  We develop a general theory of the relation between quantum phase transitions
(QPTs) characterized by nonanalyticities in the energy and bipartite
entanglement. We derive a functional relation between the matrix elements of
two-particle reduced density matrices and the eigenvalues of general two-body
Hamiltonians of $d$-level systems. The ground state energy eigenvalue and its
derivatives, whose non-analyticity characterizes a QPT, are directly tied to
bipartite entanglement measures. We show that first-order QPTs are signalled by
density matrix elements themselves and second-order QPTs by the first
derivative of density matrix elements. Our general conclusions are illustrated
via several quantum spin models.
