(Co-)type and the linear stability of Wigner's symmetry theorem

Javier Cuesta

arxiv: 1907.05733 · v1 · pith:23WMBHFOnew · submitted 2019-07-12 · 🧮 math-ph · math.MP· quant-ph

(Co-)type and the linear stability of Wigner's symmetry theorem

Javier Cuesta This is my paper

Pith reviewed 2026-05-24 22:13 UTC · model grok-4.3

classification 🧮 math-ph math.MPquant-ph

keywords Wigner's symmetry theoremBanach spacestype and cotypelinear stabilitytransition probabilitiesalmost symmetriesfinite dimensional spaces

0 comments

The pith

Any map between finite-dimensional Banach spaces preserving transition probabilities up to additive error can be approximated by a linear map whose quality depends on the spaces' type and cotype constants.

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

The paper examines the linear stability of almost-symmetries and connects it to the geometry of the underlying Banach spaces. It establishes that transformations preserving transition probabilities up to an additive error admit approximation by linear maps. The quality of that approximation is controlled by the type and cotype constants of the spaces. A sympathetic reader would care because this supplies a quantitative bridge between abstract Banach-space geometry and the robustness of symmetry statements that originate in quantum mechanics but are here considered in more general finite-dimensional settings.

Core claim

Any transformation between finite dimensional Banach spaces that preserves transition probabilities up to an additive error admits an approximation by a linear map, and the quality of the approximation depends on the type and cotype constants of the involved spaces.

What carries the argument

The type and cotype constants of the Banach spaces, which quantify their geometric averaging behavior and thereby bound the distance from an almost probability-preserving map to the nearest linear map.

If this is right

The approximation error between the given map and the nearest linear map is bounded in terms of the type and cotype constants.
The result applies uniformly to all finite-dimensional Banach spaces, not only to Hilbert space.
Spaces with smaller type or cotype constants yield quantitatively stronger linear stability for almost-symmetries.
The statement furnishes a stability version of Wigner's theorem outside the Hilbert-space setting.

Where Pith is reading between the lines

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

Numerical checks could be performed by constructing explicit almost-symmetry maps on low-dimensional l_p spaces and measuring their distance to linear maps against the known type-cotype values.
The same geometric control might extend to other preservation properties, such as approximate isometries or distance preservers, in finite-dimensional normed spaces.
In modeling physical systems, representations on spaces with favorable type and cotype could be preferred when symmetry stability under small perturbations is required.

Load-bearing premise

The transformations are defined between finite-dimensional Banach spaces.

What would settle it

An explicit map between two finite-dimensional Banach spaces that preserves transition probabilities up to a small additive error yet lies farther from every linear map than the type-cotype bound predicts.

read the original abstract

We study the relation between the linear stability of almost-symmetries and the geometry of the Banach spaces on which these transformations are defined. We show that any transformation between finite dimensional Banach spaces that preserves transition probabilities up to an additive error admits an approximation by a linear map, and the quality of the approximation depends on the type and cotype constants of the involved spaces.

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 a quantitative link between type/cotype constants and how well almost transition-probability maps on finite-dimensional Banach spaces can be linearly approximated.

read the letter

The main takeaway is that the paper establishes a quantitative connection between the type and cotype constants of finite-dimensional Banach spaces and how well almost-symmetry maps can be approximated by linear ones. The new part is the explicit dependence of the approximation quality on those geometric invariants. The classical Wigner theorem says that maps preserving transition probabilities are linear (up to phase), but here they handle the approximate case and tie the error to type/cotype. This is done well in that it gives a clear statement for finite dimensions and avoids overclaiming. The abstract is straightforward about the setting. A potential soft spot is the restriction to finite dimensions. That makes the result solid but limits its direct use in typical quantum settings where spaces are infinite-dimensional. If the proof uses compactness or dimension-dependent arguments, that would explain the choice, but it means the result is more of a starting point than a full theory. The citation pattern isn't visible from the abstract, but assuming it builds on known results in both areas, that seems fine. This paper is for readers interested in the intersection of functional analysis and quantum information or foundations. Someone working on approximate symmetries or stability questions might find the quantitative bounds useful. It deserves a serious referee because the claim is precise and the approach seems grounded, even if revisions might be needed for clarity or extensions.

