Reduced Dynamics

Reduced Dynamics are an approximation of the evolution of a quantum system, where we have applied the Partial Trace over the environment.

More formally, this is a Quantum Dynamical Semigroup $\{T_t{\}}_{t\ge 0}$ that attempts to make the diagram below commutative. This is only an attempt, as the Reduced system does not contain all the information needed for commutativity to hold.

$$\begin{array}{ccc}{\rho }_0& \underset{\text{Unitarily}}{\overset{U_t\cdot U_t^{†}}{\to }}& {\rho }_t\\ {\mathrm{Tr}}^E[\cdot ]\downarrow & & \downarrow {\mathrm{Tr}}^E[\cdot ]\\ {\rho }_{S,0}& \underset{\text{Reduced Dynamics}}{\overset{T_t}{\to }}& {\rho }_{S,t}\end{array}$$

Where we use the notation:

• ${\rho }_0$ for the initial Density Operator of the full system
• ${\rho }_t:=U_t{\rho }_0{U}_t^{†}$ for the evolving Density Operator of the full system.
• ${\rho }_{S,0}:={\mathrm{Tr}}^E[{\rho }_0]$ for the initial Density Operator of the Reduced system.
• ${\rho }_{S,t}:={\mathrm{Tr}}^E[{\rho }_t]$ for the evolving Density Operator of the Reduced system.
• ${\mathrm{Tr}}^E[\cdot ]$ represents the Partial Trace over the environment.