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A low order Bargmann invariant hierarchy for set coherence

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Pith's one-line read Fourth-order invariants give the first universal pairwise test for set coherence

desk verdict Fourth-order ordering-sensitive Bargmann invariants give a clean universal pairwise test for set coherence by detecting noncommutativity. read the letter →

arxiv 2605.10003 v1 submitted 2026-05-11 quant-ph

Yan-Ling Wang This is my paper · ORCID
classification quant-ph
keywords setcoherenceBargmanninvariantsquantumnoncommutativitytracedensityoperatorsfinitefamiliespairwisecriterion
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

Set coherence holds for a finite family of quantum states exactly when they are all diagonal in one shared basis. The paper determines the minimal Bargmann invariant order needed to decide this property. Second-order data suffice for qubits but fail for qutrits, while third-order data suffice for qutrits but fail in dimension four. Fourth-order ordering-sensitive invariants succeed universally: checking them on every unordered pair completely tests whether any finite family shares a common incoherent basis, regardless of dimension. This yields a low-order hierarchy that ties cyclic trace invariants to the noncommutativity blocking set coherence.

What carries the argument

The ordering-sensitive fourth-order Bargmann invariant, a cyclic trace product over four states that detects noncommuting relations between pairs missed by lower orders.

What would settle it

A pair of states in dimension four or higher that share a common diagonalizing basis yet fail the fourth-order invariant test, or that lack a common basis yet pass the test.

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Extended reading notes

Core claim

Fourth-order, ordering-sensitive Bargmann invariants form the first universal pairwise criterion for set coherence. For any finite family of quantum states, verifying these invariants across all unordered pairs determines exactly whether the family is set incoherent, meaning all members are diagonal in one common basis. The criterion holds in arbitrary finite dimension because the ordering-sensitive fourth-order terms capture any residual noncommutativity that lower-order data miss.

Load-bearing premise

The states are finite-dimensional density operators and the fourth-order invariants detect every instance of noncommutativity that prevents a common basis.

Editorial extensions

If this is right

  • The test is complete for any pair of states in any dimension.
  • Applying the pairwise criterion to all pairs gives a complete test for set coherence of arbitrary finite families.
  • The result supplies a dimension-independent hierarchy linking cyclic trace invariants to set coherence.

Reading between the lines

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

  • The criterion may enable numerical verification of set coherence without explicitly searching for a common basis.
  • It suggests set coherence is fundamentally a pairwise relational property detectable at low order.
  • Similar ordering-sensitive invariants could be explored for other basis-independent quantum relations such as mutual coherence.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper claims that a finite family of quantum states is set incoherent (i.e., simultaneously diagonalizable in one common basis) if and only if the states are mutually commuting. It constructs a hierarchy of Bargmann invariants (cyclic trace products of increasing order) and proves that second-order data suffice to detect commutativity for qubits but not qutrits, while third-order data suffice for qutrits but fail in dimension four. Fourth-order ordering-sensitive invariants (distinguishing distinct cyclic orderings) are shown to yield an equality condition equivalent to commutativity for any finite dimension; applying this pairwise criterion to all unordered pairs therefore gives a complete test for set coherence of arbitrary finite families.

Significance. If the derivations hold, the work supplies the first explicit low-order hierarchy linking measurable cyclic trace invariants to the noncommutativity that obstructs a common incoherent basis. The dimension-dependent completeness thresholds and the universal fourth-order criterion are technically clean, parameter-free results that strengthen the interface between coherence theory and invariant theory. The pairwise reduction to commutativity is a standard linear-algebra fact, but the explicit invariant characterization makes the test experimentally accessible without basis optimization.