Referee Report

0 major / 1 minor

Summary. The manuscript studies the linear stability of almost-symmetries in the sense of Wigner's theorem. It claims that any (not necessarily linear or bijective) map between finite-dimensional Banach spaces that preserves transition probabilities up to an additive error admits approximation by a linear map, with the approximation quality controlled by the type and cotype constants of the domain and codomain spaces.

Significance. If the result holds, it supplies a quantitative, geometry-dependent stability version of Wigner's theorem that is restricted to finite dimensions and makes the error bound explicit in terms of type/cotype constants. This links a classical result in quantum mechanics to the geometry of Banach spaces and provides a falsifiable, parameter-free relation between preservation error and approximation quality.

minor comments (1)

[Abstract] The abstract states the setting is finite-dimensional Banach spaces but does not specify the norm in which the linear approximation is measured or whether the map is assumed continuous.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report. The provided summary correctly captures the scope of our work on quantitative stability for almost-symmetries of Wigner's theorem in finite-dimensional Banach spaces, with error controlled by type and cotype constants. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in Banach space geometry

full rationale

The paper's central result is a quantitative stability theorem for almost-symmetries of transition probabilities on finite-dimensional Banach spaces, with the approximation quality controlled by the spaces' type and cotype constants. These constants are standard, externally defined invariants of Banach spaces (independent of the present work) and the statement does not reduce any prediction or uniqueness claim to a fitted parameter, self-citation chain, or definitional tautology. The abstract and setting declare the finite-dimensional restriction explicitly as the domain of the result rather than smuggling it in; no load-bearing step is shown to collapse to its own inputs by the paper's equations or citations. This is the normal case of a self-contained functional-analytic argument.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities used in the proof.

pith-pipeline@v0.9.0 · 5573 in / 1015 out tokens · 27582 ms · 2026-05-24T22:13:51.826125+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

any transformation between finite dimensional Banach spaces that preserves transition probabilities up to an additive error admits an approximation by a linear map, and the quality of the approximation depends on the type and cotype constants
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Theorem 2 ... inf_H sup ||F(x)-H(x)|| / ||x|| ≤ 2δ min{T2(Z)C2(X), ...}

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

22 extracted references · 22 canonical work pages

[1]

( 5) are the same

and Eq. ( 5) are the same. Let α = ∥z∥ be the inﬁmum of Eq. ( 5). Then there exist m ∈ N, positive real numbers (λj)m j=1, ∑m j=1 λj = 1 and ( zj)m j=1 with quasi-norm one such z = α ∑m j=1 λjzj. This is a valid decomposition of z and ∑m j=1 | | |αλj zj| | | ≤α. On the other hand, let z = ∑m j=1 zj be the decomposition that achieves the inﬁmum in Eq. (

work page
[2]

Then z ∑m k=1 | | |zk| | |= ∑ j=1 ( | | |zj| | | ∑m k=1 | | |zk| | | ) zj | | |zj| | |∈ conv(BZ )

so that ∥z∥ = ∑m j=1 | | |zj| | |. Then z ∑m k=1 | | |zk| | |= ∑ j=1 ( | | |zj| | | ∑m k=1 | | |zk| | | ) zj | | |zj| | |∈ conv(BZ ). The norm of ξ ∈ Z ∗ can be computed as ∥ξ∥ = sup z∈conv(BZ ) |ξ(z)|= sup z∈BZ |ξ(z)|= sup{|ξ(z)|: | | |z| | | ≤1}, as the supremum over a convex function is achieved at the extrema l points. Thus the dual of the quasi-Banac...

work page
[3]

It follows from induction that for n ≥ 3, T2,n(Z) ≤ 2(1 + √

work page
[4]

Hence, from Lemma 6 with n = 4d2 T2(Z) ≤ 4(1 + √

log2 n. Hence, from Lemma 6 with n = 4d2 T2(Z) ≤ 4(1 + √

work page
[5]

