slug: duschenes-garciamartin-holmes-cerezo-2026-chaar-concatenation
bibkey: duschenesgarciamartinholmescerezo2026moments
doi: 10.1103/grdp-jph9
url: https://arxiv.org/abs/2511.12700v2
triage: theorem
motivation_gids:
  - D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.result

Concatenation can increase the cHaar moment norm

Problem

M. Duschenes, D. García-Martín, Z. Holmes and M. Cerezo, Moments of Quantum Channel Ensembles, PRX Quantum 7, 033050 (2026), DOI 10.1103/grdp-jph9, arXiv:2511.12700v2, Appendix B, Conjecture 1, asks whether the cHaar moment norm is non-increasing with concatenations:

$$\Vert {\hat{T}}_{\mathrm{cHaar}}^{(t)k^{\prime }}{\Vert }_{HS}^2\le \Vert {\hat{T}}_{\mathrm{cHaar}}^{(t)k}{\Vert }_{HS}^2(k^{\prime }\ge k).$$

The verbal source clause is: “Further, it is conjectured that the cHaar norm is monotonically non-increasing with environment dimension and concatenations,”. Only the concatenation clause is refuted. Preregistration #15553 specifies the conventions and witness. The Library note contains the source definitions, norm convention and locators.

Statement echo

Motivation

CHaarConcatenationNonmonotone.result refutes the finite-dimensional concatenation assertion. A decaying eigenmode is compatible with norm growth because the limiting projection is not orthogonal in the Hilbert–Schmidt geometry.

Gap

The source and literature screen leave the finite-dimensional concatenation clause unsettled in the searched scope. M. Duschenes, Effects of Noise on Optimization, Statistics, and Simulation of Quantum Systems, University of Waterloo thesis (2026-06-23; hdl:10012/23660), restates the hierarchy as a conjecture for all $t,k,d,s$ and gives no settlement of the concatenation clause. Its numerical check at $k=1,3$ does not reach the increase, which at $d=e=2$ first appears from $k=4$ to $k=5$ ([comment on #15553](https://github.com/the-omega-institute/trureturing/issues/15553#issuecomment-6111775899)). This is not an exhaustive-literature or originality claim. Theorem “cHaar explicit”, part 3, has a large-dimension limit scope and does not assert this finite-dimensional monotonicity.

Route

Use the existing two-copy Haar average and its identity/swap commutant. twirlFormula derives its trace coefficients, and momentOp2Formula evaluates the order-two channel moment for every $d\ge 2$ and nonzero $e$. At $d=e=2$, moment22Reindex identifies the literal moment with $A=P+Q/15+2R/5$. aPow gives the power normal form; normNormal evaluates its squared norm. The exact values in instanceN4 and instanceN5 contradict claim.

Falsifier

A mismatch in the Haar measure, fixed ancilla, partial trace, tensor-copy convention, matrix-unit indexing or norm would invalidate the correspondence. The settlement compares exact fourth and fifth powers of the ensemble moment, with all factors sharing a Haar sample within each moment.

Evidence

The Lean module is D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.lean, with its canonical Blueprint Scribe. Admission is open-problem-resolution (#15553; Refuted), with certified-instance utility under refutes. The seven owner publications expose existing frozen proofs. The public result is the designated refutation result (basis=refutes) and is exempt from four-slot escape registration (CLAUDE.md §3.9). Every other newly authored public theorem would be an audit target; this module has none. No Reg file is delivered.

Experiment reading

Experiment entry, path docs/reports/duschenes-et-al-2025-chaar-concatenation-monotonicity/check.py, commit 4b35e9f007321a3364bb926919e27879aa107ad6. Command: python3 check.py. Exit $0$; bad= 0. Script SHA-256: 41bf402e6e928d46607598eca1e3ba176896fadb01f48ba2fe0d2376404dd9e8. The implementation seat reran the pinned source.

The executed exact matrix-power checks cover $(d,e)=(2,2),(2,3),(3,2)$ and $1\le k\le 8$, not every pair in $2\le d,e\le 3$. The script checks the exact $N_4,N_5$ and difference at $(2,2)$, and eventual strict increase through $k=60$ of the closed-form sequence for each $2\le d,e\le 6$. A seeded $20,000$-sample Haar Monte Carlo check at $(2,2)$ compares the second moment with the exact matrix, with tolerance $0.02$ and printed maximum deviation $0.0021$. These are finite computations, not a proof of the uniform family or an unbounded tail.

Triage

Tier 1 published open conjecture; concatenation clause Refuted. Each claim has the evidence kind stated below.

What the settlement shows

Eventual increase for every $d,e\ge 2$

Write $H=d^2{e}^2-1$, $F$ for the swap on the two system copies, $u=\mathrm{vec}(I)$ and $v=\mathrm{vec}(dF-I)$. Then $⟨u,v⟩=0$, $\Vert u{\Vert }^2=d^2$ and $\Vert v{\Vert }^2=d^2(d^2-1)$. The general order-two entry formula says that the moment $B$ acts on a matrix $X$ as

$$B(X)=\frac{e^2}{H}\left(\mathrm{tr}X-\frac{\mathrm{tr}(FX)}{de}\right)I+\frac{e}{H}\left(\mathrm{tr}(FX)-\frac{\mathrm{tr}X}{de}\right)F.$$

Thus $Bu=u+\gamma v$, $Bv=\lambda v$, and $B$ annihilates the orthogonal complement of their span, where

$$\gamma =\frac{e-1}{H},\lambda =\frac{e(d^2-1)}{H},0<\lambda <1.$$

Indeed $H-e(d^2-1)=(e-1)(d^{2}e+1)>0$. Set $P=u{u}^{†}/d^2$, $Q=v{u}^{†}/d^2$ and $R=v{v}^{†}/[d^2(d^2-1)]$. These three matrices are pairwise orthogonal for the Hilbert–Schmidt inner product, with squared norms $1,d^2-1,1$. For $k\ge 1$, the triangular action gives

$$B^k=P+\frac{\gamma (1-{\lambda }^k)}{1-\lambda }Q+{\lambda }^{k}R=P+\frac{1-{\lambda }^k}{d^{2}e+1}Q+{\lambda }^{k}R.$$

Consequently, for $c=(d^2-1)/(d^{2}e+1{)}^2>0$,

$$N_k=1+c(1-{\lambda }^k{)}^2+{\lambda }^{2k},$$

$$N_{k+1}-N_k=(1-\lambda ){\lambda }^k\left[2c-(1+c)(1+\lambda ){\lambda }^k\right].$$

Since ${\lambda }^k\to 0$, every sufficiently large $k$ satisfies ${\lambda }^k<2c/[(1+c)(1+\lambda )]$, so the displayed difference is strictly positive. At $d=e=2$, $\lambda =2/5$, $c=1/27$, and the growing $Q$ coefficient is $(1-(2/5{)}^k)/9$. The same difference formula gives strict decrease when ${\lambda }^k>2c/[(1+c)(1+\lambda )]$, equality at the threshold, and strict increase below it. The limit $N_k\to 1+c$ survives; uniform non-increase over all concatenations fails. The family derivation is prose, while Lean supplies the general entry formula and the exact finite witness.

ASSUMED-UNVERIFIED

Literature completeness and originality are unverified beyond the recorded screen. The environment-dimension clause remains open. The all-$d,e$ eventual-increase derivation is not a uniform kernel-checked theorem.