No perfect state transfer at time pi on rational paths with 2^k + 1 vertices

Abstract

Kirkland, McLaren, Pereira, Plosker and Zhang (arXiv:1708.03283) conjecture that a weighted path on at least four vertices, with or without potentials, whose edge weights are all rational has no perfect state transfer between its end vertices at readout time pi, and prove it for n = 4 and for n congruent to 3 or 5 modulo 8. For every k >= 1 the statement holds for the paths on n = 2^k + 1 vertices, with arbitrary real potentials and in both directions; for k >= 3 these sizes are congruent to 1 modulo 8.

Definition 1.1 (The weighted path Hamiltonian).

$$\forall m\in \mathbb{N},\forall r\in \mathrm{Fin}\left(m\right)\to \mathbb{R},\forall q\in \mathrm{Fin}\left(m+1\right)\to \mathbb{R},\forall i\in \mathrm{Fin}\left(m+1\right),\forall j\in \mathrm{Fin}\left(m+1\right),\mathrm{pathHamiltonian}\left(r,q\right)\left(i,j\right)=\text{if}i=j\text{then}q\left(i\right)\text{else}\text{if}i+1=j\text{then}r\left(i\right)\text{else}\text{if}j+1=i\text{then}r\left(j\right)\text{else}0$$

Formalization. D5/S3/Quantum/Dynamics/RationalWeightPathTransfer.pathHamiltonian (✓ std3).

Citation. S. Kirkland, D. McLaren, R. Pereira, S. Plosker and X. Zhang (2019). Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials. DOI: 10.1080/03081087.2018.1442810. URL: https://arxiv.org/abs/1708.03283v2.

Commentary.

The vertices are 0, …, m and the edge {t, t + 1} carries the weight r t for t in Fin m. The matrix has the real potential q i at the diagonal entry (i, i), the weight r i at the entries (i, i + 1) and (i + 1, i), and zero elsewhere; all entries are real numbers cast to complex numbers. This is the tridiagonal adjacency matrix with potentials of the source, on m + 1 vertices.

Definition 1.2 (Perfect state transfer).

$$\forall V\in Type,[\mathrm{Fintype}\left(V\right)],[\mathrm{DecidableEq}\left(V\right)],\forall H\in \mathrm{Matrix}\left(V,V,\mathbb{C}\right),\forall t\in \mathbb{R},\forall a\in V,\forall b\in V,\mathrm{HasPST}\left(H,t,a,b\right)\iff (\mathrm{normSq}\left(\mathrm{hamiltonianPropagator}\left(H,-t,a,b\right)\right)=1)$$

Formalization. D5/S3/Quantum/Dynamics/RationalWeightPathTransfer.HasPST (✓ std3).

Citation. S. Kirkland, D. McLaren, R. Pereira, S. Plosker and X. Zhang (2019). Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials. DOI: 10.1080/03081087.2018.1442810. URL: https://arxiv.org/abs/1708.03283v2.

Commentary.

Perfect state transfer from vertex a to vertex b at time t means |e_a^T exp(i t H) e_b|^2 = 1, where exp is the matrix exponential and normSq is the squared modulus of a complex number. The formal statement writes exp(i t H) as the repository propagator hamiltonianPropagator H s = exp(-i s H) at s = -t.

Theorem 1.3 (Reversal form of a unitary sending the first vertex to the last).

$$\forall m\in \mathbb{N},\forall r\in \mathrm{Fin}\left(m\right)\to \mathbb{R},\forall q\in \mathrm{Fin}\left(m+1\right)\to \mathbb{R},(\forall t\in \mathrm{Fin}\left(m\right),0<r\left(t\right))\Rightarrow (\forall U\in \mathrm{Matrix}\left(\mathrm{Fin}\left(m+1\right),\mathrm{Fin}\left(m+1\right),\mathbb{C}\right),U^{\ast }\cdot U=1\Rightarrow (U\cdot \mathrm{pathHamiltonian}\left(r,q\right)=\mathrm{pathHamiltonian}\left(r,q\right)\cdot U\Rightarrow (\forall gamma\in \mathbb{C},\mathrm{normSq}\left(gamma\right)=1\Rightarrow ((\forall x\in \mathrm{Fin}\left(m+1\right),U\left(x,0\right)=\text{if}x=\mathrm{last}\left(m\right)\text{then}gamma\text{else}0)\Rightarrow ((\forall i\in \mathrm{Fin}\left(m+1\right),\forall x\in \mathrm{Fin}\left(m+1\right),U\left(x,i\right)=\text{if}x=\mathrm{rev}\left(i\right)\text{then}gamma\text{else}0)\wedge (\forall t\in \mathrm{Fin}\left(m\right),r\left(t\right)=r\left(\mathrm{rev}\left(t\right)\right)))))))$$

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

Source. Repository-derived.