Accordingly, from C2(Sd 2 ) = 1 and Theorem 2 there exist a linear map H : Sd 2 → Sd 2 such that for all x ∈ BSd 2 ∥F (x) − H(x)∥2 ≤ 32(1 + √

log2 2d. Accordingly, from C2(Sd 2 ) = 1 and Theorem 2 there exist a linear map H : Sd 2 → Sd 2 such that for all x ∈ BSd 2 ∥F (x) − H(x)∥2 ≤ 32(1 + √

work page
[6]

Analysis in Quantum Information Theory

log2(2d)√ ε. (16) Finally, from Eq. ( 13) and the triangle inequality we obtain sup x∈BSd 2 ∥f (x) − H(x)∥2 ≤ sup x∈BSd 2 ∥f (x) − F (x)∥2 + ∥F (x) − H(x)∥2 ≤ 79√ ε (1 + log2 d). V. OUTLOOK Using Theorem 2 we are able to improve –up to some logarithmic factors– the upper bound on the dimension dependence of the linear stability of Wigner’s t heorem from d...

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N. J. Kalton, “A remark on quasi-isometries,” Proc. A.M. S. Vol 131, 1225 (2002)

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Jung, Hyers-Ulam-Rassias Stability of Functional Equations in N onlinear Analysis, Vol

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in N onlinear Analysis, Vol. 48 of Springer Optimization and Its Applications. Springer (2011)

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Castillo and M

J. Castillo and M. Gonz´ alez, Three-space problems in Banach space theory, Springer (1997)

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J. Cuesta, M. M. Wolf, “Are almost-symmetries almost lin ear?”, math-ph, arxiv/1812.10019 (2018)

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[1] [1]

( 5) are the same

and Eq. ( 5) are the same. Let α = ∥z∥ be the inﬁmum of Eq. ( 5). Then there exist m ∈ N, positive real numbers (λj)m j=1, ∑m j=1 λj = 1 and ( zj)m j=1 with quasi-norm one such z = α ∑m j=1 λjzj. This is a valid decomposition of z and ∑m j=1 | | |αλj zj| | | ≤α. On the other hand, let z = ∑m j=1 zj be the decomposition that achieves the inﬁmum in Eq. (

work page

[2] [2]

Then z ∑m k=1 | | |zk| | |= ∑ j=1 ( | | |zj| | | ∑m k=1 | | |zk| | | ) zj | | |zj| | |∈ conv(BZ )

so that ∥z∥ = ∑m j=1 | | |zj| | |. Then z ∑m k=1 | | |zk| | |= ∑ j=1 ( | | |zj| | | ∑m k=1 | | |zk| | | ) zj | | |zj| | |∈ conv(BZ ). The norm of ξ ∈ Z ∗ can be computed as ∥ξ∥ = sup z∈conv(BZ ) |ξ(z)|= sup z∈BZ |ξ(z)|= sup{|ξ(z)|: | | |z| | | ≤1}, as the supremum over a convex function is achieved at the extrema l points. Thus the dual of the quasi-Banac...

work page

[3] [3]

It follows from induction that for n ≥ 3, T2,n(Z) ≤ 2(1 + √

work page

[4] [4]

Hence, from Lemma 6 with n = 4d2 T2(Z) ≤ 4(1 + √

log2 n. Hence, from Lemma 6 with n = 4d2 T2(Z) ≤ 4(1 + √

work page

[5] [5]

Accordingly, from C2(Sd 2 ) = 1 and Theorem 2 there exist a linear map H : Sd 2 → Sd 2 such that for all x ∈ BSd 2 ∥F (x) − H(x)∥2 ≤ 32(1 + √

log2 2d. Accordingly, from C2(Sd 2 ) = 1 and Theorem 2 there exist a linear map H : Sd 2 → Sd 2 such that for all x ∈ BSd 2 ∥F (x) − H(x)∥2 ≤ 32(1 + √

work page

[6] [6]

