The quantification of a genuine tetrapartite entanglement in a mixed spin-(1/2,1) Heisenberg tetramer

Abstract

The genuine tetrapartite entanglement of a quantum mixed spin-(1/2,1) Heisenberg tetramer is quantified according to the three different approaches incorporated all seven global bisections existing within the tetrapartite system. The degree of an entanglement of each bisection is evaluated through the bipartite negativity at zero- and non-zero temperature taking into account ferromagnetic as well as antiferromagnetic type of intra- ($J$) and inter-dimer ($J_1$) exchange coupling inside the square plaquette. Three utilized quantification methods based on the generalization of (i) a genuine tripartite negativity, (ii) a Coffman, Kundu and Wootters monogamy relation and (iii) a geometric average of complete trisections, result to the qualitatively and almost quantitatively identical behavior of a genuine tetrapartite negativity. It is shown that the genuine tetrapartite negativity exclusively arises from the antiferromagnetic-inter dimer $J_1>0$ coupling, whereas the character of with respect to $J$ ($J>0$ or $J<0$) determines its zero-temperature magnitude as well as its thermal stability with respect to the magnetic field and temperature. As is demonstrated for $0<J_1/J<1$ the genuine tetrapartite negativity is dramatically reduced due to the preference of magnetic arrangement involving two separable mixed spin-(1/2,1) dimers. In an opposite limit the genuine tetrapartite negativity is significantly stable with a threshold temperature proportional to the strength of an inter-dimer coupling $J_1$. It is found, that all three quantification procedures are insufficient to correctly describe the genuine tetrapartite negativity in a specific part of the parameter space with absence of relevant dimer separable states. Finally, the thermal stability of a genuine tetrapartite negativity is discussed in detail for selected geometries motivated by the real tetranuclear bimetallic complexes with a Cu${}_2$Ni${}_2$ magnetic core.