minor comments (4)
  1. The definition of 'ordering-sensitive' fourth-order Bargmann invariants (distinguishing distinct cyclic permutations) should be stated explicitly with the relevant trace expressions in the main text rather than deferred to an appendix.
  2. Section 3 (or equivalent) on the qutrit counterexample for third-order data would benefit from an explicit 3×3 matrix pair that saturates the failure case, together with the numerical values of the invariants.
  3. The manuscript cites prior work on Bargmann invariants but does not compare the present fourth-order criterion with existing higher-order coherence witnesses; a short paragraph contrasting the two would clarify novelty.
  4. Notation for the cyclic products (e.g., the precise placement of the ordering index) is introduced gradually; a consolidated table of the invariants up to order four would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, accurate summary of the results, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper derives a hierarchy of Bargmann invariants (cyclic trace products of density operators) and proves that fourth-order ordering-sensitive combinations detect noncommutativity for arbitrary finite dimension, yielding a complete pairwise test for set coherence. This rests on the standard linear-algebra fact that a finite family of Hermitian operators shares a common eigenbasis if and only if they mutually commute; the invariants are constructed directly from traces without fitting, self-definition, or load-bearing self-citation. Lower-order sufficiency is shown by explicit counter-examples in low dimensions, and the fourth-order criterion is verified by direct algebraic reduction to the commutator condition. The derivation is self-contained and independent of its inputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The work rests on standard finite-dimensional quantum mechanics and the definition of Bargmann invariants as cyclic traces; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

assumptions (2)
  • standard math Quantum states are represented by density operators on finite-dimensional Hilbert spaces.
    Implicit in the use of Bargmann invariants and basis diagonalization.
  • standard math Bargmann invariants are well-defined cyclic traces of products of the density operators.
    Used throughout the hierarchy construction.

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Cite this review

Pith. "Pith review of A low order Bargmann invariant hierarchy for set coherence." pith.science (2026). https://pith.science/paper/2605.10003

@misc{pith2026260510003,
  author       = {Pith},
  title        = {Pith review of: A low order Bargmann invariant hierarchy for set coherence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2605.10003}},
  note         = {Machine review of arXiv:2605.10003}
}
read the original abstract

Set coherence is a basis-independent relational form of quantum coherence: a finite family of quantum states is set incoherent exactly when all its members are diagonal in one common basis. We determine how much low-order Bargmann data are needed to decide this property. For two states, second-order data are complete for qubits but fail for qutrits, while complete third-order data are sufficient for qutrits but fail already in dimension four. We then show that fourth-order, ordering-sensitive Bargmann invariants give the first universal pairwise criterion for set coherence. Applied to all unordered pairs, this criterion yields a complete test for arbitrary finite families. The result provides a low-order hierarchy connecting cyclic trace invariants with the noncommutativity that prevents a common incoherent basis.

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Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    Therefore ΦW3(ρ, σ0) = ΦW3(ρ, σ1) = 1 2 , 1 2 , 1 4 , 1 4 , 1 4 , 1 8 , 1 8 , although the first pair is commuting and the second is not

    = 1 8 . Therefore ΦW3(ρ, σ0) = ΦW3(ρ, σ1) = 1 2 , 1 2 , 1 4 , 1 4 , 1 4 , 1 8 , 1 8 , although the first pair is commuting and the second is not. For everyd >4, the same construction can be embed- ded by taking a direct sum with a zero block of sized−4. This preserves positivity, trace one, all trace words, and commutativity or noncommutativity. Hence the...

    work page

  2. [2]

    For a finite Hermitian family, simultaneous diagonalizability is equivalent to pairwise commutativ- ity

    Since the Hilbert– Schmidt norm is faithful, Γ(ρ, σ) = 0 if and only if [ρ, σ] = 0. For a finite Hermitian family, simultaneous diagonalizability is equivalent to pairwise commutativ- ity. Applying the two-state criterion to every unordered pair proves the finite-family statement. The theorem gives a decision rule. It also gives a quan- titative invariant...

    work page

  3. [3]

    Proposition 3.For everyn≥2, C3(W (n) 2 )∩I 3(W (n) 2 )̸=∅

    Pairwise second-order data for qutrit families Let W (n) 2 ={ij: 1≤i≤j≤n}, which records all pairwise Hilbert–Schmidt inner prod- ucts, including purities. Proposition 3.For everyn≥2, C3(W (n) 2 )∩I 3(W (n) 2 )̸=∅. Thus all pairwise second-order data can be identical for a jointly diagonal qutrit family and for a set-coherent qutrit family. Proof.Let D1 =...

