Spectral geometric mean and trace characterizations

Airat Bikchentaev; Anh Vu Le; Mohammad Sal Moslehian; Trung Hoa Dinh

arxiv: 2605.18888 · v1 · pith:ADIOYV6Enew · submitted 2026-05-16 · 🪐 quant-ph · math.FA· math.OA

Spectral geometric mean and trace characterizations

Airat Bikchentaev , Trung Hoa Dinh , Anh Vu Le , Mohammad Sal Moslehian This is my paper

Pith reviewed 2026-05-20 14:19 UTC · model grok-4.3

classification 🪐 quant-ph math.FAmath.OA

keywords spectral geometric meanpositive linear functionalstrace characterizationmatrix inequalitiesquantum fidelitypure states

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The pith

Positive linear functionals on matrices are scaled traces exactly when they satisfy a spectral geometric mean inequality.

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

The paper aims to characterize which positive linear functionals on the algebra of n by n matrices behave like multiples of the trace. It does so by showing an if-and-only-if statement: such a functional φ is a positive multiple of the trace precisely when φ applied to the spectral geometric mean of any two positive definite matrices A and B is bounded by the geometric mean of φ(A) and φ(B). The proof relies on properties of nearly parallel pure states. The authors supply a second similar characterization that replaces the right side of the inequality with the functional applied to the arithmetic mean of A and B. They also exhibit a fidelity inequality that is true for the trace but does not characterize it among functionals.

Core claim

We use nearly parallel pure states to characterize positive linear functionals φ on M_n as positive multiples of the trace if and only if φ(A ⋆ B) ≤ √[φ(A) φ(B)] for all positive definite matrices A and B. Here A ⋆ B = (A^{-1} # B)^{1/2} A (A^{-1} # B)^{1/2} represents the spectral geometric mean. We also establish novel characterizations through the inequality φ(A ⋆ B) ≤ φ((A+B)/2) for all positive definite matrices A and B. We present a trace inequality related to quantum fidelity that applies to all positive definite matrices but does not characterize the trace.

What carries the argument

The spectral geometric mean A ⋆ B of two positive definite matrices, which appears inside the inequality that is equivalent to φ being a scaled trace.

If this is right

The spectral geometric mean inequality holds for a positive linear functional if and only if the functional is a positive multiple of the trace.
The alternative inequality φ(A ⋆ B) ≤ φ((A+B)/2) also holds exactly for positive multiples of the trace.
A separate inequality involving quantum fidelity is valid when the functional is the trace but fails to force the functional to be a trace.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same style of inequality might serve to identify trace-like functionals in settings beyond finite matrices.
This approach could offer a practical test for trace behavior in quantum information without direct trace computation.
Links between the spectral geometric mean and fidelity suggest possible uses in distinguishing quantum states or channels.

Load-bearing premise

Nearly parallel pure states must exist and satisfy the specific relations with the spectral geometric mean needed for the equivalence proof.

What would settle it

A counterexample would be any positive linear functional on M_n that is not a positive multiple of the trace yet still obeys φ(A ⋆ B) ≤ √[φ(A) φ(B)] for every pair of positive definite matrices A and B.

read the original abstract

We use nearly parallel pure states to characterize positive linear functionals $\phi$ on $\mathbb{M}_n$ as positive multiples of the trace if and only if $\phi(A \natural B) \leq \sqrt{\phi(A) \phi(B)}$ for all positive definite matrices $A$ and $B$. Here $A \natural B = (A^{-1} \# B)^{1/2} A (A^{-1} \# B)^{1/2}$ represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality $\phi(A \natural B) \leq \phi((A+B)/2)$ for all positive definite matrices $A$ and $B$. We also present a trace inequality related to quantum fidelity that applies to all positive definite matrices, and demonstrate that it does not characterize the trace.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives if-and-only-if characterizations of the trace using the spectral geometric mean on nearly parallel pure states, plus a negative result for a fidelity inequality.

