How would I compute a density matrix of a 2 qubit mixed state?

Question

I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed states, how would I do this? $$ |00> \;with \;probability \; 2/4 \\ |01> \;with\; probability\; 0\\ |10> \;with \;probability\; 1/4 \\ |11>\; with\; probability \;1/4$$

Answer 1

$\rho = \dfrac{1}{2}|00\rangle \langle 00 | + 0|01\rangle \langle 01| + \dfrac{1}{4} |10\rangle \langle 10| + \dfrac{1}{4} |11\rangle \langle 11 | = \begin{pmatrix} 1/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 1/4 & 0\\ 0 & 0 & 0 & 1/4 \end{pmatrix}$

---

Note that $|00\rangle = |0\rangle \otimes |0\rangle = \begin{pmatrix} 1\\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1\\ 0 \end{pmatrix} = \begin{pmatrix} 1 \cdot 1 \\ 1 \cdot 0 \\0 \cdot 1 \\ 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 1\\ 0 \\0 \\ 0 \end{pmatrix}$ and similarly, you can calculate that $|01\rangle = \begin{pmatrix} 1\\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0\\ 1 \end{pmatrix}= \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} $, and $|10 \rangle = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} $, and lastly $|11\rangle = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} $.

Now, $|00\rangle \langle 00| $ represents the outer-product and hence $$|00\rangle \langle 00| = \begin{pmatrix} 1\\ 0 \\0 \\ 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} $$

you can calculate the others and verify.