Pith. sign in

REVIEW 3 major objections 6 minor 1 cited by

Quantum Wasserstein distances for quantum permutation groups

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper establishes three quantum Wasserstein distances on the tracial states of the quantum permutation group, all of which extend the classical Hamming L1-Wasserstein distance on permutations.

desk verdict First quantum Hamming-type Wasserstein distances on C(S_n^+), with solid proofs for two of the three metrics but a load-bearing unproven lemma on extreme traces of tensor products. read the letter →

arxiv 2505.19269 v2 pith:PSEVBENI submitted 2025-05-25 math.OA

Anshu , David Jekel , Therese Basa Landry This is my paper
classification math.OA MSC 46L6749Q2246L8981R60
keywords quantumpermutationgroupmetricspaceoptimaltransportHammingdistancetracialstateWassersteinC*-algebraamalgamatedfreeproduct
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 asks whether the classical Hamming distance between permutations can be quantized to a metric on the quantum permutation group $S_n^+$, the non-commutative C*-algebraic analogue of the permutation group. It defines three distances—$W_{H,\wedge}$, $W_{H,1}$, and $W_{H,\otimes}$—on the tracial state space of $C(S_n^+)$, each built from a different notion of coupling two traces. It proves that $W_{H,\wedge}$ and $W_{H,1}$ are genuine metrics, that $W_{H,\otimes}$ satisfies the triangle inequality though its self-distance may be nonzero, and that all three are subadditive under convolution. For states coming from probability measures on the classical subgroup $S_n$, all three distances coincide with the ordinary $L^1$-Wasserstein distance for the normalized Hamming metric, and the $*$-algebra generated by the matrix entries lies in the domain of the associated Lipschitz seminorms. If correct, $C(S_n^+)$ carries genuine quantum metric structures that extend Hamming geometry to the quantum setting.

What carries the argument

The central object is the Hamming cost operator $C_H = 1 - \frac{1}{n}\sum_{i,j=1}^n u_{ij} \otimes u_{ij}$ in $C(S_n^+) \otimes C(S_n^+)$, the quantum analogue of the classical expression $1 - \operatorname{tr}_n(\sigma^T \sigma')$ for the normalized Hamming distance. The argument proceeds by showing $C_H$ is positive, symmetric, and satisfies the triangle inequality $\iota_{1,3}(C_H) \le \iota_{1,2}(C_H) + \iota_{2,3}(C_H)$. The three distances use three coupling notions: arbitrary tracial couplings (embeddings of the algebra into a tracial von Neumann algebra), for which the triangle inequality is proved by the amalgamated free product construction together with the identity $\tau(p\wedge q)+\tau(p\vee q)=\tau(p)+\tau(q)$ for projections; and tracial tensor-product couplings (traces on the tensor product with the given marginals), for which the triangle inequality is proved by amalgamating couplings using the Choquet-simplex property of trace spaces—every trace has a unique barycentric decomposition into extreme traces—together with the asserted factorization of extreme traces on tensor products.

What would settle it

Find a pair of C*-algebras $A$ and $B$ with an extreme tracial state on $A \otimes B$ that is not a tensor product of extreme traces; if such a state exists, Lemma 4.7 is false and the paper's proof of the triangle inequality for $W_{H,\otimes}$ collapses, and one can then test whether $W_{H,\otimes}$ still satisfies the triangle inequality for that specific pair of traces on $C(S_n^+)$.

