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Poset-refined majorization relations

T0 review · 0 major / 4 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read When change-of-basis matrices admit an LU-approximation on a poset, classical majorization bounds for matrix sums and products sharpen to a tighter middle vector built from partially aligned eigenvalues or singular values.

desk verdict Clean poset-LU refinement of classical majorization that unifies and extends recent tensor inequalities with checkable proofs. read the letter →

arxiv 2607.28061 v1 pith:7KT726LR submitted 2026-07-30 math.FA quant-ph

Alexander E. Guterman , Michael M. Wolf This is my paper · ORCID
classification math.FAquant-ph MSC 15A4215A1806A0647A30
keywords majorizationpartialorderLU-approximationKyFaninequalitiesHornlog-majorizationtensorproductscompoundmatricesKroneckersums
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

Classical majorization says that the eigenvalues of a sum of Hermitian matrices are majorized by the sum of their individually sorted eigenvalues, and analogous statements hold for singular values of sums and products. This paper shows that if the indices of those eigenvalues or singular values carry a partial order, and the change-of-basis matrix between the two matrices can be approximated by products of lower- and upper-triangular matrices compatible with that order, then a strictly tighter majorizing vector appears in the middle: the sorted list of pointwise sums (or products) of the spectral functions on the poset. The same mechanism refines von Neumann’s trace inequality. The product order on multi-indices makes the LU condition automatic for tensor products, exterior and symmetric powers, and Kronecker sums, yielding short proofs and extensions of separable Ky Fan relations and related bounds that do not follow easily from the classical fully aligned versions.

What carries the argument

Poset-LU approximation: a matrix C admits an LU-approximation when it is a norm limit of products LU with L lower-triangular and U upper-triangular with respect to the partial order on P. That condition forces nonzero minors det C_{S,T} to be witnessed by an intermediate set R comparable to both S and T, which is what inserts the tighter diagonal vector D_{α+β} (or D_{αβ}) into the majorization chain.

What would settle it

Exhibit two Hermitian matrices whose eigenbases give a change-of-basis matrix outside the LU-closure for a chosen poset, yet λ(A+B) is still majorized by λ(D_{α+β}); or, for a product-order example where LU holds, find a numerical counterexample where the Ky Fan sums of A+B exceed those of D_{α+β}.

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Extended reading notes

Core claim

If Hermitian matrices A and B have order-decreasing spectral functions α and β on a finite poset P, and the change-of-basis matrix C lies in the norm closure of L(P)U(P), then λ(A+B) is majorized by λ(D_{α+β}), which in turn is majorized by the classical aligned sum λ(A)+λ(B). Parallel refinements hold for weak majorization of singular values of a sum and for log-majorization of singular values of a product, and they imply a refined von Neumann trace inequality.

Load-bearing premise

The change-of-basis matrix between the two matrices must be approximable by lower- and upper-triangular factors that respect the chosen partial order; without that, the tighter middle bound need not hold.

Editorial extensions

If this is right

  • Ky Fan majorization for a sum of tensor products holds for any number of factors and for arbitrary (not necessarily positive) matrices, with singular values and weak majorization.
  • Singular values of sums of exterior or symmetric powers are weakly majorized by the sorted list of corresponding products of singular values over strictly (or weakly) increasing multi-indices.
  • Products of Kronecker sums of positive-semidefinite matrices obey a refined Horn log-majorization by the sorted pointwise product of the Kronecker-sum eigenvalue vectors.
  • Convex trace functions and information quantities (relative entropy, quantum Chernoff coefficient) inherit tighter bounds from the refined majorization whenever the LU condition holds.

