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A two-level system can be described by a density operator involving the Bloch vector

$$ \vec{r}; \quad r_x = Tr(\rho X); \quad r_y = Tr(\rho Y); \quad r_z = Tr(\rho Z) $$

as

$$ \rho = \frac{I + \vec{r}\cdot \vec{\sigma}}{2} $$

where $X$, $Y$, and $Z$ are the Pauli operators.

What is the physical idea behind defining the density operator for a two level system like this, and in particular what is $\vec{\sigma}$ here?

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The density operator combines pure quantum states into a mixed quantum state. The basic idea is to take a system composed of many pure states and to represent them as a single object, which evolves in time, as a complete system.

In this example, the mixed state is represented as a Block sphere, and the $\vec{\sigma}$ is a pauli matrix. The Bloch sphere is essentially a representation of the system that can be thought of as a sphere with basis vectors X, Y, Z in your case, which each represent the pure states. The Bloch vector points somewhere in the sphere, pointing to a mixed state (which, if it were pointing only along X, would be a pure state)

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