Acknowledgement. S. Kirkland, D. McLaren, R. Pereira, S. Plosker and X. Zhang (2019). Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials. DOI: 10.1080/03081087.2018.1442810. URL: https://arxiv.org/abs/1708.03283v2.

Commentary.

Let U be unitary, commute with the path Hamiltonian H, and send the first basis vector to gamma times the last one, with |gamma| = 1. By induction on j, U e_j = gamma e_(m - j) and r t = r (m - 1 - t) for t < j. Indeed r_j U e_(j+1) = U(H e_j - q_j e_j - r_(j-1) e_(j-1)) = gamma (H e_(m-j) - q_j e_(m-j) - r_(j-1) e_(m-j+1)); by the induction hypothesis only the components at m - j - 1 and m - j remain, orthogonality of the columns j and j + 1 of U removes the second one, and the unit norm of column j + 1 together with the positivity of the weights gives r j = r (m - 1 - j). Here rev i is the mirror vertex m - i of the vertex i, rev t is the mirror index m - 1 - t of the edge t, and last(m) is the vertex m. The source records the underlying fact as known, citing Kay: a symmetric tridiagonal Hamiltonian with perfect state transfer between its end vertices is persymmetric. The statement here is its form for an arbitrary unitary commuting with the path Hamiltonian: every column of U is gamma times a basis vector, and the edge weights are mirror symmetric.

Theorem 1.4 (Transfer from the first vertex forces mirror-symmetric weights).

$$\forall m\in \mathbb{N},\forall r\in \mathrm{Fin}\left(m\right)\to \mathbb{R},\forall q\in \mathrm{Fin}\left(m+1\right)\to \mathbb{R},(\forall t\in \mathrm{Fin}\left(m\right),0<r\left(t\right))\Rightarrow (\forall U\in \mathrm{Matrix}\left(\mathrm{Fin}\left(m+1\right),\mathrm{Fin}\left(m+1\right),\mathbb{C}\right),U^{\ast }\cdot U=1\Rightarrow (U\cdot \mathrm{pathHamiltonian}\left(r,q\right)=\mathrm{pathHamiltonian}\left(r,q\right)\cdot U\Rightarrow (\forall gamma\in \mathbb{C},\mathrm{normSq}\left(gamma\right)=1\Rightarrow ((\forall x\in \mathrm{Fin}\left(m+1\right),U\left(x,0\right)=\text{if}x=\mathrm{last}\left(m\right)\text{then}gamma\text{else}0)\Rightarrow (\forall t\in \mathrm{Fin}\left(m\right),r\left(t\right)=r\left(\mathrm{rev}\left(t\right)\right))))))$$

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

Source. Repository-derived.

Acknowledgement. S. Kirkland, D. McLaren, R. Pereira, S. Plosker and X. Zhang (2019). Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials. DOI: 10.1080/03081087.2018.1442810. URL: https://arxiv.org/abs/1708.03283v2.

Commentary.

This is the second conclusion of the reversal form of a unitary sending the first vertex to the last: if U is unitary, commutes with the path Hamiltonian and sends the first basis vector to gamma times the last one, with |gamma| = 1, then r t = r (m - 1 - t) for every edge t. Here rev t is the mirror index m - 1 - t of the edge t, and last(m) is the vertex m. The source records the underlying fact as known, citing Kay: a symmetric tridiagonal Hamiltonian with perfect state transfer between its end vertices is persymmetric. The statement here is its form for an arbitrary unitary commuting with the path Hamiltonian, restricted to the edge weights.

Theorem 1.5 (Spectral weights under perfect state transfer).

