2
$\begingroup$

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? $$ |00> \;with \;probability \; 2/4 \\ |01> \;with\; probability\; 0\\ |10> \;with \;probability\; 1/4 \\ |11>\; with\; probability \;1/4$$

$\endgroup$

1 Answer 1

3
$\begingroup$

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


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.

$\endgroup$
3
  • $\begingroup$ @KAJ266 How did you calculate the bra-kets? Do you multiply them elementwise? $\endgroup$
    – lambda
    Commented Feb 23, 2021 at 0:18
  • $\begingroup$ @lambda I updated the answer to show you more details $\endgroup$
    – KAJ226
    Commented Feb 23, 2021 at 0:47
  • 1
    $\begingroup$ @KAJ266 thank you, I appreciate the help, I calculated the others and I see how it works now $\endgroup$
    – lambda
    Commented Feb 23, 2021 at 2:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.