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 for quantum lattice systems. We consider two agents interacting in disjoint spacetime regions with a spin-lattice system that evolves in time according to a local Hamiltonian. In their respective spacetime regions, the two agents apply quantum instruments to the spins. By considering a purification of the quantum instruments, and analyzing the quantum mutual information between the ancillas used to implement them, we obtain a spacetime area law bound on the amount of correlation between the agents’ measurement outcomes. Furthermore, this bound applies both to signaling correlations between the choice of operations on the side of one agent, and the measurement outcomes on the side of the other; as well as to the entanglement they can harvest from the spins by coupling detectors to them.

面积定律是物理相互作用局部性的深远影响，它们与从黑洞到量子多体系统等一系列系统都息息相关。通常情况下，这些法律涉及的纠缠熵或子系统的量子互信息在单一时间。但是，当考虑在时空中传播的信息时，尽管它是由具有本地交互作用的物理系统携带的，但可以预期，时区适用于区域法则。在这项工作中，我们证明了量子晶格系统的这种定律。我们考虑两个代理在不相交的时空区域中与根据局部哈密顿量随时间演化的自旋-晶格系统相互作用。在它们各自的时空区域中，这两种物质将量子仪器应用于自旋。通过考虑纯化量子仪器，并分析用于实现它们的小环之间的量子互信息，我们获得了一个时空面积定律，其受制于代理测量结果之间的相关量。此外，该界限既适用于发信号通知一个代理人方面的操作选择与另一代理人方面的测量结果之间的相关性；以及它们可以通过将探测器耦合到自旋而从自旋中收获的纠缠。