# Understanding the density matrix of a qubit

A density operator $\rho$ for the pure or mixed state of a qubit can be written in the following general form:

$$\rho = \begin{pmatrix} a+b & c-id \\ c+id & a-b \end{pmatrix}$$

I know $\rho$ is a Hermitian operator and can be decomposed in terms of Pauli matrices. But how is the generalized form on the right hand side of the equation produced?

• What do you mean by how it is produced? Commented Nov 8, 2016 at 7:51
• Do you mean what probabilities and what projectors to use? Commented Nov 8, 2016 at 8:18

The general density matrix $\rho$ for a qubit is a $2 \times 2$ complex Hermitian matrix, therefore it can be expanded over the basis $\{I, \sigma_x, \sigma_y, \sigma_z\}$(*), which is an orthogonal basis of the Hilbert space of $2 \times 2$ complex Hermitian matrices:

$$\rho=aI+c\sigma_x+d\sigma_y+b\sigma_z$$

where $a,b,c,d \in \mathbb R$. We therefore have $$\rho = \begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}+ \begin{pmatrix} 0 & c \\ c & 0 \end{pmatrix}+ \begin{pmatrix} 0 & -id \\ id & 0 \end{pmatrix}+ \begin{pmatrix} b & 0 \\ 0 & -b \end{pmatrix}= \begin{pmatrix} a+b & c-id \\ c+id & a-b \end{pmatrix}$$

However, the density matrix should also satisfy $Tr(\rho)=1$, i.e.

$$2a=1 \to a=1/2$$

we therefore have the more usual form

$$\rho=\frac 1 2 \begin{pmatrix} 1+z & x-iy \\ x+iy & 1-z \end{pmatrix}$$

where $z=2b$,$x=2c$,$y=2d$.

(*) $\sigma_i$ are the Pauli matrices

• For the density matrix $\rho$, are we not also required to add 1) each element of $\rho$; that is $(a+b), (c-id), (c+id),(a-b)$ must be $\in \mathbb{R}_{>0}$ and 2) the sum over the elements must equal $1$; that is $(a+b)+(c-id),+(c+id)+(a-b)=1$? Commented Mar 25, 2019 at 15:52

The way to work out the decomposition is to set your density matrix $\rho$ to be equal to a linear combination of the Pauli matrices: $$\rho = \sum_{i=0}^3\alpha_i\sigma_i.$$ where $\sigma_0=I$ and and the 1,2,3 matrices are the $x,y,z$ Pauli matrices and the alphas are real numbers. You then do some algebra to calculate the alphas.

• Try the book "The geometry of quantum states", where you find a picture of a general density matrix.
– XXDD
Commented Nov 8, 2016 at 16:06