Reading between the lines

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

  • The discussion’s weaker combinatorial condition (nonzero minors witnessed by a common comparable R) may let the method apply to posets and bases that fail full LU-density but still satisfy the minor condition.
  • Other natural partial orders—Bruhat order, dominance order on partitions, or causal orders—could systematically produce new majorization refinements once LU or the minor condition is checked.
  • The short tensor-product proof suggests the same LU-plus-product-order pattern may streamline multipartite inequalities elsewhere in matrix analysis and quantum information.
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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

0 major / 4 minor

Summary. The paper refines classical majorization relations (Ky Fan for eigenvalues and singular values, Horn log-majorization, and von Neumann’s trace inequality) by replacing perfect alignment of eigenvalues/singular values with a partial order on the index set. Under the hypothesis that the change-of-basis matrix lies in the norm closure of L(P)U(P) for a finite poset P, and that the spectral maps are order-decreasing, Theorems 1–3 establish intermediate majorizing vectors of the form λ(D_{α+β}) or λ(D_{αβ}). The proofs rely on additive compounds/Golden–Thompson, Cauchy–Binet on LU approximants, level-set integrals or order-preserving bijections, and Cordes’ inequality. Applications via the product order yield a short proof and extension of the separable Ky Fan relation to arbitrary matrices and any number of factors (Corollary 2), plus analogous statements for exterior/symmetric powers and products of Kronecker sums.

Significance. The work supplies a clean, reusable framework that unifies and strengthens several classical matrix inequalities under a verifiable structural hypothesis. Lemma 3 cleanly characterises when LU-approximability is automatic, while Lemma 4 and the triangularity arguments for compounds/permanents make the hypothesis hold for all stated applications. The short proof of the multi-factor separable Ky Fan relation (and its extension beyond positive matrices) is a concrete payoff; the same method immediately yields new statements for (anti)symmetric powers and Kronecker sums. The contribution is solid matrix analysis with clear quantum-information side applications (relative entropy, Chernoff coefficients).

minor comments (4)
  1. [Theorem 3 proof] In the proof of Theorem 3 the passage to the s→0 limit after Cordes and the Hilbert–Schmidt bound is only sketched; a one-line justification that the resulting max is attained on the support of the minors would improve readability.
  2. [Example 1 (continued)] Example 1 (continued) after Theorem 3 simply lists the three vectors; stating the concrete matrices or the value of k that realises the products would make the numerical illustration self-contained.
  3. [§5 Discussion] The discussion section correctly notes that LU-approximability can be weakened to the existence of a common R with R ⪯_k S and R ⪯_k T whenever det C_{S,T} ≠ 0; it would be helpful to record this weaker hypothesis already in the statements of Theorems 1–3.
  4. [front matter / references] Typographical consistency: the date line reads “July 31, 2026” and several arXiv identifiers in the bibliography are likewise future-dated; these should be corrected or left as placeholders consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: self-contained majorization refinements from classical lemmas under an explicit LU-approximability hypothesis

full rationale

The paper derives refined Ky Fan, Horn, and von Neumann-type majorization bounds (Theorems 1–3) from standard tools—Golden–Thompson, additive compounds, Cauchy–Binet, Cordes, exterior/symmetric powers—under the explicit structural hypothesis that the change-of-basis matrix lies in the norm closure of L(P)U(P). That hypothesis is not smuggled in by definition or fit: Lemma 3 characterizes when it holds for all matrices (iff P is linearly ordered), and Lemmas 4–5 plus triangularity of minors/permanents verify it for the product-order applications. There are no fitted parameters, no predictions that reduce to inputs by construction, and no load-bearing self-citation uniqueness claims. Motivating citations ([Alh26], [AKP26]) are external targets that the paper re-proves and extends independently. The derivation chain is therefore non-circular.

Assumptions & free parameters 0 free parameters · 7 assumptions · 1 invented entities

Pure-math paper. No empirical free parameters. Background axioms are standard matrix-analysis facts (Golden–Thompson, Cauchy–Binet, Cordes, Ky Fan variational characterisation, properties of additive compounds and exterior/symmetric powers). The only paper-specific conceptual addition is the definition of poset-triangular matrices and LU-approximability; these are definitions, not unproved physical entities.

