  We measure the equation of state in a bilayer Hubbard system for different
ratios of the two tunnelling amplitudes $t_\perp /t$. From the equation of
state we deduce the compressibility and observe its dependency on $t_\perp /t$.
Moreover, we infer thermodynamic number fluctuations from the equation of state
by employing the fluctuation-dissipation theorem. By comparing the
thermodynamic with local density fluctuations, we find that non-local density
fluctuations in our bilayer Hubbard system become more prominent for higher
$t_\perp /t$ in the low filling regime. To validate our measurements, we
compare them to Determinant Quantum Monte Carlo simulations of a bilayer
Hubbard system with 6$\times$6 lattice sites per layer.
