Pretty good state transfer at vertex 4 of the double blow-up of P_11

Abstract

The double blow-up of P_11 admits pretty good state transfer between the two copies of source vertex 4, refuting Conjecture 1 of Bhattacharjya, Monterde and Pal.

Definition 1.1 (Path adjacency).

$$\forall n\in \mathbb{N},\forall i\in \mathrm{Fin}\left(n\right),\forall j\in \mathrm{Fin}\left(n\right),\mathrm{pathAdj}\left(n\right)\left(i,j\right)=\text{if}(\mathrm{val}\left(i\right)+1=\mathrm{val}\left(j\right)\vee \mathrm{val}\left(j\right)+1=\mathrm{val}\left(i\right))\text{then}(1:\mathbb{C})\text{else}(0:\mathbb{C})$$

Formalization. D5/S3/Quantum/Dynamics/PathDoubleBlowUpPGSTRefutation.pathAdj (✓ std3).

Citation. Bikash Bhattacharjya, Hermie Monterde, Hiranmoy Pal (2024). Quantum walks on blow-up graphs. DOI: 10.1088/1751-8121/ad6653. URL: https://arxiv.org/abs/2308.13887v2.

Commentary.

Source vertices are 1 through n. A vertex j in Fin(n) represents source vertex val(j) + 1, so adjacency is the disjunction val(i) + 1 = val(j) or val(j) + 1 = val(i). Matrix entries are complex zero and one.

Definition 1.2 (The double blow-up).

$$\forall n\in \mathbb{N},\forall p\in \mathrm{Fin}\left(2\right)\times \mathrm{Fin}\left(n\right),\forall q\in \mathrm{Fin}\left(2\right)\times \mathrm{Fin}\left(n\right),\mathrm{doubleBlowUp}\left(n\right)\left(p,q\right)=\mathrm{pathAdj}\left(n\right)\left(p.2,q.2\right)$$

Formalization. D5/S3/Quantum/Dynamics/PathDoubleBlowUpPGSTRefutation.doubleBlowUp (✓ std3).

Citation. Bikash Bhattacharjya, Hermie Monterde, Hiranmoy Pal (2024). Quantum walks on blow-up graphs. DOI: 10.1088/1751-8121/ad6653. URL: https://arxiv.org/abs/2308.13887v2.

Commentary.

Section 2, page 2 states: The blow-up of G, denoted by ⊎ⁿG, is the graph with vertex set ℤ_n × V, and two vertices (l, u) and (m, v) are adjacent in ⊎ⁿG if and only if the vertices u and v are adjacent in G. Here the number of copies is two, with carrier Fin(2) times Fin(n). The copy coordinates do not affect adjacency.

Definition 1.3 (Pretty good state transfer).

$$\forall I\in Type,[\mathrm{Fintype}\left(I\right)][\mathrm{DecidableEq}\left(I\right)]\forall A\in \mathrm{Matrix}\left(I,I,\mathbb{C}\right),\forall a\in I,\forall b\in I,\mathrm{PGST}\left(A,a,b\right)\iff (\exists s\in \mathbb{N}\to \mathbb{R},\mathrm{Tendsto}\left((k:\mathbb{N}\mapsto \Vert \mathrm{ProjectionProbabilityFlow}.\mathrm{hamiltonianPropagator}\left(A,-s\left(k\right)\right)\left(a,b\right)\Vert ),atTop,\mathrm{nhds}\left(1\right)\right))$$

Formalization. D5/S3/Quantum/Dynamics/PathDoubleBlowUpPGSTRefutation.PGST (✓ std3).

Citation. Bikash Bhattacharjya, Hermie Monterde, Hiranmoy Pal (2024). Quantum walks on blow-up graphs. DOI: 10.1088/1751-8121/ad6653. URL: https://arxiv.org/abs/2308.13887v2.

Commentary.

Section 1, pages 1-2 states: A graph G exhibits PGST between u and v if there is a sequence τ_k ∈ ℝ such that lim_{k→∞} |U(τ_k){u,v}| = 1, i.e., |U(t){u,v}|² can be made arbitrarily close to one through appropriate choices of t. The matrix U(t) is exp(itA), represented by hamiltonianPropagator(A, -t). The sequence is indexed by the naturals and convergence is Tendsto atTop to nhds(1). Complex modulus is the Lean norm.

