  We present a wide array of quantum measures on numerical solutions of 1D
Bose- and Fermi-Hubbard Hamiltonians for finite-size systems with open boundary
conditions. Finite size effects are highly relevant to ultracold quantum gases
in optical lattices, where an external trap creates smaller effective regions
in the form of the celebrated "wedding cake" structure and the local density
approximation is often not applicable. Specifically, for the Bose-Hubbard
Hamiltonian we calculate number, quantum depletion, local von-Neumann entropy,
generalized entanglement or Q-measure, fidelity, and fidelity susceptibility;
for the Fermi-Hubbard Hamiltonian we also calculate the pairing correlations,
magnetization, charge-density correlations, and antiferromagnetic structure
factor. Our numerical method is imaginary time propagation via time-evolving
block decimation. As part of our study we provide a careful comparison of
canonical vs. grand canonical ensembles and Gutzwiller vs. entangled
simulations. The most striking effect of finite size occurs for bosons: we
observe a strong blurring of the tips of the Mott lobes accompanied by higher
depletion, and show how the location of the first Mott lobe tip approaches the
thermodynamic value as a function of system size.