$$\forall V\in Type,[\mathrm{Fintype}\left(V\right)],[\mathrm{DecidableEq}\left(V\right)],\forall H\in \mathrm{Matrix}\left(V,V,\mathbb{C}\right),\mathrm{IsHermitian}\left(H\right)\Rightarrow (\forall t\in \mathbb{R},\forall a\in V,\forall b\in V,\forall m\in \mathbb{N},\mathrm{card}\left(V\right)=m+1\Rightarrow (\forall P\in \mathbb{R},P\ne 0\Rightarrow ((\forall p\in \mathbb{N},p\le m\Rightarrow (\left(H^p\right)\left(a,b\right)=\text{if}p=m\text{then}P\text{else}0))\Rightarrow (\mathrm{HasPST}\left(H,t,a,b\right)\Rightarrow (\mathrm{Injective}\left(\mathrm{eigenvalues}\left(H\right)\right)\wedge \left((\forall k\in V,\mathrm{eigenvectorUnitary}\left(H\right)\left(a,k\right)\cdot \mathrm{conj}\left(\mathrm{eigenvectorUnitary}\left(H\right)\left(b,k\right)\right)=(\frac{P}{{\prod }_{l\in \mathrm{erase}\left(\mathrm{univ},k\right)}(\mathrm{eigenvalues}\left(H,k\right)-\mathrm{eigenvalues}\left(H,l\right))}:\mathbb{C}))\wedge \left((\forall k\in V,\exp \left((t:\mathbb{C})\cdot I\cdot (\mathrm{eigenvalues}\left(H,k\right):\mathbb{C})\right)\cdot (\frac{P}{{\prod }_{l\in \mathrm{erase}\left(\mathrm{univ},k\right)}(\mathrm{eigenvalues}\left(H,k\right)-\mathrm{eigenvalues}\left(H,l\right))}:\mathbb{C})=\exp \left(((t:\mathbb{C})\cdot I)\cdot H\right)\left(a,b\right)\cdot (\mathrm{normSq}\left(\mathrm{eigenvectorUnitary}\left(H\right)\left(a,k\right)\right):\mathbb{C}))\wedge \left((\forall k\in V,\left|\frac{P}{{\prod }_{l\in \mathrm{erase}\left(\mathrm{univ},k\right)}(\mathrm{eigenvalues}\left(H,k\right)-\mathrm{eigenvalues}\left(H,l\right))}\right|=\mathrm{normSq}\left(\mathrm{eigenvectorUnitary}\left(H\right)\left(a,k\right)\right))\wedge (\forall k\in V,\exp \left((t:\mathbb{C})\cdot I\cdot (\mathrm{eigenvalues}\left(H,k\right):\mathbb{C})\right)=(\text{if}0<\frac{P}{{\prod }_{l\in \mathrm{erase}\left(\mathrm{univ},k\right)}(\mathrm{eigenvalues}\left(H,k\right)-\mathrm{eigenvalues}\left(H,l\right))}\text{then}\exp \left(((t:\mathbb{C})\cdot I)\cdot H\right)\left(a,b\right)\text{else}-\exp \left(((t:\mathbb{C})\cdot I)\cdot H\right)\left(a,b\right)))\right)\right)\right))))))$$

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

Source. Repository-derived.

Acknowledgement. Alastair Kay (2010). A Review of Perfect State Transfer and its Application as a Constructive Tool. DOI: 10.1142/S0219749910006514. URL: https://arxiv.org/abs/0903.4274v3.

Commentary.

Let W be the unitary matrix of orthonormal eigenvectors of the Hermitian matrix H, eigenvectorUnitary(H), with the eigenvalues lambda_k = eigenvalues(H, k), and put gamma = exp(i t H)(a, b). In the formula I is the imaginary unit, conj is complex conjugation, (v : C) is the complex number equal to the real number v, normSq is the squared modulus, card(V) is the number of sites, univ is the set of all indices and erase(univ, k) the set of the indices different from k. Write c_k = W(a, k) conj(W(b, k)) and rho_k = P / prod_{l != k} (lambda_k - lambda_l). The hypotheses on the powers of H give sum_k c_k lambda_k^p = 0 for p < m and = P for p = m; evaluating sum_k c_k f(lambda_k) on the monic polynomial f = prod_{l != k} (X - lambda_l), of degree m, gives c_k prod_{l != k} (lambda_k - lambda_l) = P. Since P is not zero, the eigenvalues are distinct and c_k = rho_k, a nonzero real number. The spectral theorem gives gamma = sum_k W(a, k) exp(i t lambda_k) conj(W(b, k)), the pairing of the two unit vectors x_k = conj(W(a, k)) and y_k = exp(i t lambda_k) conj(W(b, k)). Perfect state transfer says |gamma| = 1, so sum_k |y_k - gamma x_k|^2 = 1 + |gamma|^2 - 2 Re(conj(gamma) gamma) = 0 and exp(i t lambda_k) conj(W(b, k)) = gamma conj(W(a, k)) for every k; this is the equality case of the Cauchy-Schwarz inequality. Multiplying by W(a, k) gives exp(i t lambda_k) rho_k = gamma |W(a, k)|^2, and taking moduli gives |rho_k| = |W(a, k)|^2. Dividing by the nonzero real number rho_k gives exp(i t lambda_k) = gamma where rho_k > 0 and exp(i t lambda_k) = -gamma where rho_k < 0; as rho_k is never zero, the second case is the one in which rho_k > 0 fails, which is how the last conjunct is written. For the Hamiltonian of a path on the vertices 0, …, m the hypotheses on the powers hold between the two end vertices with P the product of the m edge weights; this is the statement pathHamiltonian_pow_apply_ends of the same module, an instance of the column formula for the powers together with the symmetry of the Hamiltonian. The weight formula |W(a, k)|^2 = |P| / |prod_{l != k} (lambda_k - lambda_l)| is the classical Hochstadt formula for the weights of the eigenvectors of a symmetric tridiagonal matrix on its first site, which Kay’s review recalls in its subsection on inverse eigenvalue problems: with B’(lambda_n) = prod_{m != n} (lambda_n - lambda_m), the weight |alpha_n|^2 is the reciprocal of (-1)^n B’(lambda_n), normalised so that the weights sum to one. For a mirror-symmetric chain and its two end sites the phase is gamma on the symmetric and -gamma on the antisymmetric eigenvectors, and these alternate in increasing order of the eigenvalues, so the last conjunct is the perfect state transfer condition of Kay’s review, lambda_n - lambda_{n-1} = (2 m_n + 1) pi / t_0 for the ordered eigenvalues of a symmetric chain, with t_0 the transfer time and m_n an integer. Derived here is the form of these facts for an arbitrary Hermitian matrix with the stated moments between two sites a and b and perfect state transfer between them at a real time t, with no symmetry assumption: the constant of proportionality is the moment P, the cross weights W(a, k) conj(W(b, k)) are the signed numbers rho_k, and the phases exp(i t lambda_k) are gamma times the sign of rho_k.

