Can a Qubit's Mixed State Density Operator Be Expressed Using Pauli Matrices?

[mtak0114]

1. Show that an arbitrary density operator for a mixed state qubit may be written as


2. $$\rho = \frac{I+r^i\sigma_i}{2}$$, where $$||r||<1$$
(Nielsen and Chuang pg 105)

3. So my attempt was as follows
Given that a $$\rho$$ is hermitian it may be written as a linear combination of the pauli matrices and the identity.
$$\rho = aI+r^i\sigma_i$$ where r^i is arbitrary
the trace condition $$tr(\rho)=1$$ implies $$a=1/2$$ and
the positivity condition $$\langle\varphi|\rho|\varphi\rangle \geq 0$$
implies that $$2cos\theta |r||\langle\varphi|\sigma^i|\varphi\rangle| \geq -1$$ which implies that
$$|r|\leq 1/2$$, fnally redefing r above gives the result QED
is this correct or am i assuming too much?

cheers

Mark

 

[grey_earl]

Everything fine!