itian operators {Em} with any Em being a posi- tive operator (meaning (VEm/Y) 0 for any normalized state 14)), and altogether satisfying the completeness relation (17.64) The operators Em that satisfy the above conditions are then said to be POVM operators. As we have seen in the previous section, any Hermitian operator A can be represented by a diagonal matrix with nonnegative coefficients, hence, there exists a diagonal oper- ator B = A, such that B² = A, and which is also Hermitian (B2 is real, SO Bii is). Therefore, for each Em we can associate a Hermitian operator Mm = that satisfies M+M, = Em, and consequently the two above POVM conditions. According to the general definition, the set {Mm} is, thus, a truly eligible set of measurement operators, even if their number, with respect to the number n of pure states, is not specified. The projection or von-Neumann measurements, with the set {Mm = 12m) (2ml} being defined by the n eigenstates of an observable, represent a specific case of POVM set. I shall now illustrate an important application of POVM, which no other measurement type can provide. Consider a simple example in the basis V = {|0), |1)}. Assume the 17.2 Quantum measurements and types Table 17.1 Probabilities associated with different POVM measurement possibilities when the input state is either /V/> = |0) (left) or 1/4/ = 1+) (right). Input state (v)==|0) p(1) = (0|E1|0) = 0 p(1) = (+|E1|+) N/A p(2) = (0|E2|0) = N/ N p(2) = (+|E2|+) : 0 p(3) = <0|(1 E p(3) = E1 E2)|- 1 - 2/ three POVM operators: =u|-)(- (17.65) E3=1-E1-E2, with = By summation of the above three equations, we immediately observe that the completeness relation is satisfied. It is also easily verified that the three operators E1, E2, E3 are positive (it is left as an exercise to prove that in the general case, the condition U < 2/3 must be satisfied). We have, thus, defined a valid POVM set of three measurement operators from the two-element basis V. The following will show how the above POVM measurement can be applied. Assume that the