  The ground-state spin arrangement and the bipartite entanglement within
Cu$^{2+}$-Cu$^{2+}$ dimers across the magnetization process of the 4f-3d
heterometallic coordination polymer
[{Dy(hfac)$_2$(CH$_3$OH)}$_2${Cu(dmg)(Hdmg)}$_2$]$_n$ (H$_2$dmg =
dimethylglyoxime, Hhfac = 1,1,1,5,5,5-hexafluoropentane-2,4-dione) are
theoretically examined using the symmetric isotropic spin-$1/2$
Ising-Heisenberg orthogonal-dimer chain. The numerical results point to five
possible ground states of the compound with three different degrees of the
quantum entanglement within Cu$^{2+}$-Cu$^{2+}$. Besides the standard
ferrimagnetic and saturated phases without quantum entanglement of Cu$^{2+}$
ions, which are manifested in low-temperature magnetization curve as wide
plateaus at the non-saturated magnetization $16.26\mu_{\rm B}$ and at the
saturation value $20.82\mu_{\rm B}$, respectively, one also finds an intriguing
singlet-like phase with just partial entanglement within Cu$^{2+}$-Cu$^{2+}$
and two singlet phases with fully entangled Cu$^{2+}$-Cu$^{2+}$ dimers. The
former quantum phase can be identified in the low-temperature magnetization
process as very narrow intermediate plateau at the magnetization $9.27\mu_{\rm
B}$ per unit cell, while the latter ones as zero magnetization plateau and
intermediate plateau at the magnetization $18.54\mu_{\rm B}$. Non-monotonous
temperature variations of the concurrence, through which the entanglement
within cooper dimers is quantified, point to the possible temporary thermal
activation of the entangled states of Cu$^{2+}$-Cu$^{2+}$ also above
non-entangled ferrimagnetic and saturated phases.