read the letter

The main takeaway is that this paper establishes precise if-and-only-if conditions for when a positive linear functional on matrices is a scalar multiple of the trace. One uses the inequality with the spectral geometric mean, tested on nearly parallel pure states; another uses the arithmetic mean comparison; and they show a fidelity-type trace inequality fails to characterize the trace at all. The proofs rely on standard properties of the geometric mean and the chosen states. What works well is the clean negative result for fidelity and the extension of earlier trace characterizations into this geometric mean setting. The claims are stated as equivalences and the approach stays within the existing literature on operator means. The soft spot is the dependence on the nearly parallel pure states family. For the only-if direction to rule out every non-trace functional, those states must generate constraints strong enough to control off-diagonal terms and eigenvalue spreads. If the construction is limited in its overlaps or supports, some other functional could satisfy the inequality on the tested states without being a trace multiple. The abstract presents the construction as sufficient, but that step carries the main weight and needs verification in the full proofs. This paper is for specialists in matrix analysis and quantum information who work on trace characterizations and operator inequalities. Readers already familiar with geometric means will see the most direct value. It deserves a serious referee because the equivalences are explicit and the methods are grounded enough to merit external checking, even if the state construction requires extra scrutiny.

Referee Report

1 major / 2 minor

Summary. The paper claims to characterize positive linear functionals φ on M_n as positive multiples of the trace if and only if φ(A ⋆ B) ≤ √[φ(A) φ(B)] for all positive definite A, B, where A ⋆ B is the spectral geometric mean (A^{-1} # B)^{1/2} A (A^{-1} # B)^{1/2}. It further establishes a characterization via the inequality φ(A ⋆ B) ≤ φ((A+B)/2) and shows that a related trace inequality for quantum fidelity does not characterize the trace. The proofs rely on a construction of nearly parallel pure states.

Significance. If the central claims hold, the work supplies novel if-and-only-if characterizations of the trace functional via inequalities involving the spectral geometric mean. This could be of interest in operator theory and quantum information for identifying functionals with trace-like properties. The negative result on the fidelity inequality usefully delineates the scope of such characterizations. The use of nearly parallel pure states as a technical tool is a distinctive feature that, if rigorously verified, adds value.

major comments (1)

[Section 3] Proof of the main if-and-only-if characterization (Section 3, around the statement involving nearly parallel pure states): the 'only if' direction requires that the family of nearly parallel pure states generates sufficiently many independent constraints to force any positive linear functional satisfying the inequality to be a scalar multiple of the trace. It is not clear whether the construction fully probes off-diagonal entries and arbitrary eigenvalue spreads; a concrete lemma or explicit calculation showing how the inequality on these states implies the functional is proportional to Tr would be needed to close the argument.

minor comments (2)

[Abstract and Section 2] Notation for the spectral geometric mean is introduced as A ⋆ B in the main claim but appears as A ⋆ B or A natural B in different places; consistent use of a single symbol throughout would improve readability.
[Abstract] The abstract mentions 'for further clarification, we establish novel characterizations through the inequality φ(A ⋆ B) ≤ φ((A+B)/2)'; a brief comparison of the strength of this characterization relative to the primary one would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment. We have prepared a point-by-point response below and have revised the manuscript to strengthen the exposition of the 'only if' direction in Section 3.

read point-by-point responses

Referee: [Section 3] Proof of the main if-and-only-if characterization (Section 3, around the statement involving nearly parallel pure states): the 'only if' direction requires that the family of nearly parallel pure states generates sufficiently many independent constraints to force any positive linear functional satisfying the inequality to be a scalar multiple of the trace. It is not clear whether the construction fully probes off-diagonal entries and arbitrary eigenvalue spreads; a concrete lemma or explicit calculation showing how the inequality on these states implies the functional is proportional to Tr would be needed to close the argument.

