  In proposals for quantum computers using arrays of trapped ultracold polar
molecules as qubits, a strong external field with appreciable gradient is
imposed in order to prevent quenching of the dipole moments by rotation and to
distinguish among the qubit sites. That field induces the molecular dipoles to
undergo pendular oscillations, which markedly affect the qubit states and the
dipole-dipole interaction. We evaluate entanglement of the pendular qubit
states for two linear dipoles, characterized by pairwise concurrence, as a
function of the molecular dipole moment and rotational constant, strengths of
the external field and the dipole-dipole coupling, and ambient temperature. We
also evaluate a key frequency shift, $\triangle\omega$, produced by the
dipole-dipole interaction. Under conditions envisioned for the proposed quantum
computers, both the concurrence and $\triangle\omega$ become very small for the
ground eigenstate. In principle, such weak entanglement can be sufficient for
operation of logic gates, provided the resolution is high enough to detect the
$\triangle\omega$ shift unambiguously. In practice, however, for many candidate
polar molecules it appears a challenging task to attain adequate resolution.
Simple approximate formulas fitted to our numerical results are provided from
which the concurrence and $\triangle\omega$ shift can be obtained in terms of
unitless reduced variables.
