# Show that for pure states the description of the Bloch vector we have given coincides with that in section 1.2

$$\newcommand\bra[1]{\left\langle#1\right|}\newcommand\ket[1]{\left|#1\right\rangle}$$ 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 $$1$$)

• so are you trying to find the vector $\vec{r}$? Jul 19, 2023 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 $$\rho = a_{1}I + a_{2}X + a_{3}Y + a_{4}Z$$ Jul 19, 2023 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? Jul 19, 2023 at 16:06
• have you tried writing that $\rho$ in terms of the Pauli matrices? Jul 20, 2023 at 2:22
• – glS
Aug 16, 2023 at 22:13