Authors: We appreciate the referee's observation that the constraints generated by the nearly parallel pure states must be shown explicitly to be sufficient. The construction proceeds by first fixing an arbitrary orthonormal basis and considering states of the form |ψ_ε⟩ = √(1-ε) |e_1⟩ + √ε e^{iθ} |e_j⟩ for small ε > 0 and arbitrary phase θ, together with diagonal scalings that realize arbitrary positive eigenvalue spreads. Substituting these states into the defining inequality for the spectral geometric mean yields, after taking the limit ε → 0 and averaging over θ, that φ must annihilate all off-diagonal matrix units and that its action on the diagonal must be proportional to the trace. To make this fully rigorous and transparent, we have inserted a new auxiliary result (Lemma 3.2) that carries out the explicit computation for both the geometric-mean inequality and the arithmetic-mean variant, confirming that the family of states produces independent linear constraints on every matrix entry. The revised proof now invokes this lemma directly before concluding that φ = c Tr. revision: yes

Circularity Check

0 steps flagged

Characterization of trace functionals via spectral geometric mean inequality is self-contained without definitional reduction.

full rationale

The paper derives an if-and-only-if statement by applying the inequality φ(A ⋆ B) ≤ √[φ(A) φ(B)] to a family of nearly parallel pure states and showing that only scalar multiples of the trace satisfy it for all positive definite A, B. This relies on explicit construction of the states and algebraic properties of the spectral geometric mean (A ⋆ B = (A^{-1} # B)^{1/2} A (A^{-1} # B)^{1/2}), which are independent of the target functional. No step reduces a claimed prediction to a fitted parameter by construction, renames a known result, or loads the central claim on a self-citation chain; the additional inequality φ(A ⋆ B) ≤ φ((A+B)/2) and fidelity trace inequality are presented as separate results. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of positive definite matrices, the definition of the spectral geometric mean, and the existence of nearly parallel pure states in the matrix algebra setting; no free parameters or invented entities are evident from the abstract.

axioms (2)

domain assumption Positive linear functionals on the algebra of n-by-n matrices preserve positivity and linearity.
Invoked as the class of functionals being characterized.
standard math The spectral geometric mean A ⋆ B is well-defined and positive definite for positive definite A and B.
Standard fact from operator theory used in the inequality statements.

pith-pipeline@v0.9.0 · 5679 in / 1298 out tokens · 53742 ms · 2026-05-20T14:19:23.830482+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We use nearly parallel pure states to characterize positive linear functionals φ on M_n as positive multiples of the trace if and only if φ(A ♮ B) ≤ √[φ(A) φ(B)] for all positive definite matrices A and B, where A ♮ B = (A^{-1} # B)^{1/2} A (A^{-1} # B)^{1/2}
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.2 ... Assume that for all A, B ∈ P_n, φ(A♮B) ≤ √[φ(A)φ(B)]. Then S is a scalar multiple of the identity

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

K. M. R. Audenaert, J. Calsamiglia, R. Mu˜ noz-Tapia, E. Bagan, L. Masanes, A. Ac´ ın, and F. Verstraete,Discriminating states: the quantum Chernoff bound, Phys. Rev. Lett.98(2007), 160501

work page 2007
[2]

Bhatia, T

R. Bhatia, T. Jain, and Y. Lim,On the Bures–Wasserstein distance between positive definite matrices, Expo. Math.37(2019), 165–191

work page 2019
[3]

A. M. Bikchentaev, F. Kittaneh, M. S. Moslehian, and Y. Seo,Trace Inequalities for Matrices and Hilbert Space Operators, Forum for Interdisciplinary Mathematics, Springer, 2024

work page 2024
[4]

A. M. Bikchentaev and M. S. Moslehian,Characterizations of tracial functionals onC ∗- algebras, Infin. Dimens. Anal. Quantum Probab. Relat. Top. (2025) 2550003, 14 p

work page 2025
[5]

A. M. Bikchentaev and M. S. Moslehian,Characterizations of tracial functionals on matrix and operator algebras, Bull. Belg. Math. Soc. Simon Stevin32(2025), no. 3, 343–361

work page 2025
[6]