Theorem 1.6 (Spectral parity classes of a transfer at time pi).

$$\forall V\in Type,[\mathrm{Fintype}\left(V\right)],[\mathrm{DecidableEq}\left(V\right)],\forall H\in \mathrm{Matrix}\left(V,V,\mathbb{C}\right),\mathrm{IsHermitian}\left(H\right)\Rightarrow (\forall a\in V,\forall b\in V,a\ne b\Rightarrow (\forall m\in \mathbb{N},\mathrm{card}\left(V\right)=m+1\Rightarrow (\forall P\in \mathbb{R},P\ne 0\Rightarrow ((\forall p\in \mathbb{N},p\le m\Rightarrow (\left(H^p\right)\left(a,b\right)=\text{if}p=m\text{then}P\text{else}0))\Rightarrow (\mathrm{HasPST}\left(H,\pi ,a,b\right)\Rightarrow (\exists z\in V\to \mathbb{Z},\exists A\in \mathrm{Finset}\left(V\right),\mathrm{Injective}\left(z\right)\wedge \left(\mathrm{Nonempty}\left(A\right)\wedge \left(A\ne \mathrm{univ}\wedge \left(\left(\forall i\in A,\mathrm{Even}\left(z\left(i\right)\right)\right)\wedge \left(\left(\forall i\in V,\left(\neg (i\in A)\right)\Rightarrow (\mathrm{Odd}\left(z\left(i\right)\right))\right)\wedge P\cdot \sum _{i\in A}(\prod _{j\in \mathrm{erase}\left(\mathrm{univ},i\right)}((z\left(i\right)-z\left(j\right):\mathbb{Z}):\mathbb{R}){)}^{-1}=\frac{1}{2}\right)\right)\right)\right)))))))$$

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

Source. Repository-derived.

Acknowledgement. S. Kirkland, D. McLaren, R. Pereira, S. Plosker and X. Zhang (2019). Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials. DOI: 10.1080/03081087.2018.1442810. URL: https://arxiv.org/abs/1708.03283v2.

Commentary.

Let c k = W a k conj(W b k) for the orthonormal eigenvectors W of H with eigenvalues lambda k. The hypotheses on the powers of H give sum_k c k lambda_k^p = 0 for p < m and = P for p = m; evaluating sum_k c k f(lambda k) on the monic polynomial f = prod_{l != k} (X - lambda l) gives c k prod_{l != k} (lambda k - lambda l) = P, so the eigenvalues are distinct and every c k is a nonzero real number. Transfer at time pi forces exp(i pi lambda_k) conj(W b k) = gamma conj(W a k) with gamma = exp(i pi H)(a, b), since the sum of the squared moduli of their differences vanishes. Hence |c k| = |W a k|^2, whose sum is 1, while sum_k c k = 0. The class A of indices with c k > 0 has sum_{k in A} c k = 1/2, and exp(i pi lambda_k) equals gamma on A and -gamma off A. Shifting the eigenvalues by an eigenvalue of the class A gives integers z, even on A and odd off A, with lambda k - lambda l = z k - z l. The source records as known that, after a common shift, the eigenvalues of a path with perfect state transfer at time pi between its end vertices are integers that alternate between even and odd. The identity P sum_{i in A} 1 / prod_{j != i} (z i - z j) = 1/2, stated for a general Hermitian matrix with the given moment data, is derived here.

