In functional analysis and quantum information science, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalization of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalization of quantum measurement described by PVMs (called projective measurements).
In rough analogy, a POVM is to a PVM what a mixed state is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see purification of quantum state); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system.
POVMs are the most general kind of measurement in quantum mechanics, and can also be used in quantum field theory. They are extensively used in the field of quantum information.
Definition
In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, a POVM is a set of positive semi-definite Hermitian matrices on a Hilbert space that sum to the identity matrix,
In quantum mechanics, the POVM element is associated with the measurement outcome , such that the probability of obtaining it when making a measurement on the quantum state is given by
,
where is the trace operator. When the quantum state being measured is a pure state this formula reduces to
.
The simplest case of a POVM generalises the simplest case of a PVM, which is a set of orthogonal projectors that sum to the identity matrix:
The probability formulas for a PVM are the same as for the POVM. An important difference is that the elements of a POVM are not necessarily orthogonal. As a consequence, the number of elements of the POVM can be larger than the dimension of the Hilbert space they act in. On the other hand, the number of elements of the PVM is at most the dimension of the Hilbert space.
In general, POVMs can also be defined in situations where the number of elements and the dimension of the Hilbert space is not finite:
Definition. Let be measurable space; that is is a σ-algebra of subsets of . A POVM is a function defined on whose values are bounded non-negative self-adjoint operators on a Hilbert space such that and for every ,
is a non-negative countably additive measure on the σ-algebra .
Its key property is that it determines a probability measure on the outcome space, so that can be interpreted as the probability (density) of outcome when making a measurement on the quantum state .
This definition should be contrasted with that of the projection-valued measure, which is similar, except that for projection-valued measures, the values of are required to be projection operators.
Naimark's dilation theorem
Note: An alternate spelling of this is "Neumark's Theorem"
Naimark's dilation theorem shows how POVMs can be obtained from PVMs acting on a larger space. This result is of critical importance in quantum mechanics, as it gives a way to physically realize POVM measurements.
In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, Naimark's theorem says that if is a POVM acting on a Hilbert space of dimension , then there exists a PVM acting on a Hilbert space of dimension and an isometry such that for all ,
One way to construct such a PVM and isometry is to let , , and
The probability of obtaining outcome with this PVM, and the state suitably transformed by the isometry, is the same as the probability of obtaining it with the original POVM:
This construction can be turned into a recipe for a physical realisation of the POVM by extending the isometry into a unitary , that is, finding such that
This can always be done. The recipe for realizing the POVM measurement described by on a quantum state is then to prepare an ancilla in the state , evolve it together with through the unitary , and make the projective measurement on the ancilla described by the PVM .
Note that in this construction the dimension of the larger Hilbert space is given by . This is not the minimum possible, as a more complicated construction gives for rank-1 POVMs.
Post-measurement state
The post-measurement state is not determined by the POVM itself, but rather by the PVM that physically realizes it. Since there are infinitely many different PVMs that realize the same POVM, the operators alone do not determine what the post-measurement state will be. To see that, note that for any unitary the operators