assumptions (7)
  • standard math Golden–Thompson inequality: tr exp(A+B) ≤ tr(exp(A)exp(B)) for Hermitian A,B
    Used in Lemma 1 to bound λ_max(A+B).
  • standard math Cauchy–Binet formula for minors of a product
    Central to the second inequality in the proof of Theorem 1 and the triangularity arguments for exterior/symmetric powers.
  • standard math Cordes inequality ∥X^s Y^s∥_∞ ≤ ∥XY∥_∞^s for PSD X,Y and s∈(0,1]
    Used in the proof of Theorem 3 to pass from the product of singular values to a max over minors.
  • standard math Additive compound matrices satisfy (A+B)^[k]=A^[k]+B^[k] and λ_max(A^[k])=λ_[k](A)
    Invoked via Fiedler (1974) in Lemma 2 to reduce Ky Fan k-norms to ordinary max eigenvalues.
  • standard math Exterior and symmetric powers are continuous, multiplicative, and preserve unitarity
    Cited from Greub; used in Corollary 3 to transfer LU-approximability to compound matrices.
  • domain assumption Finite poset P and order-decreasing real (or nonnegative) functions α,β on P
    Standing setup of Theorems 1–3; without order-decreasing spectra the middle diagonal vector need not refine the classical bound.
  • domain assumption Change-of-basis matrix C lies in the norm closure of L(P)U(P)
    The load-bearing hypothesis of all three main theorems; Lemma 3 shows it is automatic only for linear orders.
invented entities (1)
  • Poset LU-approximation (closure of L(P)U(P)) independent evidence
    purpose: Provides the structural condition under which classical majorization can be refined by a poset-aligned intermediate vector.
    Defined in §3 via lower/upper triangular matrices compatible with the partial order; characterised for density by Lemma 3. It is a definitional tool, not an unobserved physical object.

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Pith. "Pith review of Poset-refined majorization relations." pith.science (2026). https://pith.science/paper/7KT726LR

@misc{pith2026260728061,
  author       = {Pith},
  title        = {Pith review of: Poset-refined majorization relations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7KT726LR}},
  note         = {Machine review of arXiv:2607.28061}
}
read the original abstract

Several classical majorization relations for sums or products of matrices involve a majorizing vector of perfectly aligned eigenvalues or singular values. By relaxing the order of alignment to a partial order, we show that the majorization can be strengthened, provided the change-of-basis matrices admit an LU-approximation with respect to this partial order. In this way, we obtain refined versions of Ky Fan's majorization relations, Horn's log-majorization relation, and von Neumann's trace inequality. As an application, we give a short proof of the separable Ky Fan majorization relation for an arbitrary number of tensor factors and extend it to a sum of tensor products of arbitrary matrices. Further applications concern majorization relations for sums of (anti-)symmetric powers and for products of Kronecker sums.

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Works this paper leans on

125 extracted references · 2 linked inside Pith

  1. [1]

    , title=

    Alhejji, Mohammad A. , title=. Annales Henri Poincar. 2026 , volume=

    2026

  2. [2]

    2026 , eprint=

    A majorization relation for a sum of two tensor products of positive semidefinite operators , author=. 2026 , eprint=

    2026

  3. [3]

    Additive compound matrices and an inequality for eigenvalues of symmetric stochastic matrices , url =

    Fiedler, Miroslav , journal =. Additive compound matrices and an inequality for eigenvalues of symmetric stochastic matrices , url =

  4. [4]

    An analysis of completely-positive trace-preserving maps on

    Mary. An analysis of completely-positive trace-preserving maps on. Linear Algebra and its Applications , volume =. 2002 , issn =. doi:https://doi.org/10.1016/S0024-3795(01)00547-X , url =

    doi:10.1016/s0024-3795(01)00547-x 2002

  5. [5]

    and Paulsen, Vern I

    Pandey, Satish K. and Paulsen, Vern I. and Prakash, Jitendra and Rahaman, Mizanur , title =. Journal of Mathematical Physics , volume =. 2020 , abstract =. doi:10.1063/1.5045184 , url =

    doi:10.1063/1.5045184 2020

  6. [6]

    Foundations of Physics , year=

    Appleby, Marcus and Flammia, Steven and McConnell, Gary and Yard, Jon , title=. Foundations of Physics , year=. doi:10.1007/s10701-017-0090-7 , url=

    doi:10.1007/s10701-017-0090-7

  7. [7]

