REVIEW 4 minor 3 references
Comment on 'Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices'
T0 review · 0 major / 4 minor · reviewed 2026-07-10 · grok-4.5
Pith's one-line read An explicit family of quantum states shows that the conjectured quantum Wasserstein distance fails the triangle inequality.
desk verdict Clean analytical kill of both Friedland et al. conjectures; the only soft spot is an unwritten but immediate continuity sentence. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Proposition 1, which reduces WC_E,p between any two diagonal states to an ordinary minimization over classical couplings of the expression (1/2 ∑_{i<j} E_{ij}^p (√γ_{ij}−√γ_{ji})^2)^{1/p}. Closed-form evaluation of this formula on the three states yields the explicit distances (15)–(17) whose comparison immediately produces the strict triangle violation.
What would settle it
Pick any concrete pair (s,t) inside the open triangle Δ° (for example s=0.3, t=0.1), compute the three numbers WCQ,2(ρ,σ), WCQ,2(σ,τ) and WCQ,2(ρ,τ) by the classical-coupling formula of Proposition 1, and check whether the third exceeds the sum of the first two.
Extended reading notes
Core claim
For the antisymmetric quantum cost matrix CQ and for p=2, the associated quantity WCQ,2 fails the triangle inequality on the explicit family of diagonal states ρ=diag(1−s,s,0), σ=diag(s,1−s−t,t), τ=diag(t,1−s−t,s) whenever the parameters lie in the open set Δ°={(s,t):0<t<s<1/2, 2s+t<1}. The same triples remain counterexamples for every quantum cost matrix sufficiently close to CQ, thereby refuting both Conjecture I and Conjecture II of the original paper.
Load-bearing premise
The map that sends a cost matrix to the associated quantum Wasserstein function is continuous enough that a strict open violation for CQ automatically produces a violation for every nearby cost matrix.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This Comment disproves two conjectures of Friedland et al. (PRL 129, 110402) on the quantum 2-Wasserstein semidistance W_{C,2} built from quantum cost matrices and quantum couplings. The authors first prove a general reduction (Proposition 1) expressing W_{C_E,p} between diagonal states as a minimization over classical couplings that involves the interference terms (sqrt(gamma_ij)-sqrt(gamma_ji))^2. They then exhibit an explicit two-parameter family of three-dimensional diagonal states (14) and compute the three pairwise distances in closed form (15)-(17) by elementary monotonicity and Cauchy-Schwarz arguments. On the interior of the parameter domain the triangle inequality is violated, and the same triples serve as counterexamples for every n>3 by padding with zeros. The same family is claimed to kill the neighborhood conjecture as well.
Significance. The result cleanly settles two concrete open claims from a recent PRL by means of fully explicit, hand-checkable counterexamples. The reduction formula of Proposition 1 is of independent interest: it makes precise why quantum transport between classical states is cheaper than classical transport and recovers known n=2 formulae as special cases. Because the violation is strict and open, and because the map C |-> W_{C,2} is continuous on the compact set of couplings in finite dimension, the same triples also refute the neighborhood conjecture. The calculations are elementary and reproducible; no numerical search or machine-checked proof is required, yet the algebraic verification via strict concavity of the square-root function is transparent and robust.
minor comments (4)
- [Section 2, after (17)] After equations (15)-(17) the continuity argument needed to kill Conjecture II is left implicit. A single sentence noting that |W_C,2 - W_CQ,2| is controlled by the operator-norm difference of the squared cost matrices on the compact set of couplings would make the neighborhood claim fully self-contained.
- [Figure 1] Figure 1 is useful but its caption could briefly state that the plotted quantity is exactly the triangle deficit given by (15)-(17), so that a reader can verify the figure without re-deriving the expressions.
- [Remark 2] The parenthetical remark that the original definition of W should have included a factor of 2^{1/p} (Remark 2) is interesting but slightly digressive; it could be shortened or moved to a footnote without loss of the main argument.
- [Acknowledgements] The acknowledgement that the counterexamples were found with ChatGPT assistance is transparent; no change is required, but the journal may wish to confirm that the final algebraic proofs are author-verified.
Circularity Check
No circularity: explicit algebraic counterexamples obtained by direct minimization over classical couplings, with no self-referential definitions or load-bearing self-citations.
full rationale
The paper’s central claim is a disproof of two conjectures by exhibiting an explicit three-parameter family of diagonal states (14) for which the triangle inequality of W_{C_Q,2} fails. Every step is self-contained and proceeds from the definitions of quantum cost matrices and couplings given in the cited PRL: unitary averaging reduces the quantum problem on diagonal states to a classical min (Proposition 1, eqs. (4)–(12)); the three pairwise distances are then evaluated by elementary parametrizations of Γ_cl together with monotonicity or Cauchy–Schwarz-plus-attainment (eqs. (15)–(17)); the resulting algebraic inequality is reduced to the strict concavity of r ↦ √(1+a^{2}+2ar) (eqs. (18)–(20)). No quantity is defined in terms of the target conclusion, no parameters are fitted to data, and the single concurrent self-citation ([3]) is never invoked in the proofs. Continuity of C ↦ W_{C,2} needed for the neighborhood claim is immediate from the compact set of couplings and is not circular. The derivation therefore contains no circular steps.
Assumptions & free parameters
assumptions (3)
- domain assumption Definition of quantum cost matrix CE and of the set Γ(ρ1,ρ2) of quantum couplings (partial-trace constraints) as given in Friedland et al. PRL 2022.
- standard math A density matrix is positive semi-definite with unit trace; partial traces of a bipartite density matrix recover the marginals.
- standard math Cauchy–Schwarz inequality and strict concavity of the map x ↦ √x on [0,∞).
Cite this review
Pith. "Pith review of Comment on 'Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices'." pith.science (2026). https://pith.science/paper/5PSJWCV4
@misc{pith2026260707764,
author = {Pith},
title = {Pith review of: Comment on 'Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices'},
year = {2026},
howpublished = {\url{https://pith.science/paper/5PSJWCV4}},
note = {Machine review of arXiv:2607.07764}
}
abstract
Friedland et al. [PRL 129, 110402 (2022)] proposed and studied a quantum analogue of the $p$-Wasserstein distance based on quantum cost matrices and quantum couplings. They conjectured that, despite being only a semidistance in general, this quantity is a true distance for a particular quantum cost matrix and for cost matrices in a small neighborhood of it. We disprove these conjectures by exhibiting an explicit family of triples of states for which the triangle inequality fails.
Figures
Reference graph
Works this paper leans on
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[1]
Sh. Friedland, M. Eckstein, S. Cole, K. Życzkowski,Quantum Monge–Kantorovich problem and transport distance between density matrices, Phys. Rev. Lett.129, 110402 (2022)
work page 2022
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[2]
S. Cole, M. Eckstein, Sh. Friedland, K. Życzkowski,On quantum optimal transport, Math. Phys. Anal. Geom. 26, 14 (2023)
work page 2023
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[3]
Distances between pure quantum states induced by a distance matrix
T. Miller, R. Bistroń,Distances between pure quantum states induced by a distance matrix,arXiv:2509.14727[math-ph](2025) 7
work page Pith review arXiv 2025
Reviewed July 10, 2026 · model on record in the stance chip above.
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