper bound of the form sum_j LE_{ij} ≤ (N-1) E_{i|rest}, so the bound has independent mathematical content. The skeptical observation about GHZ_N concerns whether the statement in the missing Sec. III includes an (N-1) factor or normalization; that is a correctness/ambiguity concern, not a circularity. The omission of Sec. III from the supplied text is a completeness limitation, but it does not show that the derivation reduces to its own inputs. No load-bearing self-citation chain is evident, and the numerical results are confirmatory rather than fitted. Score 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard quantum measurement theory and properties of entanglement measures. No free parameters are used. The symmetry assumptions on the states are explicitly stated domain restrictions. No new physical entities are introduced; RLE is a new measure, not an entity.

assumptions (3)
  • standard math Standard quantum mechanics: pure states, projective measurements, and the Born rule (Sec. II, Eqs. 1-2).
    The definitions of post-measurement states and probabilities rely on standard quantum measurement postulates.
  • standard math The entanglement measure used (likely concurrence) is convex and non-increasing under LOCC.
    The proofs of bounds on average entanglement typically require such properties. The specific measure is not named in the visible text but is standard in localizable entanglement studies.
  • domain assumption The states considered are pure and belong to specific symmetry classes (permutation-symmetric or a fixed magnetization sector).
    The analytical proofs are claimed for these families only; the generalization to arbitrary states is tested numerically.

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Cite this review

Pith. "Pith review of Block entanglement bounds distribution of regionally localized entanglement." pith.science (2026). https://pith.science/paper/6D6RNDQ2

@misc{pith2026250805431,
  author       = {Pith},
  title        = {Pith review of: Block entanglement bounds distribution of regionally localized entanglement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6D6RNDQ2}},
  note         = {Machine review of arXiv:2508.05431}
}
read the original abstract

In quantum networks, eliminating connections between nodes is crucial to mitigate the effects of decoherence, often achieved by performing measurements on nodes that are idle, or vulnerable to noise. To characterize the entanglement content of the resulting smaller network, we introduce the notion of ``regionally localized entanglement", defined as the average entanglement concentrated over a two-qubit region in a multi-qubit system. Hence, the total regionally localized entanglement can be obtained by considering all two-qubit regions sharing a common qubit, referred to as the ``hub". We prove that the total regionally localized entanglement corresponding to a specific hub is bounded above and below via the localizable block entanglement shared between the hub and the rest of the multi-qubit system for a number of paradigmatic pure quantum states, including permutation-symmetric states and arbitrary superposition of states from a specific magnetization sector. Numerical simulations confirm that the bounds for permutation-symmetric pure states remain valid even for Haar-uniformly generated pure states, and when each of the qubits is sent through local phase-flip channels of Markovian and non-Markovian types, except when the system-size is small. On the other hand, arbitrary states from a particular magnetization sector yield bounds on total regionally localized entanglement that are distinct from the permutation-symmetric states, highlighting the structurally unique entanglement properties of the former.

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Works this paper leans on

73 extracted references · 39 canonical work pages

  1. [1]

    author author H. J. \ Kimble ,\ 10.1038/nature07127 journal journal Nature \ volume 453 ,\ pages 1023 ( year 2008 ) NoStop

    doi:10.1038/nature07127 2008

  2. [2]

    \ Luo ,\ https://doi.org/10.1007/978-981-97-6226-2 title Quantum Networks ,\ edition 2nd \ ed.\ ( publisher Springer Singapore ,\ year 2024 ) NoStop

    author author M.-X. \ Luo ,\ https://doi.org/10.1007/978-981-97-6226-2 title Quantum Networks ,\ edition 2nd \ ed.\ ( publisher Springer Singapore ,\ year 2024 ) NoStop

    work page doi:10.1007/978-981-97-6226-2 2024

  3. [3]

    \ Breuer \ and\ author F

    author author H.-P. \ Breuer \ and\ author F. Petruccione ,\ @noop title The Theory of Open Quantum Systems \ ( publisher Oxford University Press, Oxford ,\ year 2002 ) NoStop

    work page 2002

  4. [4]

    author author D. A. \ Lidar \ and\ author T. A. \ Brun ,\ title Introduction to decoherence and noise in open quantum systems , \ in\ DOI: 10.1017/CBO9781139034807.003 booktitle Quantum Error Correction ,\ editor edited by\ editor D. A. \ Lidar \ and\ editor T. A. \ Brun \ ( publisher Cambridge University Press ,\ address Cambridge ,\ year 2013 )\ pp.\ pa...