    Osaka Mathematical Journal , number =

    Hidehiko Yamabe and Zuiman Yujob. Osaka Mathematical Journal , number =

  8. [8]

    and Hoang, Michael C

    Fuchs, Christopher A. and Hoang, Michael C. and Stacey, Blake C. , TITLE =. Axioms , VOLUME =. 2017 , NUMBER =

    2017

Show all 125 references
  1. [9]

    ON MUTUALLY UNBIASED BASES , journal =

    DURT, THOMAS and ENGLERT, BERTHOLD-GEORG and BENGTSSON, INGEMAR and \. ON MUTUALLY UNBIASED BASES , journal =. 2010 , doi =

    2010

  2. [10]

    Mutually

    McNulty, Daniel and Weigert, Stefan , journal =. Mutually. doi:10.22331/q-2026-04-01-2051 , url =

    2026 doi

  3. [11]

    Journal of Physics A: Mathematical and General , abstract =

    Scott, A J , title =. Journal of Physics A: Mathematical and General , abstract =. 2006 , volume =. doi:10.1088/0305-4470/39/43/009 , url =

    2006 doi

  4. [12]

    2016 , volume =

    Jeli\'c, Marija , title =. 2016 , volume =

    2016

  5. [13]

    2007 , eprint=

    Some Open Problems in Quantum Information Theory , author=. 2007 , eprint=

    2007

  6. [14]

    Operational criterion and constructive checks for the separability of low-rank density matrices , author =. Phys. Rev. A , volume =. 2000 , publisher =. doi:10.1103/PhysRevA.62.032310 , url =

    doi

  7. [15]

    John Milnor , journal =

  8. [16]

    Algebraic topology

    Hatcher, Allen. Algebraic topology. 2000

    2000

  9. [17]

    Blagojević and Günter M

    Pavle V.M. Blagojević and Günter M. Ziegler , keywords =. The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes , journal =. 2011 , note =. doi:https://doi.org/10.1016/j.topol.2011.05.008 , url =

    2011 doi

  10. [18]

    Sbornik: Mathematics , abstract =

    A Yu Volovikov , title =. Sbornik: Mathematics , abstract =. 1993 , volume =. doi:10.1070/SM1993v076n02ABEH003418 , url =

    1993 doi

  11. [19]

    An ideal-valued cohomological index theory with applications to

    Fadell, Edward and Husseini, Sufian , volume=. An ideal-valued cohomological index theory with applications to. Ergodic Theory and Dynamical Systems , year=. doi:10.1017/S0143385700009342 , number=

    doi

  12. [20]

    doi:doi:10.1515/9783110858372 , isbn =

    Transformation Groups , author =. doi:doi:10.1515/9783110858372 , isbn =

    doi

  13. [21]

    Mathematika , volume =

    Basu, Samik and Kundu, Bikramjit , title =. Mathematika , volume =. doi:https://doi.org/10.1112/mtk.70001 , url =

    doi

  14. [22]

    Journal of Homotopy and Related Structures , year=

    Nath, Arijit and Nath, Avijit , title=. Journal of Homotopy and Related Structures , year=. doi:10.1007/s40062-024-00357-2 , url=

    doi

  15. [23]

    Perturbation Theory for Linear Operators , url =

    Kato, Tosio , year =. Perturbation Theory for Linear Operators , url =

  16. [24]

    David Hobby and D. M. Silberger , journal =. Quotients of Primes , volume =

  17. [25]

    Periodic Jacobi Matrices on Trees , volume =

    Nir Avni and Jonathan Breuer and Barry Simon , journal =. Periodic Jacobi Matrices on Trees , volume =

  18. [26]

    , title =

    Baker, A. , title =. Mathematika , volume =

  19. [27]

    and Gamboa, J.M

    Fernando, J.F. and Gamboa, J.M. and Ruiz, J.M. , title =. J. Eur. Math. Soc. , volume =. doi:10.4171/JEMS/439 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/S0025579300008068 , year =

    doi

  20. [28]

