Wikipedia / POVM
Wikipedia / POVM
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
denote a Hilbert space a measurable space a Borel σ-algebra whose values are positive bounded self-adjoint operators
is a non-negative countably additive is the identity operator.
In quantum mechanics when measuring a quantum state.
is finite-dimensional, a POVM is equivalently a set of positive semi-definite Hermitian matrices that sum to the identity matrix,
A POVM differs from a projection-valued measure are required to be orthogonal projections.
, such that the probability of obtaining it when making a quantum measurement on the quantum state is given by
,
is the trace this formula reduces to
.
The discrete case of a POVM generalizes the simplest case of a PVM, which is a set of orthogonal projectors that sum to the identity matrix:
of the PVM is at most the dimension of the Hilbert space.
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.
and an isometry,
, this isometry can be constructed as
.
In the general case, the isometry and PVM can be constructed by defining, and
, so this is a more wasteful construction.
with this PVM, and the state suitably transformed by the isometry, is the same as the probability of obtaining it with the original POVM:
such that
. This can always be done.
.
Post-measurement state
the operators
, so that using the isometry
takes it together with the ancilla to state
to the state
, the corresponding post-measurement state is given by
.
does not have to be Hermitian.
on a second measurement is
,
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
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, at the cost of sometimes having an inconclusive result. It is possible to do this with projective measurements..
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
where
.
The probability of having a conclusive outcome is given by
with the same probability. This result is known as the Ivanović-Dieks-Peres limit, named after the authors who pioneered UQSD research.
to
to
A projective measurement 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. The realisation of the POVM with a projective measurement was slightly different from the one described here.
See also
References
POVMs
K. Kraus, States, Effects, and Operations, Lecture Notes in Physics 190, Springer (1983).
A.S. Holevo, Probabilistic and statistical aspects of quantum theory, North-Holland Publ. Cy., Amsterdam (1982).
External links