measured is in a pure state , the resulting unitary takes it together with the ancilla to state and the projective measurement on the ancilla will collapse to the state on obtaining result . When the state being measured is described by a density matrix , the corresponding post-measurement state is given by . We see therefore that the post-measurement state depends explicitly on the unitary . Note that while is always Hermitian, generally, does not have to be Hermitian. Another difference from the projective measurements is that a POVM measurement is in general not repeatable. If on the first measurement result was obtained, the probability of obtaining a different result on a second measurement is , which can be nonzero if and are not orthogonal. In a projective measurement these operators are always orthogonal and therefore the measurement is always repeatable. An example: unambiguous quantum state discrimination Suppose you have a quantum system with a 2-dimensional Hilbert space that you know is in either the state or the state , and you want to determine which one it is. If and are orthogonal, this task is easy: the set will form a PVM, and a projective measurement in this basis will determine the state with certainty. If, however, and are not orthogonal, this task is impossible, in the sense that there is no measurement, either PVM or POVM, that will distinguish them with certainty. The impossibility of perfectly discriminating between non-orthogonal states is the basis for quantum information protocols such as quantum cryptography, quantum coin flipping, and quantum money. The task of unambiguous quantum state discrimination (UQSD) is the next best thing: to never make a mistake about whether the state is or , at the cost of sometimes having an inconclusive result. It is possible to do this with projective measurements. For example, if you measure the PVM , where is the quantum state orthogonal to , and obtain result , then you know with certainty that the state was . If the result was , then it is inconclusive. The analogous reasoning holds for the PVM , where is the state orthogonal to . This is unsatisfactory, though, as you can't detect both and with a single measurement, and the probability of getting a conclusive result is smaller than with POVMs. The POVM that gives the highest probability of a conclusive outcome in this task is given