  Area laws are a far-reaching consequence of the locality of physical
interactions, and they are relevant in a range of systems, from black holes to
quantum many-body systems. Typically, these laws concern the entanglement
entropy or the quantum mutual information of a subsystem at a single time.
However, when considering information propagating in spacetime, while carried
by a physical system with local interactions, it is intuitive to expect area
laws to hold for spacetime regions; in this work, we prove such a law in the
case of quantum lattice systems. We consider a sequence of local quantum
operations performed at discrete times on a spin-lattice, such that each
operation is associated to a point in spacetime. In the time between
operations, the time evolution of the spins is governed by a finite range
Hamiltonian. By considering a purification of the quantum instruments and
analyzing the quantum mutual information between the ancillas used in the
purification, we obtain a spacetime area law bound for the correlations between
the measurement outcomes inside a spacetime region, and those outside of it.