Watch

Extended reading notes

Core claim

The central claim is that the tracial state space of $C(S_n^+)$ admits three natural analogues of the $L^1$-Wasserstein distance for the Hamming metric. $W_{H,\wedge}$ is defined by taking the infimum over tracial couplings of the cost $1 - \frac{1}{n}\sum_{i,j} \tau(\alpha_1(u_{ij}) \wedge \alpha_2(u_{ij}))$, using the intersection of the two projections in a tracial von Neumann algebra. $W_{H,1}$ uses the non-commutative $L^1$ distance between the two projections $\alpha_1(u_{ij})$ and $\alpha_2(u_{ij})$ in a tracial coupling. $W_{H,\otimes}$ is the Wasserstein distance for the cost operator $C_H = 1 - \frac{1}{n}\sum_{i,j} u_{ij} \otimes u_{ij}$ over tracial tensor-product couplings. The paper proves $W_{H,\wedge}$ and $W_{H,1}$ are metrics, $W_{H,\otimes}$ is a pseudometric satisfying the triangle inequality, all three are subadditive under convolution, and all three agree with the classical $L^1$-Wasserstein distance on states induced by probability measures on $S_n$. It also proves the density of Lipschitz elements for $W_{H,\wedge}$ and $W_{H,1}$.

Load-bearing premise

The triangle inequality for $W_{H,\otimes}$ relies on the unproved Lemma 4.7 asserting that every extreme tracial state on a minimal tensor product $A \otimes B$ is a tensor product of extreme traces, and the triangle inequality for $W_{H,1}$ is delegated to analogy with the $W_{H,\wedge}$ case, so a failure of either would leave the corresponding metric property unsupported.

Editorial extensions

If this is right

  • The tracial state space of $C(S_n^+)$ becomes a metric space for $W_{H,\wedge}$ and $W_{H,1}$, and a pseudometric space for $W_{H,\otimes}$, with convergence in any of the three implying weak-$*$ convergence.
  • Convolution of traces is non-expansive in each argument for all three distances, giving quantitative bounds on iterated convolutions and their approach to the Haar state.
  • On classical states (those factoring through the quotient map to $C(S_n)$), all three distances equal the ordinary $L^1$-Wasserstein distance for the normalized Hamming metric, so the quantum distances are faithful extensions of the classical geometry.
  • The $*$-algebra generated by the $u_{ij}$ is contained in the Lipschitz domain for $W_{H,\wedge}$ and $W_{H,1}$, so Lipschitz elements are dense in $C(S_n^+)$, a baseline requirement for a quantum compact metric space.
  • The self-distance of $W_{H,\otimes}$ vanishes exactly for classical states, so the classical states are precisely those with zero self-distance; for non-classical traces the self-distance is positive.

Reading between the lines

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

  • A natural next step is to determine whether the triangle inequality for $W_{H,\otimes}$ can be proved without the unstated Lemma 4.7; if the lemma is false, the metric property may still hold for $C(S_n^+)$ by a direct argument, but the paper's proof would not cover it.
  • The reduction of $W_{H,\otimes}$ to a classical optimal transport problem on the Birkhoff polytope (the polytope of doubly stochastic matrices) suggests the tensor distance on traces is computable in terms of doubly stochastic matrices, potentially enabling numerical experiments that the paper does not undertake.
  • Because the classical recovery result (Proposition 5.3) uses only the quotient map and a norm comparison for projections, the same argument may extend to other quantum groups that carry a classical subgroup and a compatible cost operator, though the paper does not make this generalization.
  • If $W_{H,\wedge}$ and $W_{H,1}$ generate distinct topologies, as the paper suspects, then the choice of coupling notion matters for the resulting quantum metric space structure; testing this conjecture would require explicit sequences of traces with controlled behavior in one distance but not the other.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper defines three Wasserstein-type distances on the tracial state space of the quantum permutation group C(S_n^+), inspired by the normalized Hamming metric on the classical permutation group S_n. The first two, W_{H,∧} and W_{H,1}, are defined through tracial couplings in the sense of Biane--Voiculescu, using either the meet of projections or the noncommutative L1-norm; the third, W_{H,⊗}, is defined through tracial couplings on the minimal tensor product. The main theorems assert that W_{H,∧} and W_{H,1} are metrics, that W_{H,⊗} satisfies the triangle inequality, that all three are subadditive under convolution, and that when the states are induced by classical probability measures on S_n all three distances coincide with the classical L1-Wasserstein distance for the normalized Hamming metric. The paper also proves bounds relating the distances to total variation, characterizes the maximum value, and shows density of Lipschitz elements.

