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arXiv:2507.18767

Early State Exclusion in 7-Qubit Spin Chains

Mia Gabriella Escobar, Garcia, Valentin, Anastasiia Minenkova

Published Jul 24, 2025• 0 citations
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Abstract

The existence of infinite families of $N \times N$ Jacobi matrices representing the Hamiltonians of quantum spin chains with and without early state exclusion (ESE) has been shown to exist for any even $N \geq 4$. However, their existence for odd $N \geq 7$ has remained an open problem. In Section 3, we consider a chain of qubits experiencing nearest-neighbor interactions with environmental effects and present infinite families of $7 \times 7$ Jacobi matrices with and without ESE.

DOI
10.48550/arxiv.2507.18767

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The existence of infinite families of N×NN \times NN×N Jacobi matrices representing the Hamiltonians of quantum spin chains with and without early state exclusion (ESE) has been shown to exist for any even N≥4N \geq 4N≥4. However, their existence for odd N≥7N \geq 7N≥7 has remained an open problem. In Section 3, we consider a chain of qubits experiencing nearest-neighbor interactions with environmental effects and present infinite families of 7×77 \times 77×7 Jacobi matrices with and without ESE.
Perfect state transfer (PST) in one-dimensional quantum spin chains provides a natural setting in which quantum information transport can be analyzed using spectral methods. In the single-excitation subspace, the dynamics of a chain with nearest-neighbor interactions are governed by a Jacobi matrix, allowing questions of state transfer to be formulated in terms of eigenvalue distributions and symmetry properties of eigenvectors. In this work, we investigate the phenomenon of \emph{early state exclusion} (ESE), whereby the overlap of the time-evolved state with the initial state vanishes at a time strictly earlier than the first occurrence of perfect state transfer. Building on earlier constructions of Hamiltonians exhibiting PST with and without ESE, we provide explicit Jacobi matrix realizations for arbitrary odd-length chains and establish general conditions under which ESE occurs or does not occur. We propose the process of \emph{spectral fusion} (SF) to build infinite families of such Hamiltonians. These results broaden the known class of spin chains displaying early state exclusion and further clarify the role of spectral structure of the Hamiltonians.
The spectral statistics and entanglement within the eigenstates of generic spin chain Hamiltonians are analysed. A class of random matrix ensembles is defined which include the most general nearest-neighbour qubit chain Hamiltonians. For these ensembles, and their generalisations, it is seen that the long chain limiting spectral density is a Gaussian and that this convergence holds on the level of individual Hamiltonians. The rate of this convergence is numerically seen to be slow. Higher eigenvalue correlation statistics are also considered, the canonical nearest-neighbour level spacing statistics being numerically observed and linked with ensemble symmetries. A heuristic argument is given for a conjectured form of the full joint probability density function for the eigenvalues of a wide class of such ensembles. This is numerically verified in a particular case. For many translationally-invariant nearest-neighbour qubit Hamiltonians it is shown that there exists a complete orthonormal set of eigenstates for which the entanglement present in a generic member, between a fixed length block of qubits and the rest of the chain, approaches its maximal value as the chain length increases. Many such Hamiltonians are seen to exhibit a simple spectrum so that their eigenstates are unique up to phase. The entanglement within the eigenstates contrasts the spectral density for such Hamiltonians, which is that seen for a non-interacting chain of qubits. For such non-interacting chains, their always exists a basis of eigenstates for which there is no entanglement present.
We study the Boltzmann transport equation for the Bose-Hubbard chain in the kinetic regime. The time-dependent Wigner function is matrix-valued with odd dimension due to integer spin. For nearest neighbor hopping only, there are infinitely many additional conservation laws and nonthermal stationary states. Adding longer-range hopping amplitudes entails exclusively thermal equilibrium states. Especially for small next-nearest neighbor hopping amplitudes, we observe prethermalization with two time scales, which can be related to the relative strength of the nearest and next-nearest hopping. We provide a derivation of the Boltzmann equation based on the Hubbard Hamiltonian, including general interactions beyond on-site, and illustrate the results by numerical simulations.