A symmetric, informationally complete, positive operator-valued measure (SIC-
POVM
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) an ...
) is a special case of a generalized
measurement
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, used in the field of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. A measurement of the prescribed form satisfies certain defining qualities that makes it an interesting candidate for a "standard quantum measurement", utilized in the study of foundational quantum mechanics, most notably in
QBism
In physics and the philosophy of physics, quantum Bayesianism is a collection of related approaches to the interpretation of quantum mechanics, of which the most prominent is QBism (pronounced "cubism"). QBism is an interpretation that takes an a ...
. Furthermore, it has been shown that applications exist in
quantum state tomography and
quantum cryptography
Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. The best known example of quantum cryptography is quantum key distribution which offers an information-theoretically secure solution ...
, and a possible connection has been discovered with
Hilbert's twelfth problem
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analogue ...
.
Definition
Due to the use of SIC-POVMs primarily in quantum mechanics,
Dirac notation
Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety of ...
will be used throughout this article to represent elements in a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
.
A POVM over a
-dimensional Hilbert space
is a set of
positive-semidefinite operators that sum to the
identity:
If a POVM consists of at least
operators which
span the space of self-adjoint operators
, it is said to be an informationally complete POVM (IC-POVM). IC-POVMs consisting of exactly
elements are called minimal. A set of
rank-1
projectors which have equal pairwise
Hilbert–Schmidt inner product In mathematics, Hilbert–Schmidt may refer to
* a Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H ...
s,
defines a minimal IC-POVM with elements
called a SIC-POVM.
Properties
Symmetry
The condition that the projectors
defined above have equal pairwise inner products actually fixes the value of this constant. Recall that
and set
. Then
implies that
. Thus,
This property is what makes SIC-POVMs ''symmetric''; with respect to the
Hilbert–Schmidt inner product In mathematics, Hilbert–Schmidt may refer to
* a Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H ...
, any pair of elements is equivalent to any other pair.
Superoperator
In using the SIC-POVM elements, an interesting superoperator can be constructed, the likes of which map
. This operator is most useful in considering the
relation of SIC-POVMs with spherical t-designs. Consider the map
:
This operator acts on a SIC-POVM element in a way very similar to identity, in that
:
But since elements of a SIC-POVM can completely and uniquely determine any quantum state, this linear operator can be applied to the decomposition of any state, resulting in the ability to write the following:
:
where
From here, the
left inverse can be calculated to be