    work page

  4. [4]

    Proposition 4.For everyn≥2, C4(W (n) ≤3 )∩I 4(W (n) ≤3 )̸=∅

    Pairwise third-order data for four-dimensional families Let W (n) ≤3 = [ 1≤i<j≤n {ii, jj, ij, iii, jjj, iij, ijj}. Proposition 4.For everyn≥2, C4(W (n) ≤3 )∩I 4(W (n) ≤3 )̸=∅. Thus all pairwise words of order at most three can be identical for a jointly diagonal four-dimensional family and for a set-coherent four-dimensional family. Proof.Let D1 = 1√ 2 di...

    work page

  5. [5]

    Baumgratz, M

    T. Baumgratz, M. Cramer, and M. B. Plenio, Quantify- ing coherence, Phys. Rev. Lett.113, 140401 (2014)

    work page 2014

  6. [6]

    Streltsov, G

    A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys.89, 041003 (2017)

    work page 2017

  7. [7]

    Designolle, R

    S. Designolle, R. Uola, K. Luoma, and N. Brunner, Set coherence: Basis-independent quantification of quantum coherence, Phys. Rev. Lett.126, 220404 (2021)

    work page 2021

  8. [8]

    Bargmann, Note on wigner’s theorem on symmetry operations, J

    V. Bargmann, Note on wigner’s theorem on symmetry operations, J. Math. Phys.5, 862 (1964)

    work page 1964

Show all 36 references
  1. [9]

    J. G. Kirkwood, Quantum statistics of almost classical assemblies, Phys. Rev.44, 31 (1933)

    1933

  2. [10]

    P. A. M. Dirac, On the analogy between classical and quantum mechanics, Rev. Mod. Phys.17, 195 (1945)

    1945

  3. [11]

    D. R. M. Arvidsson-Shukur, W. F. Braasch, S. De Bi` evre, J. Dressel, A. N. Jordan, C. Langrenez, M. Lostaglio, J. S. Lundeen, and N. Yunger Halpern, Properties and applications of the kirkwood–dirac distribution, New J. Phys.26, 121201 (2024)

    2024

  4. [12]

    Zhang and N

    L. Zhang and N. Li, Can imaginarity be broadcast via real operations?, Communications in Theoretical Physics 76, 115104 (2024)

    2024

  5. [13]

    Schmid, R

    D. Schmid, R. D. Baldij˜ ao, Y. Ying, R. Wagner, and J. H. Selby, Kirkwood–dirac representations beyond quantum states and their relation to noncontextuality, Phys. Rev. A110, 052206 (2024)

    2024

  6. [14]

    Dressel, Weak values as interference phenomena, Phys

    J. Dressel, Weak values as interference phenomena, Phys. Rev. A91, 032116 (2015)

    2015

  7. [15]

    Wagner and E

    R. Wagner and E. F. Galv˜ ao, Simple proof that anoma- lous weak values require coherence, Phys. Rev. A108, L040202 (2023)

    2023

  8. [16]

    E. F. Galv˜ ao and D. J. Brod, Quantum and classical bounds for two-state overlaps, Phys. Rev. A101, 062110 (2020)

    2020

  9. [17]

    Wagner, R

    R. Wagner, R. S. Barbosa, and E. F. Galv˜ ao, Inequali- ties witnessing coherence, nonlocality, and contextuality, Phys. Rev. A109, 032220 (2024)

    2024

  10. [18]

    Giordani, C

    T. Giordani, C. Esposito, F. Hoch, G. Carvacho, D. J. Brod, E. F. Galv˜ ao, N. Spagnolo, and F. Sciarrino, Wit- nesses of coherence and dimension from multiphoton in- distinguishability tests, Phys. Rev. Research3, 023031 (2021)

    2021

  11. [19]

    Giordani, R

    T. Giordani, R. Wagner, C. Esposito, A. Camillini, F. Hoch, G. Carvacho, C. Pentangelo, F. Ceccarelli, S. Piacentini, A. Crespi, N. Spagnolo, R. Osellame, E. F. Galv˜ ao, and F. Sciarrino, Experimental certifi- cation of contextuality, coherence, and dimension in a programmabl...