Significance. If the proofs are completed, this is a useful contribution to noncommutative optimal transport and quantum metric spaces: it gives the first Wasserstein-type metric structures on the trace space of C(S_n^+) that recover the classical Hamming geometry on S_n. The paper is constructive and parameter-free: the cost operator is explicit, the classical Wasserstein agreement is proved rather than assumed, and the connections to Biane--Voiculescu free Wasserstein distances, Jacelon's trace-space metrics, and recent quantum optimal transport literature are clearly drawn. The restriction to tracial states is honestly motivated. The main gaps are local and appear fillable, but as written they affect load-bearing steps of the advertised theorems.

major comments (3)
  1. [§4.1, Lemma 4.7] Lemma 4.7 asserts that every extreme tracial state on A⊗B is a tensor product τ1⊗τ2 of extreme tracial states and that ∂eT(A⊗B) ≅ ∂eT(A) × ∂eT(B). This is stated without proof or citation and is used essentially in Lemma 4.8, Lemma 4.10, Lemma 4.14, and hence in the triangle inequality for W_{H,⊗} (Theorem 1.5(1)). The statement is not a tautology: for general states on A⊗B the analogous factorization fails because of entangled states, and even for traces one needs a factoriality and splitting argument for the GNS representation. As written, the advertised triangle inequality for W_{H,⊗} does not follow from the text. Please supply a proof or a precise reference, and in Lemma 4.8 spell out the disintegration argument used to construct the three-fold measure λ from the two couplings.
  2. [§3.1, Lemmas 3.5 and 3.7] The proof of Lemma 3.5 defines f(φ) = inf_k f_k(φ), but Lemma 3.4 and the definition of W_{H,∧} require the functional sup_k f_k(φ) (equivalently lim_k f_k, since the f_k are increasing). With the infimum, f(φ) is essentially f_1(φ) and minimizing it does not compute W_{H,∧}; moreover the sentence that an infimum of lower semi-continuous functions is lower semi-continuous is false in general. This affects the proof that the infimum in (3.1) is attained (Theorem 1.4(3)) and the lower semi-continuity argument in Lemma 3.7, which refers back to 'f as in the proof of Lemma 3.5'. Please replace inf_k by sup_k and give the correct semicontinuity justification.
  3. [§3.4, Proposition 3.10(1)] The triangle inequality for W_{H,1} is delegated to 'as in the case of W_{H,∧}' after forming the amalgamated free product. Since W_{H,1} is one of the three central distances in Theorem 1.4, the estimate ∥γ1(uij) − γ3(uij)∥_{L1} ≤ ∥γ1(uij) − γ2(uij)∥_{L1} + ∥γ2(uij) − γ3(uij)∥_{L1} and the summation over i,j should be written out, or at least the exact lines of Proposition 3.6 to which the argument is analogous should be identified. This is a small gap, but it is load-bearing for the metric claim.
minor comments (6)
  1. [§3.4, Proposition 3.10(1)] The nondegeneracy proof contains an incorrect display: τ(α1(uij) − α2(uij)) is not equal to τ((α1(uij) − α2(uij))^*(α1(uij) − α2(uij))). The correct argument is that a minimizing coupling has ∥α1(uij) − α2(uij)∥_{L1} = τ(|α1(uij) − α2(uij)|) = 0, and faithfulness of τ forces α1(uij) = α2(uij).
  2. [§3.3, Proposition 3.8] In the proof, the coupling is written as (N ⊗ M, τ⊗σ, γ1, γ2) after earlier using (M ⊗ N, τ⊗σ, γ1, γ2); please use a single consistent ordering of the tensor factors.
  3. [§4.1, Lemma 4.8] The statement says 'there exists a unique σ ∈ T(A⊗A⊗A)' but the conclusion uses ρ instead of σ; please align the notation for the three-fold trace.
  4. [§4.2, Remark 4.12] The remark introduces ρ without defining it; the displayed formula should use the identity map or an explicit amplification, and the notation should be cleaned up.
  5. [§4.2, Lemma 4.13] The proof of the triangle inequality for the cost operator is clear, but the last line '1 − p1,3 ≤ 1 − p1,2p2,3 ≤ (1 − p1,2) + (1 − p2,3)' would be easier to follow if the intermediate commuting-projection identity were displayed before the final inequality.
  6. [General] There are several minor typographical issues, including 'F unding' in the acknowledgements and the use of 'infimum of lower semi-continuous functions' in Lemma 3.5; a careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the three distances are defined from explicit cost data, the classical L1-Wasserstein recovery is a proved theorem, and the only flagged issue is the unproved, non-circular Lemma 4.7.