Analysis in Quantum Information Theory

log2(2d)√ ε. (16) Finally, from Eq. ( 13) and the triangle inequality we obtain sup x∈BSd 2 ∥f (x) − H(x)∥2 ≤ sup x∈BSd 2 ∥f (x) − F (x)∥2 + ∥F (x) − H(x)∥2 ≤ 79√ ε (1 + log2 d). V. OUTLOOK Using Theorem 2 we are able to improve –up to some logarithmic factors– the upper bound on the dimension dependence of the linear stability of Wigner’s t heorem from d...

work page

[7] [7]

A remark on quasi-isometries,

N. J. Kalton, “A remark on quasi-isometries,” Proc. A.M. S. Vol 131, 1225 (2002)

work page 2002

[8] [8]

Polynomial approximation o n convex subsets of Rn,

Y. Brudnyi and N. J. Kalton, “Polynomial approximation o n convex subsets of Rn,” Construc- tive Approximation 16, 161-200 (2000)

work page 2000

[9] [9]

Twisted sums of sequence spac es and the three space problem,

N. J. Kalton and N. T. Peck, “Twisted sums of sequence spac es and the three space problem,” Trans. Amer. Math. Soc. 255, 1-30 (1979)

work page 1979

[10] [10]

Jung, Hyers-Ulam-Rassias Stability of Functional Equations in N onlinear Analysis, Vol

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in N onlinear Analysis, Vol. 48 of Springer Optimization and Its Applications. Springer (2011)

work page 2011

[11] [11]

Maps on density opera tors preserving quantum f - diver- gences,

L. Moln´ ar, G. Nagy, and P. Szokol. “Maps on density opera tors preserving quantum f - diver- gences,” Quantum Information Processing, 12(7), 2013

work page 2013

[12] [12]

Castillo and M

J. Castillo and M. Gonz´ alez, Three-space problems in Banach space theory, Springer (1997)

work page 1997

[13] [13]

Un th´ eor` eme de prolongement,

B. Maurey, “Un th´ eor` eme de prolongement,” C. R. Acad. S c. Paris, Vol 279, 329-332 (1974)

work page 1974

[14] [14]

Are almost-symmetries almost lin ear?

J. Cuesta, M. M. Wolf, “Are almost-symmetries almost lin ear?”, math-ph, arxiv/1812.10019 (2018)

work page arXiv 2018

[15] [15]

Duality and twisted su ms of Banach spaces,

F. C. S´ anchez, J. M. F. Castillo, “Duality and twisted su ms of Banach spaces,” J. of Funct. Anal. 175, 1-16 (2000)

work page 2000

[16] [16]

The dimen sion of almost spherical sections of convex bodies,

T. Figiel, J. Lindenstrauss, and V. D. Milman “The dimen sion of almost spherical sections of convex bodies,” Acta. Math. 139, 53-94 (1977)

work page 1977

[17] [17]

On the three sp ace problem,

P. Enﬂo, J. Lindenstrauss, and G. Pisier “On the three sp ace problem,” Math. Scand. 36, 199-210 (1975)

work page 1975

[18] [18]

Tomczak-Jaegermann, Banach-Mazur distances and ﬁnite-dimensional operator id eals, Longman Scientiﬁc & Technical (1989)

N. Tomczak-Jaegermann, Banach-Mazur distances and ﬁnite-dimensional operator id eals, Longman Scientiﬁc & Technical (1989)

work page 1989

[19] [19]

Benyamini, J

Y. Benyamini, J. Lindenstrauss, Geometric nonlinear functional analysis, Volume 1, Amer. Math. Soc., Providence, Rhode Island (2000)

work page 2000

[20] [20]

Albiac and N

F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Springer (2000)

work page 2000

[21] [21]

N. J. Kalton, N. T. Peck and J. Roberts, An F-space sampler, London Math. Soc. Lecture Notes, Vol. 89, Cambridge University Press (1984)

work page 1984

[22] [22]

Aubrun and S

G. Aubrun and S. J. Szarek, Alice and Bob Meet Banach: The Interface of Asymptotic Geo- metric Analysis and Quantum Information Theory , Amer. Math. Soc. (2017)

work page 2017