    2023

  12. [20]

    Fernandes, R

    C. Fernandes, R. Wagner, L. Novo, and E. F. Galv˜ ao, Unitary-invariant witnesses of quantum imaginarity, Phys. Rev. Lett.133, 190201 (2024)

    2024

  13. [21]

    Li and Y.-X

    M.-S. Li and Y.-X. Tan, Bargmann invariants for quan- tum imaginarity, Phys. Rev. A111, 022409 (2025)

    2025

  14. [22]

    Zhang, B

    L. Zhang, B. Xie, and B. Li, Geometry of sets of bargmann invariants, Phys. Rev. A111, 042417 (2025)

    2025

  15. [23]

    Zhang, B

    L. Zhang, B. Xie, and Y. Tao, Bargmann-invariant frame- work for local unitary equivalence and entanglement, Phys. Rev. A112, 052426 (2025)

    2025

  16. [24]

    Oszmaniec, D

    M. Oszmaniec, D. J. Brod, and E. F. Galv˜ ao, Measur- ing relational information between quantum states, and applications, New J. Phys.26, 013053 (2024)

    2024

  17. [25]

    M.-S. Li, R. Wagner, and L. Zhang, Multistate imaginar- ity and coherence in qubit systems, Phys. Rev. A113, 012428 (2026)

    2026

  18. [26]

    Wagner, Z

    R. Wagner, Z. Schwartzman-Nowik, I. L. Paiva, A. Te’eni, A. Ruiz-Molero, R. S. Barbosa, E. Cohen, and E. F. Galv˜ ao, Quantum circuits for measuring weak val- ues, kirkwood–dirac quasiprobability distributions, and state spectra, Quantum Sci. Technol.9, 015030 (2024)

    2024

  19. [27]

    Y. Quek, E. Kaur, and M. M. Wilde, Multivariate trace estimation in constant quantum depth, Quantum8, 1220 (2024)

    2024

  20. [28]

    Simonov, R

    K. Simonov, R. Wagner, and E. F. Galv˜ ao, Estimation of multivariate traces of states given partial classical in- formation, Phys. Rev. A112, 062435 (2025)

    2025

  21. [29]

    M. Shin, J. Lee, S. Lee, and K. Jeong, Resource-efficient algorithm for estimating the trace of quantum state pow- 8 ers, Quantum9, 1832 (2025)

    2025

  22. [30]

    Wagner, Bargmann scenarios (2026), arXiv:2604.18833 [quant-ph]

    R. Wagner, Bargmann scenarios (2026), arXiv:2604.18833 [quant-ph]

    2026 arXiv

  23. [31]

    Procesi, The invariant theory ofn×nmatrices, Adv

    C. Procesi, The invariant theory ofn×nmatrices, Adv. Math.19, 306 (1976)

    1976

  24. [32]

    Y. P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR Izv.8, 727 (1974)

    1974

  25. [33]

    Formanek, Generating the ring of matrix invariants, inRing Theory, Lecture Notes in Mathematics, Vol

    E. Formanek, Generating the ring of matrix invariants, inRing Theory, Lecture Notes in Mathematics, Vol. 1197 (Springer, Berlin, 1986) pp. 73–82

    1986

  26. [34]

    R. Uola, A. C. S. Costa, H. C. Nguyen, and O. G¨ uhne, Quantum steering, Rev. Mod. Phys.92, 015001 (2020)

    2020

  27. [35]

    Heinosaari, T

    T. Heinosaari, T. Miyadera, and M. Ziman, An invitation to quantum incompatibility, J. Phys. A: Math. Theor.49, 123001 (2016)

    2016

  28. [36]

    Wagner and E

    R. Wagner and E. F. Galv˜ ao, Commutativity from a single bargmann invariant equality (2026), arXiv:2605.07405v1 [quant-ph]

    2026 arXiv

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