Positive-Operator Valued Measure (POVM)

A positive-operator valued measure (POVM), in earlier days also refereed to as probability-operator measure (POM), is the most general formulation of a measurement in the theory of quantum physics. The need for POVMs arises from the fact that usual projective measurements or projection-valued measure (PVM) on sub-systems of larger systems cannot be described by a PVM acting on the subsystem alone. So in very simple terms a POVM is to a PVM what a density matrix is to a pure state. A POVM determines only the probability, that is course positive, of each measurement outcome, ignoring the postmeasurement state. A POVM is a set of Hermitian positive semidefinite operators on a Hilbert space that sum to the identity operator, i.e., . For a state (pure with or mixed), the probability of obtaining the result associated with is and .

Let’s take a look at two simple examples: Consider a measurement apparatus that measures a single qubit. However, the apparatus may fail with probability . In that case it does not interact with the qubit and therefore returns no measurement result. There is no projective description of the measurement, instead it is described a POVM consiting of three elements given by , and (they obviously satisfy the completness relation ). The second example is a little bit more complicated an taken from the famous textbook Nielsen & Chuang ¹, p.92: , and . It is easy to calculate that the three elements sum up to identity. Furthermore, we have a set of Hermitian positive semidefinite operators and therefore a legitimate POVM. Although it might seem strange at first sight, this POVM has a very practical purpose, namely it allows one to discriminate between the state and , since these two states are not orthogonal there is no projective measurement that can completely distinguishes them). If we calculate the probabilities when was sent we get and if was sent we get . So if the resulte is we can safely say that was sent. The cost for this discrimination is that if we get a result that corresponds to , which is the most likely result for more than half of our trails, then the measurement tells us nothing at all.

1. Chuang, Michael A. Nielsen & Isaac L. (2001). Quantum computation and quantum information Cambridge Univ. Press. ISBN 978-0521635035 ↩.