    work page doi:10.1017/cbo9781139034807.003 2013

  5. [5]

    Horodecki , author P

    author author R. Horodecki , author P. Horodecki , author M. Horodecki , \ and\ author K. Horodecki ,\ 10.1103/RevModPhys.81.865 journal journal Rev. Mod. Phys. \ volume 81 ,\ pages 865 ( year 2009 ) NoStop

    doi:10.1103/revmodphys.81.865 2009

  6. [6]

    Das , author T

    author author S. Das , author T. Chanda , author M. Lewenstein , author A. Sanpera , author A. Sen De , \ and\ author U. Sen ,\ title The separability versus entanglement problem , \ in\ https://doi.org/10.1002/9783527805785.ch8 booktitle Quantum Information \ ( publisher Wiley ,\ year 2016 )\ pp.\ pages 127--174 NoStop

    work page doi:10.1002/9783527805785.ch8 2016

  7. [7]

    author author G. D. \ Chiara \ and\ author A. Sanpera ,\ 10.1088/1361-6633/aabf61 journal journal Rep. Prog. Phys. \ volume 81 ,\ pages 074002 ( year 2018 ) NoStop

    work page doi:10.1088/1361-6633/aabf61 2018

  8. [8]

    author author D. P. \ DiVincenzo , author C. A. \ Fuchs , author H. Mabuchi , author J. A. \ Smolin , author A. Thapliyal , \ and\ author A. Uhlmann ,\ https://arxiv.org/abs/quant-ph/9803033 journal journal arXiv:quant-ph/9803033 \ ( year 1998 ) NoStop

    arXiv 1998

Show all 73 references
  1. [9]

    Laustsen , author F

    author author T. Laustsen , author F. Verstraete , \ and\ author S. J. \ Van Enk ,\ https://dl.acm.org/doi/abs/10.5555/2011508.2011514 journal journal Quantum Info. Comput. \ volume 3 ,\ pages 64 ( year 2003 ) NoStop

    2003

  2. [10]

    Gour \ and\ author R

    author author G. Gour \ and\ author R. W. \ Spekkens ,\ 10.1103/PhysRevA.73.062331 journal journal Phys. Rev. A \ volume 73 ,\ pages 062331 ( year 2006 ) NoStop

    2006 doi

  3. [11]

    Verstraete , author M

    author author F. Verstraete , author M. Popp , \ and\ author J. I. \ Cirac ,\ 10.1103/PhysRevLett.92.027901 journal journal Phys. Rev. Lett. \ volume 92 ,\ pages 027901 ( year 2004 a ) NoStop

    2004 doi

  4. [12]

    Verstraete , author M

    author author F. Verstraete , author M. A. \ Mart\' n-Delgado , \ and\ author J. I. \ Cirac ,\ 10.1103/PhysRevLett.92.087201 journal journal Phys. Rev. Lett. \ volume 92 ,\ pages 087201 ( year 2004 b ) NoStop

    2004 doi

  5. [13]

    Popp , author F

    author author M. Popp , author F. Verstraete , author M. A. \ Mart\' n-Delgado , \ and\ author J. I. \ Cirac ,\ 10.1103/PhysRevA.71.042306 journal journal Phys. Rev. A \ volume 71 ,\ pages 042306 ( year 2005 ) NoStop

    2005 doi

  6. [14]

    Sadhukhan , author S

    author author D. Sadhukhan , author S. S. \ Roy , author A. K. \ Pal , author D. Rakshit , author A. Sen(De) , \ and\ author U. Sen ,\ 10.1103/PhysRevA.95.022301 journal journal Phys. Rev. A \ volume 95 ,\ pages 022301 ( year 2017 ) NoStop

    2017 doi

  7. [15]

    author author D. M. \ Greenberger , author M. A. \ Horne , \ and\ author A. Zeilinger ,\ http://inis.iaea.org/search/search.aspx?orig_q=RN:22064349 title Bell's theorem, quantum theory and conceptions of the universe \ ( publisher Kluwer ,\ address Netherlands ,\ year 1989 ) NoStop

    1989

  8. [16]

    Hein , author W

    author author M. Hein , author W. D\" u r , author J. Eisert , author R. Raussendorf , author M. Van den Nest , \ and\ author H. J. Briegel ,\ https://arxiv.org/abs/quant-ph/0602096 journal journal arXiv:quant-ph/0602096 \ ( year 2006 ) NoStop

