  The spectrum of atomic hydrogen has long been viewed as a Rosetta stone that
bears the key to decode the writings of quantum mechanics in a vast variety of
physical, chemical, and biological systems. Here, we show that, in addition to
its role as a basic model of quantum mechanics, the hydrogen atom provides a
fundamental building block of quantum information. Through its electron and
nuclear spin degrees of freedom, the hydrogen atom is shown to lend a
physically meaningful frame and a suitable Hilbert space for bipartite
entanglement, whose two-qubit concurrence and quantum coherence can be
expressed in terms of the fundamental physical constants -- the Planck and
Boltzmann constants, electron and proton masses, the fine-structure constant,
as well as the Bohr radius and the Bohr magneton. The intrinsic, natural
entanglement that the hyperfine-structure (HFS) states of the H atom store at
low temperatures rapidly decreases with a growth in temperature, vanishing
above a $\tau_c$ $\approx$ 5.35 $\mu$eV threshold. An external magnetic field,
however, can overcome this thermal loss of HFS entanglement. As one of the
central findings of this work, we show that an external magnetic field can
induce and sustain an HFS entanglement, against all the odds of thermal
effects, at temperatures well above the $\tau_c$ threshold, thus enabling
magnetic-field-assisted entanglement engineering in low-temperature gases and
solids.
