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 pj. 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 ai is described by the different density operator
This is true assuming that is the only eigenket (up to phase) with eigenvalue ai; more generally, Pi in this expression would be replaced by the projection operator into the eigenspace corresponding to eigenvalue ai.
Read more about this topic: Density Matrix
Famous quotes containing the word measurement:
“That’s the great danger of sectarian opinions, they always accept the formulas of past events as useful for the measurement of future events and they never are, if you have high standards of accuracy.”
—John Dos Passos (1896–1970)