    Real Algebraic Geometry , doi =

    Jacek Bochnak and Michel Coste and Marie-Francoise Roy , year =. Real Algebraic Geometry , doi =

  21. [29]

    Journal of Differential Geometry , number =

    Luigi Ambrosio and Halil Mete Soner , title =. Journal of Differential Geometry , number =. 1996 , doi =

    1996

  22. [30]

    Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable, Third Edition , publisher =

    Lars Ahlfors , year =. Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable, Third Edition , publisher =

  23. [31]

    Singularities of Plane Curves , publisher =

    Eduardo Casas-Alvero , year =. Singularities of Plane Curves , publisher =

  24. [32]

    Griffiths , year =

    Phillip A. Griffiths , year =. Introduction to algebraic curves , publisher =

  25. [33]

    Stewart and Ji-Guang Sun , year =

    G.W. Stewart and Ji-Guang Sun , year =. Matrix Perturbation Theory , publisher =

  26. [34]

    Farenick , year =

    Douglas R. Farenick , year =. Algebras of Linear Transformations , publisher =

  27. [35]

    Notes on algebraic functions , volume =

    Guan, KY and Lei, Jinzhi , year =. Notes on algebraic functions , volume =. International Journal of Mathematics and Mathematical Sciences , doi =

  28. [36]

    Defining the almost-entropic regions by algebraic inequalities , volume =

    Gomez, Arley and Mejia Moreno, Carolina and Montoya, Juan , year =. Defining the almost-entropic regions by algebraic inequalities , volume =. International Journal of Information and Coding Theory , doi =

  29. [37]

    Mathematica Aeterna , number =

    Johan Nixon , title =. Mathematica Aeterna , number =. 2013 , doi =

    2013

  30. [38]

    , number =

    Michel Coste , title =. , number =. 2002 , doi =

    2002

  31. [39]

    Advances in Mathematics , volume =

    Nir Avni and Jonathan Breuer and Barry Simon , keywords =. Advances in Mathematics , volume =. 2020 , issn =. doi:https://doi.org/10.1016/j.aim.2020.107241 , url =

    2020

  32. [40]

    Werner , title =

    Mark Fannes and Bruno Nachtergaele and Reinhard F. Werner , title =. Communications in Mathematical Physics , number =. 1992 , doi =

    1992

  33. [41]

    Wilde , title =

    Soorya Rethinasamy and Mark M. Wilde , title =. Phys. Rev. Research , number =. 2020 , doi =

    2020

  34. [42]

    M\"uller and Henrik Wilming , title =

    Paul Boes and Jens Eisert and Rodrigo Gallego and Markus P. M\"uller and Henrik Wilming , title =. Phys. Rev. Lett. , number =. 2019 , doi =

    2019

  35. [43]

    Tulja Varun Kondra and Chandan Datta and Alexander Streltsov , title =. Phys. Rev. Lett. , number =. 2021 , doi =

    2021

  36. [44]

    Patryk Lipka-Bartosik and Paul Skrzypczyk , title =. Phys. Rev. Lett. , number =. 2021 , doi =

    2021

  37. [45]

    , number =

    Roberto Rubboli and Marco Tomamichel , title =. , number =. 2021 , doi =

    2021

  38. [46]

    , number =

    Henrik Wilming , title =. , number =. 2020 , doi =

    2020

  39. [47]

    and Datta,Nilanjana , title =

    Hanson,Eric P. and Datta,Nilanjana , title =. Journal of Mathematical Physics , volume =. 2018 , doi =

    2018

  40. [48]

    Alvaro Yanguez Bachiller , title =

  41. [49]

    Journal of Mathematical Physics , volume =

    Kukulski,Ryszard and Nechita,Ion and Pawela,Lukasz and Puchala,Zbigniew and Zyczkowski,Karol , title =. Journal of Mathematical Physics , volume =. 2021 , doi =

    2021

  42. [50]

    2010 , publisher=

    An Introduction to Manifolds , author=. 2010 , publisher=

    2010

  43. [51]