Definition 1.7 (The rational weights statement on 2^k + 1 vertices).

$$claim\iff (\forall k\in \mathbb{N},1\le k\Rightarrow (\forall r\in \mathrm{Fin}\left(2^k\right)\to \mathbb{R},\forall q\in \mathrm{Fin}\left(2^k+1\right)\to \mathbb{R},(\forall j\in \mathrm{Fin}\left(2^k\right),\exists x\in \mathbb{Q},r\left(j\right)=x)\Rightarrow ((\forall j\in \mathrm{Fin}\left(2^k\right),0<r\left(j\right))\Rightarrow (\left(\neg \mathrm{HasPST}\left(\mathrm{pathHamiltonian}\left(r,q\right),\pi ,0,\mathrm{last}\left(2^k\right)\right)\right)\wedge \left(\neg \mathrm{HasPST}\left(\mathrm{pathHamiltonian}\left(r,q\right),\pi ,\mathrm{last}\left(2^k\right),0\right)\right)))))$$

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

Source. Repository-derived.

Acknowledgement. S. Kirkland, D. McLaren, R. Pereira, S. Plosker and X. Zhang (2019). Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials. DOI: 10.1080/03081087.2018.1442810. URL: https://arxiv.org/abs/1708.03283v2.

Commentary.

For k >= 1 and the path on n = 2^k + 1 vertices with edge weights r in Fin(2^k) -> R and potentials q in Fin(2^k + 1) -> R: if every weight is rational and positive, there is no perfect state transfer at time pi from the first vertex 0 to the last vertex 2^k, nor from the last vertex to the first. The rationality hypothesis is written as the existence of a rational number equal to each weight. The source states its conjecture for all n >= 4 and does not single out this family; the statement is the restriction of that conjecture to the sizes n = 2^k + 1, with k = 1 added.

Theorem 1.8 (Rational paths on 2^k + 1 vertices have no transfer at time pi).

$$claim$$

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

Source. Repository-derived.

Acknowledgement. S. Kirkland, D. McLaren, R. Pereira, S. Plosker and X. Zhang (2019). Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials. DOI: 10.1080/03081087.2018.1442810. URL: https://arxiv.org/abs/1708.03283v2.

Commentary.

The propagator exp(i pi H) is symmetric, so both directions reduce to transfer from the last vertex to the first. The entries of H^p in the column of the first vertex vanish beyond distance p, and the entry at the last vertex of H^m is P = prod_t r t with m = 2^k. The spectral parity classes give integers z, even on a class A and odd off A, with P sum_{i in A} 1 / prod_{j != i} (z i - z j) = 1/2, and A is neither empty nor everything. The column of the propagator at the first vertex is gamma times the last basis vector, so the weights are mirror symmetric and P is the square Q^2 of a rational number Q. Since C(2^k - 1, j) is odd for every j < 2^k, the partial divided-difference sum over A has 2-adic valuation 0. Then 2 v_2(Q) = v_2(1/2) = -1, which is impossible. This is partial progress on the conjecture of the source: k = 1 gives n = 3, below the range n >= 4 of the conjecture; k = 2 gives n = 5, one of the cases proved in the source; the new sizes are those with k >= 3, namely n = 9, 17, 33, …; the other sizes are not addressed.

References

• Truth anchor: D5/S3/Quantum/Dynamics/RationalWeightPathTransfer.HasPST
• Truth anchor: D5/S3/Quantum/Dynamics/RationalWeightPathTransfer.claim
• Truth anchor: D5/S3/Quantum/Dynamics/RationalWeightPathTransfer.pathHamiltonian
• Truth anchor: D5/S3/Quantum/Dynamics/RationalWeightPathTransfer.persymmetric_weights
• Truth anchor: D5/S3/Quantum/Dynamics/RationalWeightPathTransfer.pst_parity_classes
• Truth anchor: D5/S3/Quantum/Dynamics/RationalWeightPathTransfer.pst_spectral_weights
• Truth anchor: D5/S3/Quantum/Dynamics/RationalWeightPathTransfer.result
• Truth anchor: D5/S3/Quantum/Dynamics/RationalWeightPathTransfer.reversal_columns
• Dependency: D5/S3/Quantum/Dynamics/ParityNodeDividedDifference
• Dependency: D5/S3/Quantum/Dynamics/ProjectionProbabilityFlow