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SIAM Undergraduate Research Online

Early State Exclusion in 7-Qubit Spin Chains

Valentin Garcia, Mia Gabriella Escobar

Published Jan 1, 2025• 0 citations
Publisher

Abstract

The existence of infinite families of (N + 1) × (N + 1) 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 + 1) ≥ 4.However, their existence for odd (N + 1) ≥ 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 × 7 Jacobi matrices with and without ESE.

DOI
10.1137/25s1781917

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arXiv:1410.1666•Oct 7, 2014•4 citations
The existence of infinite families of (N + 1) × (N + 1) 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 + 1) ≥ 4.However, their existence for odd (N + 1) ≥ 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 × 7 Jacobi matrices with and without ESE.
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.
Most of the existing dynamical studies in one dimension on magnetic insulators have considered the simplest spin models with nearest-neighbor interactions. In real systems, however, it is possible that longer range interactions are not entirely negligible. It is expected that the inclusion of next-nearest-neighbor interactions between spins in one-dimensional spin models will introduce a multitude of new frequencies in addition to the ones already present in the dynamics that arises due to nearest-neighbor interactions. We first present an exact solution for the dynamical xx-spin-pair correlations in an Ising chain with both nearest- and next-nearest-neighbor interactions to confirm our expectation. We next show, via an approximate analytical calculation, that the dynamical zz-spin-pair correlations in the next-nearest-neighbor transverse Ising chain when plotted as a function of time is noticeably different with respect to the exactly solvable nearest-neighbor transverse Ising chain at T\ensuremath{\rightarrow}\ensuremath{\infty} when the next-nearest-neighbor interaction is \ensuremath{\gtrsim}1/2 of the magnitude of the nearest-neighbor interaction. The effects could be fairly subtle in the time domain representation and in the spectral function when these additional interactions are weak (i.e., 1/2 of the nearest-neighbor interaction magnitude). The general conclusions reached in this work are expected to be valid for other simple quantum spin models such as the XY and XXZ models in one dimension. \textcopyright{} 1996 The American Physical Society.
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.