full rationale

No circularity is present in the derivation chain. Each distance (WH,∧, WH,1, WH,⊗) is defined directly from the explicit cost operator CH and from explicit classes of tracial couplings; the metric axioms, convolution subadditivity, and the comparison with the classical L1-Wasserstein distance are then proved rather than assumed. In particular, Proposition 5.3 derives the equality with the classical W1 by constructing classical couplings from tracial couplings and vice versa, so the classical Hamming geometry is an output, not an input. The paper's uses of [BV01] and [BR24] are external support for standard techniques and are not self-referential; the one self-citation ([GJNS22]) appears only in motivational remarks and an open question and is not load-bearing for Theorems 1.4, 1.5, or Proposition 1.6. The load-bearing Lemma 4.7 (extreme traces on minimal tensor products factor as tensor products of extreme traces) is stated without proof or citation, and it is used essentially in Lemma 4.8 and hence in Theorem 1.5(1); this is a genuine omitted-support gap that should be filled, but it is not circularity because it is not an assumption that the paper is trying to prove, not a fitted parameter, and not a renamed version of the paper's own conclusion. Similarly, the triangle-inequality proof for WH,1 in Proposition 3.10 is delegated to analogy with WH,∧; that is an incomplete argument, not a circular one.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters, no fitted values, and no new physical entities are introduced. The construction lives inside standard operator-algebraic objects, namely the universal C*-algebra C(S_n^+), its trace space, tensor products, and amalgamated free products. The ledger documents the background facts the proofs rely on, several of which are stated without full proof.

assumptions (6)
  • standard math Amalgamated free product of tracial von Neumann algebras exists and is unique (Fact 2.6).
    Used in the triangle inequality for W_{H,∧} and W_{H,1} (Propositions 3.6 and 3.10); cited to Popa, VDN, and Brown-Ozawa.
  • standard math Every extreme tracial state on a minimal tensor product A⊗B is of the form τ1⊗τ2 with τ1, τ2 extreme (Lemma 4.7).
    Used to amalgamate tensor couplings in Lemma 4.8, giving the triangle inequality of W_{H,⊗}; stated without proof or citation.
  • standard math The trace space T(A) is a Choquet simplex, and each trace has a unique barycentric measure (Lemma 4.6).
    Used in Lemmas 4.8 and 4.10; cited to Blackadar-Rordam.
  • standard math Cuntz-Pedersen special Jordan decomposition of tracial functionals (CP79, Proposition 2.8).
    Used in Proposition 5.2 to bound W_{H,∧} by half the norm distance; cited.
  • domain assumption The quotient map θ: C(S_n^+) → C(S_n) is a quantum subgroup embedding (Theorem 2.14).
    Used in Proposition 5.3 and 5.4 to compare with classical permutation measures.
  • domain assumption Concentration on tracial states: all distances are defined on T(C(S_n^+)).
    The paper deliberately restricts to traces; extension to general states is left open.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Wasserstein distances for quantum permutation groups." pith.science (2026). https://pith.science/paper/PSEVBENI