    2006 arXiv

  9. [17]

    Gottesman ,\ in\ https://arxiv.org/abs/0904.2557 booktitle Quantum information science and its contributions to mathematics, Proceedings of Symposia in Applied Mathematics ,\ Vol

    author author D. Gottesman ,\ in\ https://arxiv.org/abs/0904.2557 booktitle Quantum information science and its contributions to mathematics, Proceedings of Symposia in Applied Mathematics ,\ Vol. volume 68 \ ( year 2010 )\ pp.\ pages 13--58 NoStop

    2010 arXiv

  10. [18]

    author author S. M. \ Girvin ,\ 10.21468/SciPostPhysLectNotes.70 journal journal SciPost Phys. Lect. Notes \ ,\ pages 70 ( year 2023 ) NoStop

    2023 doi

  11. [19]

    Amaro , author M

    author author D. Amaro , author M. M\" u ller , \ and\ author A. K. \ Pal ,\ 10.1088/1367-2630/aac485 journal journal New J. Phys. \ volume 20 ,\ pages 063017 ( year 2018 ) NoStop

    2018 doi

  12. [20]

    Amaro , author M

    author author D. Amaro , author M. M\" u ller , \ and\ author A. K. \ Pal ,\ 10.1088/1367-2630/ab84b3 journal journal New J. Phys. \ volume 22 ,\ pages 053038 ( year 2020 ) NoStop

    2020 doi

  13. [21]

    author author H. K. \ J. \ and\ author A. K. \ Pal ,\ 10.1103/PhysRevA.108.032404 journal journal Phys. Rev. A \ volume 108 ,\ pages 032404 ( year 2023 ) NoStop

    2023 doi

  14. [22]

    \ Jin \ and\ author V

    author author B.-Q. \ Jin \ and\ author V. E. \ Korepin ,\ 10.1103/PhysRevA.69.062314 journal journal Phys. Rev. A \ volume 69 ,\ pages 062314 ( year 2004 ) NoStop

    2004 doi

  15. [23]

    author author S. O. \ Skr vseth \ and\ author S. D. \ Bartlett ,\ 10.1103/PhysRevA.80.022316 journal journal Phys. Rev. A \ volume 80 ,\ pages 022316 ( year 2009 ) NoStop

    2009 doi

  16. [24]

    Smacchia , author L

    author author P. Smacchia , author L. Amico , author P. Facchi , author R. Fazio , author G. Florio , author S. Pascazio , \ and\ author V. Vedral ,\ 10.1103/PhysRevA.84.022304 journal journal Phys. Rev. A \ volume 84 ,\ pages 022304 ( year 2011 ) NoStop

    2011 doi

  17. [25]

    Montes \ and\ author A

    author author S. Montes \ and\ author A. Hamma ,\ 10.1103/PhysRevE.86.021101 journal journal Phys. Rev. E \ volume 86 ,\ pages 021101 ( year 2012 ) NoStop

    2012 doi

  18. [26]

    author author H. K. J. \ and\ author A. K. \ Pal ,\ 10.1103/PhysRevA.105.052421 journal journal Phys. Rev. A \ volume 105 ,\ pages 052421 ( year 2022 ) NoStop

    2022 doi

  19. [27]

    K J \ and\ author A

    author author H. K J \ and\ author A. K. \ Pal ,\ 10.1103/PhysRevA.111.032401 journal journal Phys. Rev. A \ volume 111 ,\ pages 032401 ( year 2025 ) NoStop

    2025 doi

  20. [28]

    author author J. G. \ Krishnan , author H. K J , \ and\ author A. K. \ Pal ,\ https://doi.org/10.1016/j.physleta.2024.130170 journal journal Phys. Lett. A \ volume 531 ,\ pages 130170 ( year 2025 ) NoStop

    2024

  21. [29]

    Ac \'i n , author J

    author author A. Ac \'i n , author J. I. \ Cirac , \ and\ author M. Lewenstein ,\ https://doi.org/10.1038/nphys549 journal journal Nat. Phys. \ volume 3 ,\ pages 256 ( year 2007 ) NoStop

    2007 doi

  22. [30]

    author author C. H. \ Bennett , author G. Brassard , author C. Cr\'epeau , author R. Jozsa , author A. Peres , \ and\ author W. K. \ Wootters ,\ 10.1103/PhysRevLett.70.1895 journal journal Phys. Rev. Lett. \ volume 70 ,\ pages 1895 ( year 1993 ) NoStop

