The existence of infinite families of
N×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≥4. However, their existence for odd
N≥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.
A method to systematically construct the
XX quantum spin chains with nearest-neighbor interactions that allow perfect state transfer (PST) is shown. Sets of orthogonal polynomials (OPs) are in correspondence with such systems. The key observation is that for any admissible one-excitation energy spectrum, the weight function of the associated OPs is uniquely prescribed. This entails the complete characterization of these PST models with the mirror-symmetry property arising as a corollary. A simple and efficient algorithm to obtain the corresponding Hamiltonians is presented. A model connected to a special case of the symmetric
q-Racah polynomials is offered. An explanation of how additional models with PST can be derived from a parent system by removing energy levels from the one-excitation spectrum of the latter is given. This is achieved through Christoffel transformations and is also completely constructive in regards to the Hamiltonians.
I construct a family of time-independent, excitation-preserving spin Hamiltonians realising perfect state transfer from a localised two-excitation state to the symmetric two-excitation Dicke state, for every
N≥4. The Hamiltonian has the physical form
H=∑i<jJij(σi+σj−+σj+σi−)+∑iεini with real couplings, and satisfies
e−iHt∣110⋯0⟩=eiφ∣DN(2)⟩ at a finite time. An
SN−2 permutation symmetry on the unoccupied spins reduces the dynamics to a four-dimensional invariant subspace. Requiring
(∣ψ0⟩+∣DN(2)⟩)/2 to be a zero eigenvector fixes the on-site energies in closed form and leaves three coupling parameters free. The inverse spectral problem then becomes two polynomial equations in two coupling ratios; eliminating one gives a degree-six reciprocal polynomial, which
z=x+x−1 converts to a cubic. For the spectral family
(−n,−1,1) with odd
n, factorising the cubic's leading coefficient and evaluating it at
z=−2 shows that some odd
n always produces a real root below
−2, which a subresultant lifts back to the original system. The existence argument is symbolic and uses no numerical optimisation. Since that coefficient contains no odd powers of
n, the required
n comes with an explicit threshold, not an asymptotic guarantee. I also cost the construction: couplings grow as
N1/2 and the on-site range as
N3/2, the transfer time stays within a factor 2.3-3.0 of the Mandelstam-Tamm limit at every size, and the fidelity is sensitive to systematic drift of the spectator-spectator coupling class but tolerant of independent bond disorder, which self-averages. This is a constrained analogue of perfect state transfer: for general real states an unconstrained real symmetric matrix suffices, whereas here the Hamiltonian must have excitation-preserving spin-network form.
In this paper we build on the ideas presented in previous works for perfectly transferring a quantum state between opposite ends of a spin chain using a fixed Hamiltonian. While all previous studies have concentrated on nearest-neighbor couplings, we demonstrate how to incorporate additional terms in the Hamiltonian by solving an inverse eigenvalue problem. We also explore issues relating to the choice of the eigenvalue spectrum of the Hamiltonian, such as the tolerance to errors and the rate of information transfer.