Density Matrix - Measurement

Measurement

Let A be an observable of the system, and suppose the ensemble is in a mixed state such that each of the pure states occurs with probability p_j. Then the corresponding density operator is:

The expectation value of the measurement can be calculated by extending from the case of pure states (see Measurement in quantum mechanics):

where denotes trace. Moreover, if A has spectral resolution

where, the corresponding density operator after the measurement is given by:

Note that the above density operator describes the full ensemble after measurement. The sub-ensemble for which the measurement result was the particular value a_i is described by the different density operator

This is true assuming that is the only eigenket (up to phase) with eigenvalue a_i; more generally, P_i in this expression would be replaced by the projection operator into the eigenspace corresponding to eigenvalue a_i.