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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?

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  • $\begingroup$ If it's any consolation, there are orthonormal bases in which your density matrix looks like $\rho = \mathrm{diag} \{1,0,0,0\}$. $\endgroup$ – secavara Jan 26 '18 at 16:34
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I can't think of a shortcut to this, but I would recommend symbolic calculus software, Mathematica or Maple, these are not free but Sympy library in Python is, and you only need to know very basic Python to compute these sort of things.

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    $\begingroup$ 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 $v=\frac{1}{\sqrt{30}} \{3 I, 2, -I, 4\}$. $\endgroup$ – secavara Jan 26 '18 at 16:48

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