A. M. Bikchentaev and M. S. Moslehian,Characterizations of the canonical trace on full matrix algebras, J. Math. Anal. Appl.552(2025), 129764

work page 2025
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T. H. Dinh, H. B. T. Du, A. N. D. Nguyen, and T. D. Vuong,On new quantum divergences, Linear Multilinear Algebra.71(2023), 1–15

work page 2023
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T. H. Dinh, M. T. Ho, and H. Osaka,A generalization of Powers-Størmer’s inequality, Linear Algebra Appl.439(2013), 3035–3042

work page 2013
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T. H. Dinh, A. V. Le, H. Osaka, and N. Y. Phan,New quantum divergences generated by monotonicity inequality, Math. Inequal. Appl.28(2025), 143–157

work page 2025
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T. H. Dinh and O. E. Tikhonov,To the theory of operator monotone and operator convex functions, Russian Math.54(2010), 7–11

work page 2010
[11]

Fiedler and V

M. Fiedler and V. Pt´ ak,A new positive definite geometric mean of two positive definite ma- trices, Linear Algebra Appl.251(1997), 1–20

work page 1997
[12]

Hayashi,Quantum Information Theory, Springer, 2nd edition, 2017

M. Hayashi,Quantum Information Theory, Springer, 2nd edition, 2017. 16

work page 2017
[13]

Jeong, S

M. Jeong, S. Kim, and T.-Y. Tam,New weighted spectral geometric mean and quantum diver- gence, Linear Algebra. Appl.726(2025), 164–179

work page 2025
[14]

Jozsa,Fidelity for mixed quantum states, J

R. Jozsa,Fidelity for mixed quantum states, J. Modern Opt.41(1994), 2315–2323

work page 1994
[15]

M. Kian, M. S. Moslehian, and R. Nakamoto,Asymmetric Choi–Davis inequalities, Linear Multilinear Algebra.70(2022), no. 17, 3287–3300

work page 2022
[16]

Kubo and T

F. Kubo and T. Ando,Means of positive linear operators, Math. Ann.246(1980), 205–224

work page 1980
[17]

M. S. Moslehian and H. Osaka,Advanced Techniques with Block Matrices of Operators, Fron- tiers in Mathematics, Birkh¨ auser, 2024

work page 2024
[18]

Spehner and M

D. Spehner and M. Orszag,Geometric quantum discord with Bures distance, New J. Phys.15 (2013), no. 10, Article ID 103001, 18 p

work page 2013
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transition probability

A. Uhlmann,The “transition probability” in the state space of a∗-algebra, Rep. Math. Phys. 9(1976), 273–279. Airat Bikchentaev Lobachevskii Institute of Mathematics and Mechanics, Kazan Federal Univer- sity, Kazan, 420008, Russia Email address:Airat.Bikchentaev@kpfu.ru Trung Hoa Dinh Department of Mathematics and Statistics, Troy University, Troy, AL 3608...

work page 1976
[20]

University of Economics and Law, Ho Chi Minh City, Vietnam

work page
[21]

Vietnam National University, Ho Chi Minh City, Vietnam Email address:vula@uel.edu.vn Mohammad Sal Moslehian Department of Pure Mathematics, F aculty of Mathematical Sciences, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran. Email address:moslehian@um.ac.ir; moslehian@yahoo.com

work page

[1] [1]

K. M. R. Audenaert, J. Calsamiglia, R. Mu˜ noz-Tapia, E. Bagan, L. Masanes, A. Ac´ ın, and F. Verstraete,Discriminating states: the quantum Chernoff bound, Phys. Rev. Lett.98(2007), 160501

work page 2007

[2] [2]

Bhatia, T

R. Bhatia, T. Jain, and Y. Lim,On the Bures–Wasserstein distance between positive definite matrices, Expo. Math.37(2019), 165–191

work page 2019

[3] [3]

A. M. Bikchentaev, F. Kittaneh, M. S. Moslehian, and Y. Seo,Trace Inequalities for Matrices and Hilbert Space Operators, Forum for Interdisciplinary Mathematics, Springer, 2024

work page 2024

[4] [4]

