Density Operator for 2 Systems: Pure State

[PhysiSmo]

We are given that 2 systems can only be found in the states $$|00\rangle, |01\rangle, |10\rangle, |11\rangle$$. We are also given that the density operator is

$$\rho=\frac{1}{2}\left(|00\rangle \langle 00|+|11\rangle \langle 00|+|00\rangle \langle 11|+|11\rangle \langle 11|\right)$$.

a)Write the matrix form of the density operator. Prove that it describes a pure state. Which one?
b)By taking the partial trace of one system, show that the yielding state is a mixed state. Which is the matrix for this new state?

Solution.
a)We express the matrix form by taking inner products, so that the $$\rho_{nm}$$ element of the matrix is $$\rho_{nm}=\langle n|\rho |m \rangle$$. We find then

$$\rho= \begin{bmatrix} 1/2 & 0 \\ 0 & 1/2 \end{bmatrix}$$

which clearly describes a pure state, since $$Tr(\rho^2)=1$$.

How can one find then the state vectors? Since $$\rho=|\psi \rangle \langle \psi |$$ in general, is it true to say that $$\rho |\psi \rangle = |\psi \rangle$$?

b)Unfortunately, I don't have a clue for this one. What do we mean by taking the partial trace of one system? Any help please?

 

[genneth]

a) Indeed it is. The pure state is the eigenvector of $$\rho$$ which has eigenvalue of 1 (think about what eigenvalues of the density matrix mean, what the eigenstates mean, and it should be obvious why this must be so, beyond just the form of the matrix representation).

b) http://beige.ucs.indiana.edu/M743/node80.html

 

[PhysiSmo]

Thank you very much!