@misc{pith2026250519269,
  author       = {Pith},
  title        = {Pith review of: Quantum Wasserstein distances for quantum permutation groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PSEVBENI}},
  note         = {Machine review of arXiv:2505.19269}
}
abstract

We seek an analog for the quantum permutation group $S_n^+$ of the normalized Hamming distance for permutations. We define three distances on the tracial state space of $C(S_n^+)$ that generalize the $L^1$-Wasserstein distance of probability measures on $S_n$ equipped with the normalized Hamming metric, for which we demonstrate basic metric properties, subadditivity under convolution, and density of the Lipschitz elements in the $\mathrm{C}^{\ast}$-algebra.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Transportation cost and contraction coefficient for channels on von Neumann algebras

    math.OA 2025-06 conditional novelty 8.0 of 10

    A new framework defines channel cost and contraction via Lipschitz seminorms, proving duality, tensor properties, and applications to word length, Carnot-Carathéodory distance, and mixing times.

Reference graph

Works this paper leans on

13 extracted references · 11 canonical work pages · cited by 1 Pith paper

  1. [1]

    The Hamming Distance and the Fell Topology on AF Algebras

    [AB] Konrad Aguilar and Zoe Batterman, The hamming distance and the fell topology on af algebras , preprint, arXiv:2310.10752. [ABOW25] Konrad Aguilar, Karina Behera, Tron Omland, and Nicole Wu, On the bures metric, c∗-norm and quantum metric, Bulletin of the Australian Mathematical Society (2025), 1–11. [ACJW23] Roy Araiza, Yidong Chen, Marius Junge, and...

    work page Pith review arXiv 2025

  2. [5]

    Carlen and Jan Maas, Gradient flow and entropy inequalities for quantum markov semigroups with detailed balance, Journal of Functional Analysis 273 (2017), no

    [CM17] Eric A. Carlen and Jan Maas, Gradient flow and entropy inequalities for quantum markov semigroups with detailed balance, Journal of Functional Analysis 273 (2017), no. 5, 1810–1869. [Con89] A. Connes, Compact metric spaces, fredholm modules, and hyperfiniteness , Ergodic Theory Dynam. Systems 9 (1989), 207–220. [Con94] , Noncommutative geometry, Ac...

    work page 2017

  3. [1971]

    [Avi82] Daniel Avitzour, Free produts of C∗-algebras, Transactions of the American Mathematical Society 271 (1982), 423–35. QUANTUM W ASSERSTEIN DISTANCES FOR QUANTUM PERMUTATION GROUPS 29 [BBC08] Teodor Banica, Julien Bichon, and Beno ˆ ıt Collins, Quantum permutation groups: a survey , Banach Center Pub- lications 78 (2008), no. 1, 13–34. [Bla85] Bruce ...

    work page 1982

  4. [1990]

    Quantum Hamming Metrics

    [Pop93] Sorin Popa, Markov traces on universal jones algebras and subfactors of finite index , Invent Math 111 (1993), 375–405. [PX03] Gilles Pisier and Quanhua Xu, Non-commutative Lp-spaces, Handbook of the geometry of Banach spaces (Williams B. Johnson and Joram Lindenstrauss, eds.), vol. 2, Elsevier, 2003, pp. 1459–1517. [Rie98] M. A. Rieffel, Metrics ...

    work page Pith review arXiv 1993

  5. [1992]

    Moore, S ¸erban-Valentin Stratila, and Dan-Virgil Voiculescu, eds.), Springer Berlin Heidelberg, Berlin, Heidelberg, 1985, pp

    [Voi85] Dan-Virgil Voiculescu, Symmetries of some reduced free product C∗-algebras, Operator Algebras and their Con- nections with Topology and Ergodic Theory (Huzihiro Araki, Calvin C. Moore, S ¸erban-Valentin Stratila, and Dan-Virgil Voiculescu, eds.), Springer Berlin Heidelberg, Berlin, Heidelberg, 1985, pp. 556–588. MR 0799593 [Wan98] S. Wang, Quantum...

