Quantum expectation value

Let $\mathcal H$ be a finite-dimensional Hilbert space and let $A$ be an observable, represented as a self-adjoint element of the observable algebra (observable algebra ). Let $\rho$ be a quantum state given by a density operator (density-operator state ; compare density operator ).

The expectation value of $A$ in state $\rho$ is

where $\operatorname{Tr}$ is the operator trace (trace ).

In thermal equilibrium at inverse temperature $\beta>0$ (inverse temperature β ), one typically uses the Gibbs state (quantum Gibbs state ) $\rho_\beta$ and writes

This is the quantum analogue of the classical ensemble average (ensemble average ).

Physical interpretation

$\langle A\rangle_\rho$ is the theoretical prediction for the average outcome of many measurements of the observable $A$ on identically prepared systems in state $\rho$. In equilibrium, $\langle A\rangle_\beta$ is the thermal average at temperature $T=(k_B\beta)^{-1}$ (see temperature and Boltzmann constant ).

Key properties

1. Linearity.
   For scalars $a,b$ and observables $A,B$,

   $$\langle aA+bB\rangle_\rho \;=\; a\langle A\rangle_\rho + b\langle B\rangle_\rho.$$

2. Normalization and reality.
   $\langle I\rangle_\rho = 1$. If $A$ is self-adjoint, then $\langle A\rangle_\rho$ is real.

3. Positivity.
   If $A$ is positive semidefinite, then $\langle A\rangle_\rho \ge 0$. In particular, for a projector $P$, $\langle P\rangle_\rho$ is the probability that the corresponding yes/no measurement yields “yes”.

4. Bounds by operator norm.
   In finite dimension,

   $$|\langle A\rangle_\rho|\;\le\;\|A\|,$$

   where $\|A\|$ is the operator norm (see bounded operator for context).

5. Pure-state specialization.
   If $\rho=|\psi\rangle\langle\psi|$ is a pure state (pure state ), then

   $$\langle A\rangle_\rho \;=\;\langle\psi|A|\psi\rangle.$$

6. Parameter derivatives in Gibbs equilibrium.
   If the Hamiltonian depends on a parameter $\lambda$ via $H(\lambda)=H_0+\lambda V$, then the quantum partition function (quantum partition function ) satisfies

   $$\frac{\partial}{\partial\lambda}\log Z(\beta,\lambda)\;=\;-\beta\,\langle V\rangle_{\beta,\lambda}.$$

   Equivalently, the equilibrium free energy (statistical free energy ) obeys

   $$\frac{\partial}{\partial\lambda}F(\beta,\lambda)\;=\;\langle V\rangle_{\beta,\lambda}.$$

7. Gateway to correlations.
   Expectations of products and time-evolved observables define correlation functions (quantum correlation function ), which quantify fluctuations and response beyond mean values.