    1993 doi

  23. [31]

    Bouwmeester , author J.-W

    author author D. Bouwmeester , author J.-W. \ Pan , author K. Mattle , author M. Eibl , author H. Weinfurter , \ and\ author A. Zeilinger ,\ 10.1038/37539 journal journal Nature \ volume 390 ,\ pages 575 ( year 1997 ) NoStop

    1997 doi

  24. [32]

    author author C. H. \ Bennett \ and\ author S. J. \ Wiesner ,\ 10.1103/PhysRevLett.69.2881 journal journal Phys. Rev. Lett. \ volume 69 ,\ pages 2881 ( year 1992 ) NoStop

    1992 doi

  25. [33]

    Mattle , author H

    author author K. Mattle , author H. Weinfurter , author P. G. \ Kwiat , \ and\ author A. Zeilinger ,\ 10.1103/PhysRevLett.76.4656 journal journal Phys. Rev. Lett. \ volume 76 ,\ pages 4656 ( year 1996 ) NoStop

    1996 doi

  26. [34]

    Sen(De) \ and\ author U

    author author A. Sen(De) \ and\ author U. Sen ,\ @noop journal journal Phys. News \ volume 40 ,\ pages 17 ( year 2011 ) ,\ http://arxiv.org/abs/arXiv:1105.2412 arXiv:1105.2412 NoStop

    2011 arXiv

  27. [35]

    author author A. K. \ Ekert ,\ 10.1103/PhysRevLett.67.661 journal journal Phys. Rev. Lett. \ volume 67 ,\ pages 661 ( year 1991 ) NoStop

    1991 doi

  28. [36]

    Jennewein , author C

    author author T. Jennewein , author C. Simon , author G. Weihs , author H. Weinfurter , \ and\ author A. Zeilinger ,\ 10.1103/PhysRevLett.84.4729 journal journal Phys. Rev. Lett. \ volume 84 ,\ pages 4729 ( year 2000 ) NoStop

    doi

  29. [37]

    Raussendorf \ and\ author H

    author author R. Raussendorf \ and\ author H. J. \ Briegel ,\ 10.1103/PhysRevLett.86.5188 journal journal Phys. Rev. Lett. \ volume 86 ,\ pages 5188 ( year 2001 ) NoStop

    2001 doi

  30. [38]

    Raussendorf , author D

    author author R. Raussendorf , author D. E. \ Browne , \ and\ author H. J. \ Briegel ,\ 10.1103/PhysRevA.68.022312 journal journal Phys. Rev. A \ volume 68 ,\ pages 022312 ( year 2003 ) NoStop

    2003 doi

  31. [39]

    author author H. J. \ Briegel , author D. E. \ Browne , author W. D \"u r , author R. Raussendorf , \ and\ author M. Van den Nest ,\ https://doi.org/10.1038/nphys1157 journal journal Nat. Phys. \ volume 5 ,\ pages 19 ( year 2009 ) NoStop

    2009 doi

  32. [40]

    Coffman , author J

    author author V. Coffman , author J. Kundu , \ and\ author W. K. \ Wootters ,\ 10.1103/PhysRevA.61.052306 journal journal Phys. Rev. A \ volume 61 ,\ pages 052306 ( year 2000 ) NoStop

    doi

  33. [41]

    author author B. M. \ Terhal ,\ 10.1147/rd.481.0071 journal journal IBM J. Res. Dev. \ volume 48 ,\ pages 71 ( year 2004 ) NoStop

    2004 doi

  34. [42]

    Christandl \ and\ author A

    author author M. Christandl \ and\ author A. Winter ,\ 10.1063/1.1643788 journal journal J. Math. Phys. \ volume 45 ,\ pages 829 ( year 2004 ) NoStop

    2004 doi

  35. [43]

    author author T. J. \ Osborne \ and\ author F. Verstraete ,\ 10.1103/PhysRevLett.96.220503 journal journal Phys. Rev. Lett. \ volume 96 ,\ pages 220503 ( year 2006 ) NoStop

    2006 doi

  36. [44]

    author author J. S. \ Kim , author G. Gour , \ and\ author B. C. \ Sanders ,\ 10.1080/00107514.2012.725560 journal journal Contemp. Phys. \ volume 53 ,\ pages 417 ( year 2012 ) NoStop

    2012

  37. [45]

