Density Operator

In this section we show that ensemble averages for both pure and mixed states can be computed as follows,

$$A = Tr\{\hat{\rho} \hat{A}\},$$ (10)

where $\hat{\rho}$ is the density operator

$$\hat{\rho} = \sum_k p_k \vert\phi_k><\phi_k\vert.$$ (11)

Note that, in particular, the density operator of an ensemble where all of the replica systems are described by the same state vector $\vert\psi>$ (i.e., a pure state) is

$$\hat{\rho} = \vert\psi><\psi \vert.$$ (12)

Eq. (10) can be proved first for a pure state $\vert\psi> = \sum_k a_k \vert\phi_k >$, where $\mid \phi_k \rangle$ constitute a complete basis set of orthonormal states (i.e., $<\phi_{k'}\vert\phi_k>=\delta_{kk'}$), by computing the Tr$\{\hat{\rho} \hat{A}\}$ in such representation as follows,

$$A = \sum_{k'} <\phi_{k'}\vert\psi><\psi\vert\hat{A}\vert\phi_{k'}>.$$ (13)

Substituting the expansion of $\mid \psi \rangle$ into Eq. (13) we obtain,

$$A = \sum_{k'} \sum_j \sum_k <\phi_{k'}\vert\phi_k> a_k a_j^* <\phi_j\vert\hat{A}\vert\phi_{k'}>,$$ (14)

and since $<\phi_{k'}\vert\phi_k>=\delta_{kk'}$,

$$A = \sum_k p_k <\phi_k\vert\hat{A}\vert\phi_k> + \sum_k \sum_{j \neq k} \sqrt{p_k p_j} e^{i(\theta_k - \theta_j)} <\phi_j\vert\hat{A}\vert\phi_k>,$$ (15)

where we have substituted the expansion coefficients $a_j$ in accord with Eq. (4). Equation (15) is identical to Eq. (6) and, therefore, Eq. (10) is identical to Eq. (6) which defines an ensemble average for a pure state. Eq. (10) can also be proved for an arbitrary mixed state defined by the density operator introduced by Eq. (11), by computing the Tr$\{\hat{\rho} \hat{A}\}$ as follows,

$$A = \sum_{k'} \sum_k p_k <\phi_{k'}\vert\phi_k> <\phi_k\vert\hat{A}\vert\phi_{k'}> = \sum_k p_k <\phi_k \vert\hat{A}\vert \phi_k>,$$ (16)

which is identical to Eq. (7).

Exercise 2:
(A) Show that Tr$\{\hat{\rho}\}$=1 for both mixed and pure states.
(B) Show that Tr$\{\hat{\rho}^2\}$=1 for pure states.
(C) Show that Tr$\{\hat{\rho}^2\} \leq 1$ for mixed states.

Note that the Tr$\{\hat{\rho}^2\}$ is, therefore, a measurement of decoherence (i.e., lost of interference between the various different states in the ensemble). When the system is in a coherent superposition state, such as the one described by Eq. (2), Tr$\{\hat{\rho}^2\}$=1. However, Tr$\{\hat{\rho}^2\} \le$1 when the system is in an incoherent superposition of states such as the one described by Eq. (9).