  We consider families of invertible many-body quantum states in $d$ spatial
dimensions that are parameterized over some parameter space $X$. The space of
such families is expected to have topologically distinct sectors classified by
the cohomology group $\mathrm{H}^{d+2}(X;\mathbb{Z})$. These topological
sectors are distinguished by a topological invariant built from a
generalization of the Berry phase, called the higher Berry phase. In the
previous work, we introduced a generalized inner product for three
one-dimensional many-body quantum states, (``triple inner product''). The
higher Berry phase for one-dimensional invertible states can be introduced
through the triple inner product and furthermore the topological invariant,
which takes its value in $\mathrm{H}^{3}(X;\mathbb{Z})$, can be extracted. In
this paper, we introduce an inner product of four two-dimensional invertible
quantum many-body states. We use it to measure the topological nontriviality of
parameterized families of 2d invertible states. In particular, we define a
topological invariant of such families that takes values in
$\mathrm{H}^{4}(X;\mathbb{Z})$. Our formalism uses projected entangled pair
states (PEPS). We also construct a specific example of non-trivial
parameterized families of 2d invertible states parameterized over
$\mathbb{R}P^4$ and demonstrate the use of our formula. Applications for
symmetry-protected topological phases are also discussed.