    Gour , author D

    author author G. Gour , author D. A. \ Meyer , \ and\ author B. C. \ Sanders ,\ 10.1103/PhysRevA.72.042329 journal journal Phys. Rev. A \ volume 72 ,\ pages 042329 ( year 2005 ) NoStop

    2005 doi

  38. [46]

    Gour , author S

    author author G. Gour , author S. Bandyopadhyay , \ and\ author B. C. \ Sanders ,\ 10.1063/1.2435088 journal journal J. Math. Phys. \ volume 48 ,\ pages 012108 ( year 2007 ) NoStop

    2007 doi

  39. [47]

    Buscemi , author G

    author author F. Buscemi , author G. Gour , \ and\ author J. S. \ Kim ,\ 10.1103/PhysRevA.80.012324 journal journal Phys. Rev. A \ volume 80 ,\ pages 012324 ( year 2009 ) NoStop

    2009 doi

  40. [48]

    author author J. S. \ Kim ,\ 10.1103/PhysRevA.85.062302 journal journal Phys. Rev. A \ volume 85 ,\ pages 062302 ( year 2012 ) NoStop

    2012 doi

  41. [49]

    author author J. S. \ Kim ,\ 10.1140/epjd/e2014-50411-1 journal journal Eur. Phys. J. D \ volume 68 ,\ pages 324 ( year 2014 ) NoStop

    2014 doi

  42. [50]

    author author J. S. \ Kim ,\ 10.1103/PhysRevA.97.012334 journal journal Phys. Rev. A \ volume 97 ,\ pages 012334 ( year 2018 ) NoStop

    2018 doi

  43. [51]

    Guo ,\ 10.1007/s11128-018-1996-5 journal journal Quant

    author author Y. Guo ,\ 10.1007/s11128-018-1996-5 journal journal Quant. Info. Proc. \ volume 17 ,\ pages 222 ( year 2018 ) NoStop

    1996 doi

  44. [52]

    \ Yang , author B

    author author L.-M. \ Yang , author B. Chen , author S.-M. \ Fei , \ and\ author Z.-X. \ Wang ,\ 10.1007/s11128-019-2513-1 journal journal Quant. Info. Proc. \ volume 19 ,\ pages 4 ( year 2019 ) NoStop

    2019 doi

  45. [53]

    \ Jin , author S.-M

    author author Z.-X. \ Jin , author S.-M. \ Fei , \ and\ author X. Li-Jost ,\ 10.1007/s10773-019-04053-3 journal journal Int. J. Theor. Phys. \ volume 58 ,\ pages 1576 ( year 2019 a ) NoStop

    2019 doi

  46. [54]

    \ Jin , author S.-M

    author author Z.-X. \ Jin , author S.-M. \ Fei , \ and\ author C.-F. \ Qiao ,\ 10.1007/s11128-019-2220-y journal journal Quant. Info. Proc. \ volume 18 ,\ pages 105 ( year 2019 b ) NoStop

    2019 doi

  47. [55]

    Shen , author D

    author author Z. Shen , author D. Xuan , author W. Zhou , author Z. Wang , \ and\ author S.-M. \ Fei ,\ https://www.mdpi.com/1099-4300/26/2/127 journal journal Entropy \ volume 26 ,\ pages 2 ( year 2024 ) NoStop

    2024

  48. [56]

    Pollock , author G

    author author K. Pollock , author G. Wang , \ and\ author E. Chitambar ,\ 10.1103/PhysRevA.103.032428 journal journal Phys. Rev. A \ volume 103 ,\ pages 032428 ( year 2021 ) NoStop

    2021 doi

  49. [57]

    author author J. G. \ Krishnan , author H. K. J. , \ and\ author A. K. \ Pal ,\ 10.1103/PhysRevA.107.042411 journal journal Phys. Rev. A \ volume 107 ,\ pages 042411 ( year 2023 ) NoStop

    2023 doi

  50. [58]

    D\"ur , author G

    author author W. D\"ur , author G. Vidal , \ and\ author J. I. \ Cirac ,\ 10.1103/PhysRevA.62.062314 journal journal Phys. Rev. A \ volume 62 ,\ pages 062314 ( year 2000 ) NoStop

    doi

  51. [59]

    Sen(De) , author U

    author author A. Sen(De) , author U. Sen , author M. Wie s \' s niak , author D. Kaszlikowski , \ and\ author M. Z \. Z ukowski ,\ 10.1103/PhysRevA.68.062306 journal journal Phys. Rev. A \ volume 68 ,\ pages 062306 ( year 2003 ) NoStop

