Mediated entanglement and correlations in a star network of interacting spins

We investigate analytically a star network of spins, in which all spins interact exclusively and continuously with a central spin through Heisenberg $XX$ couplings of equal strength. We find that the central spin correlates and entangles the other spins at zero temperature to a degree that depends on the total number of spins. We find that the entanglement mediating capability of the central spin depends on the evenness or oddness of this number. In the limit of an infinite collection of spins, the difference between entanglement and correlations in terms of divisibility among multiple parties is clearly demonstrated. We also show that with a significant probability one can maximally entangle any two noncentral spins by measuring all the other spins (a process related to the recently introduced notion of localizable entanglement). This probability depends on the evenness and oddness of the total number of spins and remains substantial even for an infinite collection of spins. We show how symmetric multiparty states for optimal sharing and splitting of entanglement can be obtained as ground states of this system using a magnetic field. These states can then be mapped on to flying qubits for transmission to distant parties. We discuss a number of advantages of this mode of generation and distribution of entanglement over other standard methods.

Figure 1

This figure depicts the star configuration of spins. The spin labeled 0 is the central spin, which interacts with spins 1–5 around it.

Figure 2

This figure plots the probability of obtaining an maximally entangled state between two outer spins by projective measurements on all the other spins as $N$ increases.

Figure 3

This figure compares entanglement distribution between pairs of distant users using the scheme of the current paper with a more standard alternate scheme. Part (a) of the figure depicts entanglement distribution using quantum telephone exchanges based on entanglement swapping. Part (b) depicts entanglement distribution using the ground states of the spin star (the scheme proposed in the current paper).