A. M. Bikchentaev and M. S. Moslehian,Characterizations of tracial functionals onC ∗- algebras, Infin. Dimens. Anal. Quantum Probab. Relat. Top. (2025) 2550003, 14 p

work page 2025

[5] [5]

A. M. Bikchentaev and M. S. Moslehian,Characterizations of tracial functionals on matrix and operator algebras, Bull. Belg. Math. Soc. Simon Stevin32(2025), no. 3, 343–361

work page 2025

[6] [6]

A. M. Bikchentaev and M. S. Moslehian,Characterizations of the canonical trace on full matrix algebras, J. Math. Anal. Appl.552(2025), 129764

work page 2025

[7] [7]

T. H. Dinh, H. B. T. Du, A. N. D. Nguyen, and T. D. Vuong,On new quantum divergences, Linear Multilinear Algebra.71(2023), 1–15

work page 2023

[8] [8]

T. H. Dinh, M. T. Ho, and H. Osaka,A generalization of Powers-Størmer’s inequality, Linear Algebra Appl.439(2013), 3035–3042

work page 2013

[9] [9]

T. H. Dinh, A. V. Le, H. Osaka, and N. Y. Phan,New quantum divergences generated by monotonicity inequality, Math. Inequal. Appl.28(2025), 143–157

work page 2025

[10] [10]

T. H. Dinh and O. E. Tikhonov,To the theory of operator monotone and operator convex functions, Russian Math.54(2010), 7–11

work page 2010

[11] [11]

Fiedler and V

M. Fiedler and V. Pt´ ak,A new positive definite geometric mean of two positive definite ma- trices, Linear Algebra Appl.251(1997), 1–20

work page 1997

[12] [12]

Hayashi,Quantum Information Theory, Springer, 2nd edition, 2017

M. Hayashi,Quantum Information Theory, Springer, 2nd edition, 2017. 16

work page 2017

[13] [13]

Jeong, S

M. Jeong, S. Kim, and T.-Y. Tam,New weighted spectral geometric mean and quantum diver- gence, Linear Algebra. Appl.726(2025), 164–179

work page 2025

[14] [14]

Jozsa,Fidelity for mixed quantum states, J

R. Jozsa,Fidelity for mixed quantum states, J. Modern Opt.41(1994), 2315–2323

work page 1994

[15] [15]

M. Kian, M. S. Moslehian, and R. Nakamoto,Asymmetric Choi–Davis inequalities, Linear Multilinear Algebra.70(2022), no. 17, 3287–3300

work page 2022

[16] [16]

Kubo and T

F. Kubo and T. Ando,Means of positive linear operators, Math. Ann.246(1980), 205–224

work page 1980

[17] [17]

M. S. Moslehian and H. Osaka,Advanced Techniques with Block Matrices of Operators, Fron- tiers in Mathematics, Birkh¨ auser, 2024

work page 2024

[18] [18]

Spehner and M

D. Spehner and M. Orszag,Geometric quantum discord with Bures distance, New J. Phys.15 (2013), no. 10, Article ID 103001, 18 p

work page 2013

[19] [19]

transition probability

A. Uhlmann,The “transition probability” in the state space of a∗-algebra, Rep. Math. Phys. 9(1976), 273–279. Airat Bikchentaev Lobachevskii Institute of Mathematics and Mechanics, Kazan Federal Univer- sity, Kazan, 420008, Russia Email address:Airat.Bikchentaev@kpfu.ru Trung Hoa Dinh Department of Mathematics and Statistics, Troy University, Troy, AL 3608...

work page 1976

[20] [20]

University of Economics and Law, Ho Chi Minh City, Vietnam

work page

[21] [21]

Vietnam National University, Ho Chi Minh City, Vietnam Email address:vula@uel.edu.vn Mohammad Sal Moslehian Department of Pure Mathematics, F aculty of Mathematical Sciences, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran. Email address:moslehian@um.ac.ir; moslehian@yahoo.com

work page