Concatenation can increase the cHaar moment norm

Abstract

The second cHaar moment at system and environment dimension two has larger squared Hilbert-Schmidt norm after five concatenations than after four.

Definition 1.1 (The cHaar channel).

$$\forall d\in \mathbb{N},\forall e\in \mathbb{N},[\mathrm{NeZero}\left(e\right)]\forall U\in \mathrm{Matrix}\left((\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(e\right)),(\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(e\right)),\mathbb{C}\right),\forall X\in \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\mathrm{cHaarChannel}\left(d,e,U,X\right)=D5.S3.Quantum.Information.PartialTraceMutualInformation.partialTraceRight\left(U\cdot Matrix.kroneckerMap\left(funa\mapsto funb\mapsto a\cdot b,X,(Matrix.single\left(0,0,1\right):\mathrm{Matrix}\left(\mathrm{Fin}\left(e\right),\mathrm{Fin}\left(e\right),\mathbb{C}\right))\right)\cdot \mathrm{star}\left(U\right)\right)$$

Formalization. D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.cHaarChannel (✓ std3).

Citation. Matthew Duschenes, Diego García-Martín, Zoë Holmes, and M. Cerezo (2026). Moments of Quantum Channel Ensembles. DOI: 10.1103/grdp-jph9. URL: https://arxiv.org/abs/2511.12700v2.

Commentary.

Page 5: “Thus the action of any quantum channel is” followed by Lambda(X) = Tr_E[U_Lambda(X tensor nu_E)U_Lambda dagger]. Appendix B, Definition 4, page 25: “Using the Stinespring formalism, we can re-write this average as a twirl with respect to the Haar ensemble of unitaries acting on t copies of a composite space, of a system and environment H ⊗ E of dimensions d,d_E, and a local pure environment state ν_E ∈ B[E],” The system and environment are Fin d and Fin e; Matrix.single 0 0 1 is the fixed pure environment state. The existing partialTraceRight is precisely the sum over the environment coordinate.

Definition 1.2 (The literal moment operator).

$$\forall d\in \mathbb{N},\forall e\in \mathbb{N},\forall t\in \mathbb{N},[\mathrm{NeZero}\left(e\right)]\forall r\in ((\mathrm{Fin}\left(t\right)\to \mathrm{Fin}\left(d\right))\times (\mathrm{Fin}\left(t\right)\to \mathrm{Fin}\left(d\right))),\forall c\in ((\mathrm{Fin}\left(t\right)\to \mathrm{Fin}\left(d\right))\times (\mathrm{Fin}\left(t\right)\to \mathrm{Fin}\left(d\right))),\mathrm{momentOp}\left(d,e,t,r,c\right)={\int }_{U:Matrix.unitaryGroup\left((\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(e\right)),\mathbb{C}\right)}(\prod _{s:\mathrm{Fin}\left(t\right)}(\mathrm{cHaarChannel}\left(d,e,\mathrm{val}\left(U\right),Matrix.single\left((c).1\left(s\right),(c).2\left(s\right),1\right),(r).1\left(s\right),(r).2\left(s\right)\right)))d(Measure.haarMeasure\left((Top.top:TopologicalSpace.PositiveCompacts\left(Matrix.unitaryGroup\left((\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(e\right)),\mathbb{C}\right)\right))\right))$$

Formalization. D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.momentOp (✓ std3).

Citation. Matthew Duschenes, Diego García-Martín, Zoë Holmes, and M. Cerezo (2026). Moments of Quantum Channel Ensembles. DOI: 10.1103/grdp-jph9. URL: https://arxiv.org/abs/2511.12700v2.

Commentary.

Page 3: “We can then define the t-th order moment operator as the superoperator of the t-th order twirl,” followed by equation (9). The entry at r,c integrates the product of the t channel matrix-unit entries with the same Haar unitary U. This defines every order t, not only the second order.

