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$\newcommand\bra[1]{\left\langle#1\right|}\newcommand\ket[1]{\left|#1\right\rangle} $ I am having a little bit of difficulty with part (4) of Exercises 2.72 from Nielsen and Chuang's "Quantum Computation and Quantum Information".

It asks us to show that "for pure states the description of the Bloch vector we have given ($\rho=\frac{I + r.\sigma}{2}$ ) coincides with that in section 1.2 (($\ket{\psi} = \cos(\frac{\theta}{2})\ket{0} + e^{i \phi} \sin(\frac{\theta}{2}) \ket{1}$, for $\ket{\psi}$ is on the surface of the Bloch sphere))

I have found that $\rho = \cos^{2}(\frac{\theta}{2}) \ket{0} \bra{0} + \frac{e^{-i \phi}\sin(\theta)}{2}\ket{0}\bra{1} + \frac{ e^{i \phi} \sin{\theta} }{2} \ket{1} \bra{0} + \sin^{2}(\frac{\theta}{2}) \ket{1} \bra{1}$

This is where I am stuck. A couple of the solutions online have said that

$\rho = \frac{I + r. \sigma}{2}$ can be expressed (in the computational basis) as: $$ \rho = \frac{1 + r_{z}}{2} \ket{0} \bra{0} + \frac{r_{x} - i r_{y}}{2} \ket{0} \bra{1} + \frac{r_{x} + i r_{y}}{2} \ket{1} \bra{0} + \frac{1 - r_{z}}{2} \ket{1} \bra{1}$$ I really don't understand how we get the above representation?

(Note that here $r$ is a three dimensional vector whose norm is less than or equal to $1$)

am567
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    so are you trying to find the vector $\vec{r}$? – BeauGeste Jul 19 '23 at 13:56
  • I think I found that the vector r is $<2a_{2}, 2a_{3}, 2a_{4}>$, where $a_{2}, a_{3}, a_{4}$ are the coeffiecients of the $X, Y, Z$ respectively for an arbitrary density matrix $$\rho = a_{1}I + a_{2}X + a_{3}Y + a_{4}Z$$ – am567 Jul 19 '23 at 16:05
  • $\newcommand\bra[1]{\left\langle#1\right|}\newcommand\ket[1]{\left|#1\right\rangle} $The part I am confused on is where did the coefficients of $$\ket{0}\bra{0}, \ket{0}\bra{1}, \ket{1}\bra{0}, \ket{1}\bra{1}$$ come from? – am567 Jul 19 '23 at 16:06
  • have you tried writing that $\rho$ in terms of the Pauli matrices? – BeauGeste Jul 20 '23 at 02:22
  • $\newcommand\bra[1]{\left\langle#1\right|}\newcommand\ket[1]{\left|#1\right\rangle} $I did write $\rho$ in terms of the Pauli matrices in order to prove that $\rho$ can be expressed as $\rho=\frac{I + r.\sigma}{2}$, however I cannot seem to get from $$\rho = \cos^{2}(\frac{\theta}{2}) \ket{0} \bra{0} + \frac{e^{-i \phi}\sin(\theta)}{2}\ket{0}\bra{1} + \frac{ e^{i \phi} \sin{\theta} }{2} \ket{1} \bra{0} + \sin^{2}(\frac{\theta}{2}) \ket{1} \bra{1}$$ to the expression of $\rho$ in the computational basis (shown in question). – am567 Jul 20 '23 at 09:00
  • So you want to find the components of $\vec{r}$ now? Set up as a system of equations and solve for each $r_i$. This will give you $\vec{r}$ in terms of $\theta$ and $\phi$. – BeauGeste Jul 20 '23 at 11:03
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    related: https://quantumcomputing.stackexchange.com/a/4121/55 and https://quantumcomputing.stackexchange.com/q/5574/55 – glS Aug 16 '23 at 22:13

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