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The measurement statistics for spatial and temporal quantum processes areproduced through distinct mechanisms. Measurements that are space-likeseparated exhibit non-signaling behavior. However, time-like separatedmeasurements can only result in one-way non-signaling, as the past isindependent of the future, but the opposite is not true. This work presents thedoubled density operator as a comprehensive framework for studying quantumprocesses in space-time. It effectively captures all the physical informationof the process, with the measurement and Born rule showing uniformity for bothspatial and temporal cases. We demonstrate that the equal-time density operatorcan be derived by performing a partial trace operation on the doubled densityoperator. Furthermore, the temporality of the quantum process can be detectedby conducting a partial trace operation on either the left or right half of thedoubled density operator.

The spatiotemporal doubled density operator: a unified framework for analyzing spatial and temporal quantum processes