Schwinger Geometry, Bethe Ansatz, and a Magnonic Qudit

Abstract

Schwinger approach of unitary geometry for a finite-dimensional Hilbert space is interpreted in terms of a magnonic qudit — a hypothetic elementary unit of memory of a quantum computer. The space is interpreted within the Heisenberg model for a magnetic ring, its calculational basis as the classical configuration space for a single spin deviation, treated as a Bethe pseudoparticle, and the dual basis corresponds to quasimomenta, so that the classical phase space spans the quantum algebra of observables. Effects of the Schur-Weyl duality and Bethe ansatz exact eigenstates of the Heisenberg Hamiltonian for the XXX model on properties of the magnonic qudit are presented.

A complete set of commuting operators for the Bethe ansatz

“…e.g. [12]), when comparing l with the size of the memory of a computer or a quantum gate. It is worth mentioning, however, an attempt to find local constants of motion [19,20].…”

“…We use the algebraic Bethe ansatz approach of Faddeev and Takhtajan [9,10] to derive explicitly the set of N − 1 mutually commuting operators (involving the Heisenberg Hamiltonian for the XXX model) from the corresponding monodromy matrix, point out some their general properties and special cases, and compare on some examples the resulting eigenstates with the exact Bethe ansatz solutions, classified by rigged string configurations, as defined by Kerov, Kirillov and Reshetikhin [7]. Some preliminary results on these operators have been reported in our previous papers [11,12].…”

Ground-state and magnetocaloric properties of a coupled spin–electron double-tetrahedral chain (exact study at the half filling)

“…As a result, two 1 × 1 and two 9 × 9 matrices corresponding to the sub-spaces H ∓3/2 ke and H ∓1/2 ke , respectively, are generated. The eigenvalues corresponding to H ∓3/2 ke can be found directly (see Table 1), while for finding the eigenvalues corresponding to H ∓1/2 ke one should apply the so-called basis of wavelets [46][47][48] on the orbits O | f ini kj ⊂ H ∓1/2 ke , as the aftermath of the cyclic translational symmetry of triangular clusters. In our notation, the appropriate amplitude takes the form Table 1).…”

“…As a result, two 1 × 1 and two 9 × 9 matrices corresponding to the sub-spaces H ∓3/2 ke and H ∓1/2 ke , respectively, are generated. The eigenvalues corresponding to H ∓3/2 ke can be found directly (see Table 1), while for finding the eigenvalues corresponding to H ∓1/2 ke one should apply the so-called basis of wavelets [46][47][48] on the orbits…”

Entanglement of one-magnon Schur-Weyl states

“…+ . Such states form a basis which spans a space of all quantum states of so called magnonic qudit [29]. In this simple case, states are uniquely labelled only by standard Young tableau y (the Weyl tableau t is unnecessary in this case) (c.f.…”