Density Operator

A density Operator is a generalization of Quantum states, allowing us to represent mixed states. This is useful when we have classically uncertain quantum states, or are working in a Reduced system.

Properties

The statistical operator $\hat{\rho }$ is

1. Linear: $(\alpha \hat{\rho }+\beta )|\psi ⟩=\alpha \hat{\rho }|\psi ⟩+\beta |\psi ⟩$
2. Positive: $⟨\psi |\hat{\rho }|\psi ⟩\ge 0\forall |\psi ⟩$.
3. Has Trace 1: $\mathrm{Tr}[\hat{\rho }]=1$

My understanding of the boundaries between Statistical Operator / Density Operator / Statistical Matrix / Density Matrix is vague:

• Matrix vs Operator corresponds to whether we made a choice of basis or not.
• Statistical vs Density does not have a clear distinction AFAICT. Help me ❓

Examples

1. We have a system which is in state $|{\psi }_{\alpha }⟩$ with probability $p_{\alpha }$, ${\sum }_{\alpha }{p}_{\alpha }=1$, and make a measurement $\hat{O}={\sum }_n{O}_n|O_n⟩⟨O_n|$. Then $\mathbb{P}[O_n]={\sum }_{\alpha }{p}_{\alpha }|⟨p_{\alpha }|O_n⟩{|}^2$. This can be rewritten as $\mathbb{P}[O_n]=⟨O_n|\hat{\rho }|O_n⟩$, where $\hat{\rho }$ is our density matrix.