It is known that for density matrix, the von Neumann equation holds. $$\dot{\rho} = -i[H,\rho]$$ and thus $$\rho(t) = U(t)\rho_0 U^{*}(t)$$ where $U$ refers to a unitary matrix. But what happens for reduced density matrix of $A$ in a bipartite system $AB$? Would above be satisfied? That is, for $\rho_A = Tr_B\{\rho_{AB}\}$, would $\rho_A(t) = U_A(t)\rho_0 U_A^{*}(t)$ be satisfied, where $U_A$ is some unitary matrix?

  • $\begingroup$ Are the subsystems A and B interacting, and if not are they also initially decorrelated? $\endgroup$
    – Ian
    Sep 12 '19 at 5:55
  • $\begingroup$ @Ian $A$ and $B$ are interacting and are entangled/correlated. $\endgroup$ Sep 12 '19 at 6:01

The equation of motion for a reduced density matrix is not the Liouville von Neumann equation that you mention, but rather a generalized quantum master equation known as the Nakajima-Zwanzig equation. The interactions between the subsystems are captured in an object called the memory kernel. Alternatively, one can also encode the time dependence of the reduced density matrix directly rather than in differential equation form in terms of path integrals by using Feynman’s influence functional formalism.


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