POVMs and Measurement
As in the theory of projective measurement, the probability that the outcome associated with measurement of operator occurs is
where is the density matrix of the measured system.
Such a measurement can be carried out by doing a projective measurement in a larger Hilbert space. Let us extend the Hilbert space to and perform the measurement defined by the projection operators . The probability of the outcome associated with is
where is the orthogonal projection taking to . In the original Hilbert space, this is a POVM with operators given by . Neumark's dilation theorem guarantees that any POVM can be implemented in this manner.
In practice, POVMs are usually performed by coupling the original system to an ancilla. For an ancilla prepared in a pure state, this is a special case of the above; the Hilbert space is extended by the states where .
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Famous quotes containing the word measurement:
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