# Finding the density matrix

I'm trying to calculate the density matrix of the state $$|\psi \rangle=4|00\rangle + 3i|11\rangle -i|01\rangle + 2|10\rangle$$ And I've approached the problem by multiplying out $|\psi\rangle \langle\psi|$ with all the sixteen terms that gives me, and then worked out the matrix corresponding to the various different combinations like $|00\rangle\langle 00|$, $|01\rangle \langle 11|$ and so on, and turned the problem into matrix addition.

As far as I know that works just fine, but it's exceptionally tedious to perform sixteen matrix multiplications! I do need to end up with a matrix, but is there a better way of approaching the problem?

• If it's any consolation, there are orthonormal bases in which your density matrix looks like $\rho = \mathrm{diag} \{1,0,0,0\}$. – secavara Jan 26 '18 at 16:34

• Yes. For instance, if you pick the order $|11\rangle,|10\rangle,|01\rangle,|00\rangle$ for your basis, then you can compute your density matrix in Mathematica as KroneckerProduct[$v$, Conjugate[$v$]] , with $v=\frac{1}{\sqrt{30}} \{3 I, 2, -I, 4\}$. – secavara Jan 26 '18 at 16:48