    Milnor, John , publisher =

  44. [52]

    On the dimension of the compact invariant sets of certain non-linear maps

    Ma \ n \'e , Ricardo. On the dimension of the compact invariant sets of certain non-linear maps. Dynamical Systems and Turbulence, Warwick 1980. 1981

    1980

  45. [53]

    Robinson, James C. , year=. Dimensions, Embeddings, and Attractors , DOI=

  46. [54]

    , title=

    Heinosaari, Teiko and Mazzarella, Luca and Wolf, Michael M. , title=. Communications in Mathematical Physics , year=

  47. [55]

    and Johnson, Charles R

    Horn, Roger A. and Johnson, Charles R. , publisher =

  48. [56]

    Hatcher, Allen , publisher =

  49. [57]

    , title = "

    Evans, David E. , title = ". The Quarterly Journal of Mathematics , volume =. 1977 , month =. doi:10.1093/qmath/28.3.271 , url =

    1977 doi

  50. [58]

    Detecting strange attractors in turbulence

    Takens, Floris. Detecting strange attractors in turbulence. Dynamical Systems and Turbulence, Warwick 1980. 1981

    1980

  51. [59]

    and Casdagli, Martin , title=

    Sauer, Tim and Yorke, James A. and Casdagli, Martin , title=. Journal of Statistical Physics , year=

  52. [60]

    Assessing Quantum Dimensionality from Observable Dynamics , author =. Phys. Rev. Lett. , volume =. 2009 , month =. doi:10.1103/PhysRevLett.102.190504 , url =

    2009 doi

  53. [61]

    Quantum process tomography of unitary maps from time-delayed measurements , journal=

    L. Quantum process tomography of unitary maps from time-delayed measurements , journal=. 2023 , month=. doi:10.1007/s11128-023-04008-y , url=

    2023 doi

  54. [62]

    Mityagin, B. S. , title=. Mathematical Notes , year=. doi:10.1134/S0001434620030189 , url=

    doi

  55. [63]

    Parlett , keywords =

    Beresford N. Parlett , keywords =. The (matrix) discriminant as a determinant , journal =. 2002 , issn =. doi:https://doi.org/10.1016/S0024-3795(02)00335-X , url =

    2002 doi

  56. [64]

    Electron

    Uwe Luther and Karla Rost , title =. Electron. Trans. Numer. Anal. , volume =. 2004 , pages =

    2004

  57. [65]

    , title =

    Hedemann, Samuel R. , title =. 2017 , eprint =. doi:10.48550/ARXIV.1709.05257 , url =

    2017 doi arXiv

  58. [66]

    2001 , doi =

    Linear Algebra and its Applications , volume =. 2001 , doi =

    2001

  59. [67]

    Linear Quantum State Observers , year=

    Clouâtré, Maison and Balas, Mark and Gehlot, Vinod and Valasek, John , journal=. Linear Quantum State Observers , year=

  60. [68]

    Totally Positive Matrices , DOI=

    Pinkus, Allan , year=. Totally Positive Matrices , DOI=

  61. [69]

    1988 , publisher=

    Dynamic Systems Control: Linear Systems Analysis and Synthesis , author=. 1988 , publisher=

    1988

  62. [70]

    1960 , note =

    On the general theory of control systems , journal =. 1960 , note =. doi:https://doi.org/10.1016/S1474-6670(17)70094-8 , author =

    1960 doi

  63. [71]

    SIAM Journal on Control and Optimization , volume =

    Aeyels, Dirk , title =. SIAM Journal on Control and Optimization , volume =

  64. [72]

    , booktitle=

    Sussmann, Hector J. , booktitle=. Generic single-input observability of continuous-time polynomial systems , year=

  65. [73]

    and Kleinmann, Matthias and Sentís, Gael , keywords =

    de Almeida, Jessica O. and Kleinmann, Matthias and Sentís, Gael , keywords =. Comparison of confidence regions for quantum state tomography , publisher =. doi:10.48550/ARXIV.2303.07136 , url =

    doi arXiv

  66. [74]

    doi:10.1088/1751-8121/ab8111 , year =

    M Guţă and J Kahn and R Kueng and J A Tropp , title =. doi:10.1088/1751-8121/ab8111 , year =

    doi

  67. [75]

