Quantum expectation value

Let $\rho$ be a density operator on a finite-dimensional Hilbert space and $A$ a self-adjoint element of the observable algebra. The expectation value of $A$ in the state $\rho$ is

\langle A\rangle_\rho=\operatorname{Tr}(\rho A).

For a pure state $\rho=|\psi\rangle\langle\psi|$ , this is $\langle A\rangle_\rho=\langle\psi|A|\psi\rangle$ .

Properties

The map $A\mapsto\langle A\rangle_\rho$ is linear, positive, and normalized by $\langle I\rangle_\rho=1$ . For self-adjoint $A$ , its value is real and satisfies

|\langle A\rangle_\rho|\le \|A\|.

If $P$ is an orthogonal projection, then $\langle P\rangle_\rho$ is the Born probability of the outcome represented by $P$ .

Thermal expectation

At inverse temperature $\beta>0$ , the thermal expectation is computed with the quantum Gibbs state $\rho_\beta$ :

\langle A\rangle_\beta=\operatorname{Tr}(\rho_\beta A).