Definition 1.3 (The squared Hilbert-Schmidt norm).

$$\forall i\in Type,[\mathrm{Fintype}\left(i\right)]\forall M\in \mathrm{Matrix}\left(i,i,\mathbb{C}\right),\mathrm{hsNormSq}\left(M\right)=\sum _r\sum _{c}Complex.normSq\left(M\left(r,c\right)\right)$$

Formalization. D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.hsNormSq (✓ std3).

Citation. Matthew Duschenes, Diego García-Martín, Zoë Holmes, and M. Cerezo (2026). Moments of Quantum Channel Ensembles. DOI: 10.1103/grdp-jph9. URL: https://arxiv.org/abs/2511.12700v2.

Commentary.

The squared Hilbert-Schmidt norm is the sum of Complex.normSq over all matrix entries. The concatenation assertion compares this quantity for two powers of the moment operator.

Definition 1.4 (The concatenation monotonicity assertion).

$$\mathrm{claim}\left(\right)=\left(\forall d\in \mathbb{N},\forall e\in \mathbb{N},\forall t\in \mathbb{N},[\mathrm{NeZero}\left(e\right)](1\le d)\Rightarrow ((1\le t)\Rightarrow (\forall k\in \mathbb{N},\forall j\in \mathbb{N},(1\le k)\Rightarrow ((k\le j)\Rightarrow (\mathrm{hsNormSq}\left((\mathrm{momentOp}\left(d,e,t\right){)}^j\right)\le \mathrm{hsNormSq}\left((\mathrm{momentOp}\left(d,e,t\right){)}^k\right)))))\right)$$

Formalization. D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.claim (✓ std3).

Citation. Matthew Duschenes, Diego García-Martín, Zoë Holmes, and M. Cerezo (2026). Moments of Quantum Channel Ensembles. DOI: 10.1103/grdp-jph9. URL: https://arxiv.org/abs/2511.12700v2.

Commentary.

Appendix B, Conjecture 1, page 32: “it is conjectured that such a hierarchy holds for all concatenations k, orders t, and dimensions d,d_E. Further, it is conjectured that the cHaar norm is monotonically non-increasing with environment dimension and concatenations,” followed by the displayed concatenation inequality for every k′ ≥ k. Here d,d_E are d,e; the order is t; k and j are positive concatenation counts with k ≤ j. Page 3: “The choice of norm ∥·∥ is task-dependent, and unless noted, we use the Hilbert-Schmidt norm as it is overall the simplest to compute.” The squared Hilbert-Schmidt norm of the Liouville matrix is the existing RHLinalg.frobSq, equal to the double sum of Complex.normSq. The environment-dimension clause is a separate assertion.

Theorem 1.5 (The concatenation assertion is false).

$$\neg \mathrm{claim}\left(\right)$$

Proof. Machine-checked in Lean as D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.result (✓ std3). ∎

Resolves. Problems/duschenes-garciamartin-holmes-cerezo-2026-chaar-concatenation (refuted) by D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.result.

Source. Repository-derived.

Acknowledgement. Matthew Duschenes, Diego García-Martín, Zoë Holmes, and M. Cerezo (2026). Moments of Quantum Channel Ensembles. DOI: 10.1103/grdp-jph9. URL: https://arxiv.org/abs/2511.12700v2.

Commentary.

At d = e = t = 2 the fourth power has squared norm 1213852/1171875, and the fifth has squared norm 30362908/29296875. Their difference is 5536/9765625, which is positive. In the rational identity/swap basis, the decaying eigenmode coexists with a growing coefficient towards a non-orthogonal limiting projection.

References

• Truth anchor: D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.cHaarChannel
• Truth anchor: D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.claim
• Truth anchor: D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.hsNormSq
• Truth anchor: D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.momentOp
• Truth anchor: D5/S3/Quantum/RandomCircuits/CHaarConcatenationNonmonotone.result
• Dependency: D5/S3/Quantum/GNSMatrix
• Dependency: D5/S3/Quantum/Information/PartialTraceMutualInformation
• Dependency: D5/S3/Quantum/RandomCircuits/HaarTwoCopyTwirl