    Journal of Physics A: Mathematical and General , abstract =

    Janusz Grabowski and Marek Kuś and Giuseppe Marmo , title =. Journal of Physics A: Mathematical and General , abstract =. 2005 , month =. doi:10.1088/0305-4470/38/47/011 , url =

    2005 doi

  68. [76]

    and Cirac, J

    Wolf, Michael M. and Cirac, J. Ignacio , title=. Communications in Mathematical Physics , year=. doi:10.1007/s00220-008-0411-y , url=

    doi

  69. [77]

    Flanders, Harley , title =. Proc. Amer. Math. Soc , year = 1951, volume = 2, pages =

    1951

  70. [78]

    , author=

    On the similarity transformation between a matirx and its transpose. , author=. Pacific Journal of Mathematics , year=

  71. [79]

    Fillmore , title =

    P.A. Fillmore , title =. The American Mathematical Monthly , year = 1969, volume = 76, pages =

    1969

  72. [80]

    Kenjiro Shoda , journal=

  73. [81]

    , title=

    Rastegin, Alexey E. , title=. Journal of Statistical Physics , year=

  74. [82]

    Electronic Journal of Linear Algebra , year = 2018, doi=

    Katarzyna Filipiak and Daniel Klein and Erika Vojtková , title =. Electronic Journal of Linear Algebra , year = 2018, doi=

    2018

  75. [83]

    Bhatia , title =

    R. Bhatia , title =. Linear Algebra and its Applications , year = 1984, doi=

    1984

  76. [84]

    Marcus and H

    M. Marcus and H. Minc , title =

  77. [85]

    Greub, Werner , title =

  78. [86]

    Cordes , title=

    H.O. Cordes , title=

  79. [87]

    Nonpositive-transpose entanglement and bound entanglement: From distillability sets to inequalities and multivariable insights , author =. Phys. Rev. A , volume =

  80. [88]

    and Olkin, Ingram and Arnold, Barry C

    Marshall, Albert W. and Olkin, Ingram and Arnold, Barry C. , title =

  81. [89]

    , title =

    Mirsky, L. , title =. The Quarterly Journal of Mathematics , volume =. 1960 , month =

    1960

  82. [90]

    Audenaert , title =

    Koenraad M.R. Audenaert , title =. Journal of Mathematical Physics , year =

  83. [91]

    Letters in Mathematical Physics , year=

    Rico, Pablo Costa , title=. Letters in Mathematical Physics , year=

  84. [92]

    and Riley, Richard D

    Jackson, Dan and White, Ian R. and Riley, Richard D. , title =. Biometrical Journal , volume =

  85. [93]

    and Markiewicz, A

    Filipiak, K. and Markiewicz, A. , title=. Metrika , year=

  86. [94]

    2025 , eprint=

    Symmetry reduction for testing k -block-positivity via extendibility , author=. 2025 , eprint=

    2025

  87. [95]

    2025 , issn =

    A note on a conjecture from distillability of quantum entanglement , journal =. 2025 , issn =

    2025

  88. [96]

    Linear and Multilinear Algebra , volume =

    Yongtao Li , title =. Linear and Multilinear Algebra , volume =

  89. [97]

    Canadian Mathematical Bulletin , author=

    Inequalities on partial traces of positive semidefinite block matrices , volume=. Canadian Mathematical Bulletin , author=. 2021 , pages=

    2021

  90. [98]

    ILAS Conference , year=

    Matrix inequalities involving partial traces , author=. ILAS Conference , year=

  91. [99]

    Blagojević, Pavle V. M. and Lück, Wolfgang and Ziegler, Günter M. , title =. Journal of Topology , volume =. 2015 , abstract =

    2015

  92. [100]

    2017 , issn =

    Linear Algebra and its Applications , volume =. 2017 , issn =. doi:https://doi.org/10.1016/j.laa.2017.06.043 , author =