    work page 1998

  6. [1994]

    Cuntz and G

    [CP79] J. Cuntz and G. K. Pedersen, Equivalence and traces on C∗-algebras, Journal of Functional Analysis 33 (1979), no. 2, 135–164. [CR17] Michael Christ and Marc A. Rieffel, Nilpotent group C∗-algebras as compact quantum metric spaces, Canad. Math. Bull 60 (2017), 77–94. [Dav96] Kenneth R. Davidson, C ∗-algebras by example , Fields Institute Monographs,...

    work page 1979

  7. [1996]

    10, 6627–6643

    [DPMTL21] Giacomo De Palma, Milad Marvian, Dario Trevisan, and Seth Lloyd, The quantum wasserstein distance of order 1, IEEE Transactions on Information Theory 67 (2021), no. 10, 6627–6643. [DT21] Giacomo De Palma and Dario Trevisan, Quantum optimal transport with quantum channels , Ann. Henri Poincar´ e 22 (2021), 3199–3234. [Duv22] Rocco Duvenhage, Quad...

    work page 2021

  8. [1997]

    Roberson, Nonlocal games and quantum permutation groups , Journal of Functional Analysis 279 (2020), no

    [LMR20] Martino Lupini, Laura Manˇ cinska, and David E. Roberson, Nonlocal games and quantum permutation groups , Journal of Functional Analysis 279 (2020), no. 5, 108592. [MR21] Magdalena Musat and Mikael Rørdam, Factorizable maps and traces on the universal free product of matrix algebras, International Mathematics Research Notices 2021 (2021), no. 23, ...

    work page 2020

Show all 13 references
  1. [2008]

    [BPTV24] Gergely Bunth, J´ ozsef Pitrik, Tam´ as Titkos, and D´ aniel Virosztek,Metric property of quantum wasserstein di- vergences, Phys. Rev. A 110 (2024), 022211. [BR24] Bruce Blackadar and Mikael Rørdam, The space of tracial states on a C∗-algebra, Expositiones Mathematic...

    2024 arXiv

  2. [2016]

    Theory Relat

    [Fre19] Amaury Freslon, Cut-off phenomenon for random walks on free orthogonal quantum groups , Probab. Theory Relat. Fields 174 (2019), 731–760. [FSW25] Amaury Freslon, Adam Skalski, and Simeng Wang, Tracial central states on compact quantum groups , Journal of Functional Ana...

    2019

  3. [2023]

    [AK18] Konrad Aguilar and Jens Kaad, The podles sphere as a spectral metric space , Journal of Geometry and Physics 133 (2018)

    [AK] Are Austad and David Kyed, Quantum metrics from length functions on quantum groups , preprint, arXiv:2503.01501. [AK18] Konrad Aguilar and Jens Kaad, The podles sphere as a spectral metric space , Journal of Geometry and Physics 133 (2018). [AKK22a] Konrad Aguilar, Jens K...

    2018

  4. [2024]

    Noncommut

    [JB15] Joachim Zacharias Jyotishman Bhowmick, Christian Voigt, Compact quantum metric spaces from quantum groups of rapid decay, J. Noncommut. Geom. 9 (2015), 1175–1200. [JS97] V. Jones and V. S. Sunder, Introduction to subfactors , London Mathematical Society Lecture Note Ser...

    2015

  5. [2025]

    Todorov, and Lyudmila Turowska, Coupling capacity in c*-algebras , Proceedings of the Royal Society of Edinburgh: Section A Mathematics 155 (2025), no

    30 ANSHU, DA VID JEKEL, AND THERESE BASA LANDRY [STT25] Adam Skalski, Ivan G. Todorov, and Lyudmila Turowska, Coupling capacity in c*-algebras , Proceedings of the Royal Society of Edinburgh: Section A Mathematics 155 (2025), no. 1, 81–103. [VDN92] Dan Voiculescu, Ken Dykema, ...

    2025

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.