# Writing a Density matrix in terms of the magnitude of the Bloch Vector

Working with the density matrix and the Bloch sphere, I have been attempting to complete an exercise in Entangled Systems; New Directions in Quantum Physics. If anyone has the book it is Question 4.3 on Pg 87 of the English Edition.

In summary the question asks me to rewrite the desity matrix, $$\rho(\vec{r})$$, in terms of $$|\vec{r}|$$ instead. To start I have $$\rho(\vec{r})=\frac1{2}\left( \begin{array}{cc } 1+r_3 & r_1-ir_2 \\ r_1+ir_2 & 1-r_3 \end{array} \right)$$ with $$\vec{r} = (r_1, r_2, r_3)$$

The final answer it is expecting is $$\rho(|\vec{r}|) = \left( \begin{array}{cc } 1-|\vec{r}|^2 & 0 \\0 & 1+|\vec{r}|^2 \end{array} \right),$$ as it is stated as a part of the question.

Now, the question leaves the hint of finding the eigenvalues of $$\rho(\vec{r})$$. I believe it says this as we could then use the eigenvalues to write the diagonalized version of the density matrix by placing the eigenvalues on the diagonal. Having done so I got $$\rho(|\vec{r}|) = \left( \begin{array}{cc } \frac1{2}(1-|\vec{r}|) & 0 \\0 & \frac1{2}(1+|\vec{r}|) \end{array} \right).$$ This is similar, but not the same, and even though I could factor out the 1/2's and still maintain $$tr[\rho] = 1$$, I'm not sure how this needs to be manipulated to get the version he portrayed in the question.

So I ask, is this a typo? (There have been others) Or is there something I have not thought of to manipulate this to get the correct result?

• There is a 1/2 missing in the original $\rho(\vec r)$. Sep 7, 2022 at 20:25
• Checking the book google.fr/books/edition/Entangled_Systems/… there is no other way to know what was expected. Your result seems right, maybe a typo? Sep 7, 2022 at 20:37
• I was definitely missing the 1/2 on the original. Will add it in! Thanks! Sep 8, 2022 at 11:39
• I'm glad we agree that it might be a typo, but was the typo that he lost his 1/2 in the one using magnitude, as that would preserve the trace, or is his typo that he squared each magnitude... I'm playing with another idea. Will post if it comes to anything. Sep 8, 2022 at 11:45
• I suspect that the typo is the 1/2 that became an exponent ${}^2$ Sep 8, 2022 at 11:57