Relationship between probabilities of the state transfers and entanglements in spin systems with simple geometrical configurations

TL;DR

This paper derives analytical relations linking state transfer probabilities and entanglements in simple spin systems, demonstrating the possibility of high-probability state transfers and identifying entanglements responsible for them.

Contribution

It introduces analytical relations between transfer probabilities and entanglements in spin systems with specific geometries and applies them to demonstrate high-probability state transfers.

Findings

High-probability state transfers are possible in rectangular and parallelepiped geometries.

Entanglements responsible for these transfers have been identified.

Analytical relations between transfer probabilities and entanglements are established.

Abstract

In this paper we derive analytical relations between probabilities of the excited state transfers and entanglements calculated by both the Wootters and positive partial transpose (PPT) criteria for the arbitrary spin system with single excited spin in the external magnetic field and Hamiltonian commuting with $I_z$. We apply these relations to study the arbitrary state transfers and entanglements in the simple systems of nuclear spins having two- and three-dimensional geometrical configurations with $XXZ$ Hamiltonian. It is shown that High-Probability State Transfers (HPSTs) are possible among all four nodes placed in the corners of the rectangle with the proper ratio of sides as well as among all eight nodes placed in the corners of the parallelepiped with the proper ratio of sides. Entanglements responsible for these HPSTs have been identified.