In functional analysis and quantum measurement theory, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalisation 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.
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,: 90
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 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.: 285
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 ,
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.: 285
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
will also have the property that , so that using the isometry
in the above construction will also implement the same POVM. In the case where the state being 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: 84
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.: 87 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 by 
Note that , so when outcome is obtained we are certain that the quantum state is , and when outcome is obtained we are certain that the quantum state is .
The probability of having a conclusive outcome is given by
Using the above construction we can obtain a projective measurement that physically realises this POVM. The square roots of the POVM elements are given by
Labelling the three possible states of the ancilla as , , , and initializing it on the state , we see that the resulting unitary takes the state together with the ancilla to
and similarly it takes the state together with the ancilla to
A measurement on the ancilla then gives the desired results with the same probabilities as the POVM.
This POVM has been used to experimentally distinguish non-orthogonal polarisation states of a photon, using the path degree of freedom as an ancilla. The realisation of the POVM with a projective measurement was slightly different from the one described here.
- Quantum measurement
- Mathematical formulation of quantum mechanics
- Density matrix
- Quantum operation
- Projection-valued measure
- Vector measure
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