# How does the measure of purity of a mixed state evolve with time in quantum mechanics?

We know that the Tr() is invariant with respect to unitary transformation. So does the density matrix $$\rho(t)$$ does not evolve with time?

\begin{align} \ \rho(t) =&|\psi(t)\rangle \langle \psi(t)| \\ =& U (t,t_o)|\psi(t_o)\rangle \ \langle \psi(t_o)| U^\dagger (t,t_o) \\ =& U (t,t_o) \rho(t_o) U^\dagger (t,t_o) \end{align}

Now, if consider the measure for purity $$Tr(\rho^2)$$ then, \begin{align} Tr(\rho^2) =& Tr( U (t,t_o)|\psi(t_o)\rangle \ \langle \psi(t_o)| U^\dagger (t,t_o) .U (t,t_o)|\psi(t_o)\rangle \ \langle \psi(t_o)| U^\dagger (t,t_o)) \\ =& Tr(U (t,t_o)|\psi(t_o)\rangle \ \langle \psi(t_o) |\psi(t_o)\rangle \ \langle \psi(t_o)| U^\dagger (t,t_o)) )\\ =& Tr(U (t,t_o) \rho^2(t_o) U^\dagger (t,t_o) )\\ =& \sum_{n} \langle n| U (t,t_o) \rho^2(t_o) U^\dagger (t,t_o)|n\rangle \end{align} In one of the lecture notes[1] a statement is made that, 'the mixedness or measure for purity $$Tr(\rho^2)$$ of a density matrix is time independent'.

Questions

1. Does the measure of purity evolve with time for a density matrix?

2. Will this change based on the picture of the quantum mechanics we consider? Whether it is Schrodinger picture or Heisenberg picture or the Interaction picture ?

EDIT

1. What is the physical significance of this result that, mixedness of a statistical mixture is independent of time evolution? Also, could we prove the result in Heisenberg picture?

• Yes, as long as the evolution is unitary(Hamiltonian is hermitian). Since, $Tr(\rho^2(t)) = Tr(U (t,t_o) \rho^2(t_o) U^\dagger (t,t_o) ) = Tr(U^\dagger (t,t_o) U (t,t_o) \rho^2(t_o) ) = Tr(I\rho^2(t_o) ) = Tr(\rho^2(t_o) )$ using $Tr(ABC) = Tr(CAB)$. Commented May 19, 2019 at 9:58