    2003 doi

  52. [60]

    author author R. H. \ Dicke ,\ 10.1103/PhysRev.93.99 journal journal Phys. Rev. \ volume 93 ,\ pages 99 ( year 1954 ) NoStop

    1954 doi

  53. [61]

    Bengtsson \ and\ author K

    author author I. Bengtsson \ and\ author K. \.Z yczkowski ,\ @noop title Geometry of Quantum States: An Introduction to Quantum Entanglement ,\ edition 2nd \ ed.\ ( publisher Cambridge University Press ,\ address Cambridge, UK; New York, NY ,\ year 2017 )\ p.\ pages 619 NoStop

    2017

  54. [62]

    Peres ,\ 10.1103/PhysRevLett.77.1413 journal journal Phys

    author author A. Peres ,\ 10.1103/PhysRevLett.77.1413 journal journal Phys. Rev. Lett. \ volume 77 ,\ pages 1413 ( year 1996 ) NoStop

    1996 doi

  55. [63]

    Horodecki , author P

    author author M. Horodecki , author P. Horodecki , \ and\ author R. Horodecki ,\ https://doi.org/10.1016/S0375-9601(96)00706-2 journal journal Phys. Lett. A \ volume 223 ,\ pages 1 ( year 1996 ) NoStop

    1996 doi

  56. [64]

    author author K. Z \. Z yczkowski , author P. Horodecki , author A. Sanpera , \ and\ author M. Lewenstein ,\ 10.1103/PhysRevA.58.883 journal journal Phys. Rev. A \ volume 58 ,\ pages 883 ( year 1998 ) NoStop

    1998 doi

  57. [65]

    Vidal \ and\ author R

    author author G. Vidal \ and\ author R. F. \ Werner ,\ 10.1103/PhysRevA.65.032314 journal journal Phys. Rev. A \ volume 65 ,\ pages 032314 ( year 2002 ) NoStop

    2002 doi

  58. [66]

    Lee , author M

    author author J. Lee , author M. S. \ Kim , author Y. J. \ Park , \ and\ author S. Lee ,\ 10.1080/09500340008235138 journal journal J. Mod. Opt. \ volume 47 ,\ pages 2151 ( year 2000 ) NoStop

    doi

  59. [67]

    \ Chen \ and\ author L.-z

    author author X.-y. \ Chen \ and\ author L.-z. \ Jiang ,\ 10.1103/PhysRevA.101.012308 journal journal Phys. Rev. A \ volume 101 ,\ pages 012308 ( year 2020 ) NoStop

    2020 doi

  60. [68]

    author author M. A. \ Nielsen \ and\ author I. L. \ Chuang ,\ @noop title Quantum Computation and Quantum Information \ ( publisher Cambridge University Press ,\ year 2010 ) NoStop

    2010

  61. [69]

    author author A. S. \ Holevo \ and\ author V. Giovannetti ,\ 10.1088/0034-4885/75/4/046001 journal journal Rep. Prog. Phys. \ volume 75 ,\ pages 046001 ( year 2012 ) NoStop

    2012 doi

  62. [70]

    Yu \ and\ author J

    author author T. Yu \ and\ author J. H. \ Eberly ,\ 10.1126/science.1167343 journal journal Science \ volume 323 ,\ pages 598 ( year 2009 ) NoStop

    2009 doi

  63. [71]

    Daffer , author K

    author author S. Daffer , author K. W\'odkiewicz , author J. D. \ Cresser , \ and\ author J. K. \ McIver ,\ 10.1103/PhysRevA.70.010304 journal journal Phys. Rev. A \ volume 70 ,\ pages 010304 ( year 2004 ) NoStop

    2004 doi

  64. [72]

    Shrikant , author R

    author author U. Shrikant , author R. Srikanth , \ and\ author S. Banerjee ,\ 10.1103/PhysRevA.98.032328 journal journal Phys. Rev. A \ volume 98 ,\ pages 032328 ( year 2018 ) NoStop

    2018 doi

  65. [73]

    Gupta , author S

    author author R. Gupta , author S. Gupta , author S. Mal , \ and\ author A. Sen(De) ,\ 10.1103/PhysRevA.105.012424 journal journal Phys. Rev. A \ volume 105 ,\ pages 012424 ( year 2022 ) NoStop

    2022 doi

Pith tools

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