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A central aim of physics is to describe the dynamics of physical systems.Schrodinger's equation does this for isolated quantum systems. Describing thetime evolution of a quantum system that interacts with its environment, in itsmost general form, has proved to be difficult because the dynamics is dependenton the state of the environment and the correlations with it. For discreteprocesses, such as quantum gates or chemical reactions, quantum processtomography provides the complete description of the dynamics, provided that theinitial states of the system and the environment are independent of each other.However, many physical systems are correlated with the environment at thebeginning of the experiment. Here, we give a prescription of quantum processtomography that yields the complete description of the dynamics of the systemeven when the initial correlations are present. Surprisingly, our method alsogives quantitative expressions for the initial correlation.

Operational approach to open dynamics and quantifying initial correlations