Density Operators on the Bloch Ball

The Bloch ball turns a qubit density matrix into a real three-component vector. Its direction describes the preferred measurement axis; its length describes purity. Bloch-sphere geometry develops the spinors and rotations, while this page explains how to read a mixed state’s matrix.

Helpful background. Density operators explains positivity and trace; the Pauli matrices provide the coordinate basis.

Reading the Bloch Vector

Every trace-one Hermitian qubit matrix has the unique form

$$\rho=\frac12(I+\mathbf r\cdot\boldsymbol\sigma) =\frac12\begin{pmatrix} 1+r_z&r_x-ir_y\\ r_x+ir_y&1-r_z \end{pmatrix}.$$

Thus $r_x=2\operatorname{Re}\rho_{01}$, $r_y=-2\operatorname{Im}\rho_{01}$, and $r_z=\rho_{00}-\rho_{11}$. The minus sign in $r_y$ follows from the standard $Y$ matrix.

The eigenvalues are $(1\pm\|\mathbf r\|)/2$. Positivity is therefore exactly $\|\mathbf r\|\leq1$. Pure states lie on the sphere $\|\mathbf r\|=1$, mixed states lie inside, and $I/2$ is the center. The purity is

$$\operatorname{Tr}\rho^2=\frac{1+\|\mathbf r\|^2}{2}.$$

For a Pauli measurement along a unit vector $\mathbf n$,

$$p(\pm\mid\mathbf n)=\frac12(1\pm\mathbf r\cdot\mathbf n).$$

A shorter vector reduces the difference between the two outcome probabilities. At the center they are equal along every axis.

Quick Check

Find the Bloch vector, purity, and $Y$-measurement probabilities for

$$\rho=\begin{pmatrix}1/2&-i/4\\i/4&1/2\end{pmatrix}.$$

Solution

The vector is $(0,1/2,0)$, so its length is $1/2$ and its purity is $(1+1/4)/2=5/8$. The state is mixed. Along $+y$, the outcome probabilities are $p(+)=3/4$ and $p(-)=1/4$. The negative imaginary upper-right entry corresponds to positive $r_y$.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010, ch. 2.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018, ch. 2. Author’s manuscript.