    2017 doi

  93. [101]

    2016 , eprint=

    On Faces of the set of Quantum Channels , author=. 2016 , eprint=

    2016

  94. [102]

    Journal of Mathematical Physics , volume =

    Iten, Raban and Colbeck, Roger , title =. Journal of Mathematical Physics , volume =. 2018 , abstract =

    2018

  95. [103]

    1975 , issn =

    Completely positive linear maps on complex matrices , journal =. 1975 , issn =. doi:https://doi.org/10.1016/0024-3795(75)90075-0 , url =

    1975 doi

  96. [104]

    Linear and Multilinear Algebra , volume =

    Daeshik Choi , title =. Linear and Multilinear Algebra , volume =

  97. [105]

    Proof of a Conjectured

    Song, Zhiwei and Chen, Lin and Sun, Yize and Hu, Mengyao , journal=. Proof of a Conjectured. 2023 , volume=

    2023

  98. [106]

    and Hsieh, Min-Hsiu , title=

    Lin, Ting-Chun and Kim, Isaac H. and Hsieh, Min-Hsiu , title=. Letters in Mathematical Physics , year=

  99. [107]

    Schmidt-number witnesses and bound entanglement , author =. Phys. Rev. A , volume =

  100. [108]

    Schmidt number for density matrices , author =. Phys. Rev. A , volume =. 2000 , month =. doi:10.1103/PhysRevA.61.040301 , url =

    2000 doi

  101. [109]

    PRX Quantum , volume =

    Five Open Problems in Quantum Information Theory , author =. PRX Quantum , volume =

  102. [110]

    Evidence for bound entangled states with negative partial transpose , author =. Phys. Rev. A , volume =

  103. [111]

    Distillability and partial transposition in bipartite systems , author =. Phys. Rev. A , volume =

  104. [112]

    , title =

    Müller-Hermes, Alexander and Reeb, David and Wolf, Michael M. , title =. Journal of Mathematical Physics , volume =

  105. [113]

    Activating Distillation with an Infinitesimal Amount of Bound Entanglement , author =. Phys. Rev. Lett. , volume =

  106. [114]

    Forrest Stinespring , journal =

    W. Forrest Stinespring , journal =. Positive Functions on

  107. [115]

    and Ruskai, Mary Beth , title =

    Lieb, Elliott H. and Ruskai, Mary Beth , title =. Journal of Mathematical Physics , volume =

  108. [116]

    Quantum Info

    Bravyi, Sergey , title =. Quantum Info. Comput. , pages =. 2004 , issue_date =

    2004

  109. [117]

    One-Qubit Reduced States of a Pure Many-Qubit State: Polygon Inequalities , author =. Phys. Rev. Lett. , volume =

  110. [118]

    2003 , eprint=

    On the one-particle reduced density matrices of a pure three-qutrit quantum state , author=. 2003 , eprint=

    2003

  111. [119]

    Journal of Lie Theory , year =

    Tobias Franz , title =. Journal of Lie Theory , year =

  112. [120]

    2004 , eprint=

    Quantum marginal problem and representations of the symmetric group , author=. 2004 , eprint=

    2004

  113. [121]

    A Few Steps More Towards NPT Bound Entanglement , year=

    Pankowski, Lukasz and Piani, Marco and Horodecki, Michał and Horodecki, Paweł , journal=. A Few Steps More Towards NPT Bound Entanglement , year=

  114. [122]

    2021 , author =

    A matrix inequality for entanglement distillation problem , journal =. 2021 , author =

    2021

  115. [123]

    Entropy , VOLUME =

    Shen, Yi and Chen, Lin , TITLE =. Entropy , VOLUME =. 2018 , NUMBER =

    2018

  116. [124]

    2022 , volume =

    Wang, Yijin and Zhang, Ao and Zhang, Hong , title =. 2022 , volume =

    2022

  117. [125]

    , TITLE =

    Doković, Dragomir Z. , TITLE =. Entropy , VOLUME =. 2016 , NUMBER =

    2016

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