The Spatiotemporal Doubled-Density Operator: A Unified Framework for Analyzing Spatial and Temporal Quantum Processes

The measurement statistics for spatial and temporal quantum processes are produced through distinct mechanisms. Measurements that are space-like separated exhibit non-signaling behavior. However, time-like separated measurements can only result in one-way non-signaling, as the past is independent of the future, but the opposite is not true. This work presents the doubled-density operator as a comprehensive framework for studying quantum processes in space-time. It effectively captures all the physical information of the process, with the measurement and Born rule showing uniformity for both spatial and temporal cases. It is demonstrated that the equal-time density operator can be derived by performing a partial trace operation on the doubled-density operator. Furthermore, the temporality of the quantum process can be detected by conducting a partial trace operation on either the left or right half of the doubled-density operator.