# 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 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?

## 1 Answer

The answer to any question in QM will not change based on the picture you work in. Which picture you choose to work in, is convenience. Density matrix evolves with time but not it's trace.

• So no matter what the state will not change its purity? Will that happen even when it is being subjected to an external perturbation. – EverydayFoolish May 19 '19 at 9:25
• 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 $Tr(ABC) = Tr(CAB)$. – levitt May 19 '19 at 9:58
• What could be the physical significance of this result? One more question if I may ask? Could you give any physical examples for the statistical mixture of states(mixed states for which we need density matrix formalism) and pure states? – EverydayFoolish May 19 '19 at 10:13
• I'd recommend reading, sakurai Modern quantum mechanics, 2nd edition. Chapter 3. Theory of angular momentum. Section 4 on "Density Operators and Pure Vs Mixed Ensembles." – levitt May 19 '19 at 10:36