Definition 1.4 (Conjecture 1).

$$claim\iff (\forall t\in \mathbb{N},\forall r\in \mathbb{N},(2\le t)\Rightarrow ((\mathrm{Nat}.\mathrm{Prime}\left(r\right))\Rightarrow ((\mathrm{Odd}\left(r\right))\Rightarrow (\forall u\in \mathrm{Fin}\left(\mathrm{Nat}.\mathrm{sub}\left(2^t\cdot r,1\right)\right),(2^{\mathrm{Nat}.\mathrm{sub}\left(t,1\right)}\mid \mathrm{val}\left(u\right)+1)\Rightarrow (\neg \mathrm{PGST}\left(\mathrm{doubleBlowUp}\left(\mathrm{Nat}.\mathrm{sub}\left(2^t\cdot r,1\right)\right),(0,u),(1,u)\right))))))$$

Formalization. D5/S3/Quantum/Dynamics/PathDoubleBlowUpPGSTRefutation.claim (✓ std3).

Citation. Bikash Bhattacharjya, Hermie Monterde, Hiranmoy Pal (2024). Quantum walks on blow-up graphs. DOI: 10.1088/1751-8121/ad6653. URL: https://arxiv.org/abs/2308.13887v2.

Commentary.

Section 8, page 8, Conjecture 1 states: Let n = 2^t r − 1, where t ≥ 2 and r is an odd prime number. If u is a multiple of 2^{t−1}, then PGST does not occur between (0, u) and (1, u) in the double blow-up of P_n. The letters t and r retain their source meaning. The source vertex u is represented by u in Fin(2^t*r - 1), whose one-based label is val(u) + 1. Both subtractions in the displayed formula are Nat.sub, the truncated natural-number subtraction.

Theorem 1.5 (Refutation by source vertex 4 of P_11).

$$\neg claim$$

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

Resolves. Problems/bhattacharjya-monterde-pal-2024-double-blow-up-pgst (refuted) by D5/S3/Quantum/Dynamics/PathDoubleBlowUpPGSTRefutation.result.

Source. Repository-derived.

Acknowledgement. Bikash Bhattacharjya, Hermie Monterde, Hiranmoy Pal (2024). Quantum walks on blow-up graphs. DOI: 10.1088/1751-8121/ad6653. URL: https://arxiv.org/abs/2308.13887v2.

Commentary.

Take t = 2, r = 3 and u = 3 in Fin(11), representing source vertex 4. At that vertex the diagonal amplitude is one quarter of the sum of the four cosines with positive frequencies (sqrt(6)+sqrt(2))/2, sqrt(3), 1 and (sqrt(6)-sqrt(2))/2. Each pair of opposite eigenvalues carries weight one eighth on each eigenvalue. Biquadratic independence and the finite-torus character criterion give times pi/2 + pim(k) along which all four cosines tend to minus one. The general twin-amplitude identity then gives modulus tending to one. The source’s relation theta_5 - theta_9 + theta_11 = 0 uses theta_9 = -sqrt(2); this eigenvalue has zero weight at source vertex 4, since sin(3pi) = 0, and therefore does not obstruct transfer there.

References

• Truth anchor: D5/S3/Quantum/Dynamics/PathDoubleBlowUpPGSTRefutation.PGST
• Truth anchor: D5/S3/Quantum/Dynamics/PathDoubleBlowUpPGSTRefutation.claim
• Truth anchor: D5/S3/Quantum/Dynamics/PathDoubleBlowUpPGSTRefutation.doubleBlowUp
• Truth anchor: D5/S3/Quantum/Dynamics/PathDoubleBlowUpPGSTRefutation.pathAdj
• Truth anchor: D5/S3/Quantum/Dynamics/PathDoubleBlowUpPGSTRefutation.result
• Dependency: D5/S3/Fourier/BiquadraticKroneckerTimes
• Dependency: D5/S3/Quantum/Dynamics/EnergyEigenstateStationarity
• Dependency: D5/S3/Quantum/Dynamics/ProjectionProbabilityFlow
• Dependency: D5/S3/QuantumBounds/ReferenceFrame/TopEigenspace