  We begin by deriving bounds for the entanglement of a spin with an (adjacent
and non-adjacent) interval of spins in an arbitrary pure Finitely Correlated
States (FCS). The bounds we derive become exact in the case where one considers
the entanglement of a single spin with a half-infinite chain to the right of
it. Our result permits a more efficient calculation, numerically and in some
cases even analytically, of the entanglement in arbitrary finitely correlated
quantum spin chains. We continue the study of entanglement in the setting of
ground states of Hamiltonians with a spectral gap. In particular, for $V$ a
finite subset of $Z^d$, we let $H_V$ denote a Hamiltonian on $V$ with finite
range, finite strength interactions and a unique ground state with a
non-vanishing spectral gap. For a density matrix $\rho_A$ that describes the
finite-volume restriction to a region $A$ of the unique ground state, we
provide a detailed version of a proof by M. Hastings, that the entropy of
$\rho_A$ is bounded by a uniform constant $C$. Moreover, we provide a detailed
generalization of the 1-dimensional construction of Hastings' approximation to
the ground state in dimensions 2 and higher. Finally, we turn our attention to
the study of a conjecture central to Quantum Information Theory, the
multiplicativity of the maximal output Schatten $p$-norm of quantum channels,
for $p > 1$. In particular, we study the output 2-norm for a special class of
quantum channels, the depolarized Werner-Holevo channels, and show that
multiplicativity holds for a product of two identical channels in this class.
