  The frustrated spin-1/2 Ising-Heisenberg ladder with Heisenberg intra-rung
and Ising inter-rung interactions is exactly solved in a longitudinal magnetic
field by taking advantage of the local conservation of the total spin on each
rung and the transfer-matrix method. We have rigorously calculated the
ground-state phase diagram, magnetization process, magnetocaloric effect and
basic thermodynamic quantities for the model, which can be alternatively viewed
as an Ising-Heisenberg tetrahedral chain. It is demonstrated that a stepwise
magnetization curve with an intermediate plateau at a half of the saturation
magnetization is also reflected in respective stepwise changes of the
concurrence serving as a measure of bipartite entanglement. The ground-state
phase diagram and zero-temperature magnetization curves of the Ising-Heisenberg
tetrahedral chain are contrasted with the analogous results of the purely
quantum Heisenberg tetrahedral chain, which have been obtained through
density-matrix renormalization group (DMRG) calculations. While both
ground-state phase diagrams fully coincide in the regime of weak inter-rung
interaction, the purely quantum Heisenberg tetrahedral chain develops Luttinger
spin-liquid and Haldane phases for strongly coupled rungs which are absent in
the Ising-Heisenberg counterpart model.
