Superoperator
Encyclopedia
In physics
, a superoperator is a linear operator acting on a vector space
of linear operators.
Sometimes the term refers more specially to a completely positive map which does not increase or preserves the trace
of its argument
.
This specialized meaning is used extensively in the field of quantum computing, especially quantum programming
, as they characterise mappings between density matrices
.
The use of the super- prefix here is in no way related to its other use in mathematical physics. That is to say superoperators have no connection to supersymmetry
and superalgebra
which are extensions of the usual mathematical concepts defined by extending the ring
of numbers to include Grassmann number
s. Since superoperators are themselves operators the use of the super- prefix is used to distinguish them from the operators upon which they act.
the Schrodinger Equation
,
expresses the time evolution of the state vector 
by the action of the Hamiltonian 
which is an operator mapping state vectors to state vectors.
In the more general formulation of John von Neumann
, statistical states and ensembles are expressed by density operators rather than state vectors.
In this context the time evolution of the density operator is expressed via the von Neumann equation in which density operator is acted upon by a superoperator
mapping operators to operators. It is defined by taking the commutator
with respect to the Hamiltonian operator:
where
As commutator brackets are used extensively in QM this explicit superoperator presentation of the Hamiltonian's action is typically omitted.
as for example when we define the quantum mechanical Hamiltonian of a particle as a function of the position and momentum operators, we may (for whatever reason) define an “Operator Derivative” 
as a superoperator mapping an operator to an operator.
For example if
then its operator derivative is the superoperator defined by:
This “operator derivative” is simply the Jacobian matrix of the function (of operators) where one simply treats the operator input and output as vectors and expands the space of operators in some basis. The Jacobian matrix is then an operator (at one higher level of abstraction) acting on that vector space (of operators).
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a superoperator is a linear operator acting on a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
of linear operators.
Sometimes the term refers more specially to a completely positive map which does not increase or preserves the trace
Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
of its argument
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
.
This specialized meaning is used extensively in the field of quantum computing, especially quantum programming
Quantum programming
Quantum programming is a set of computer programming languages that allow the expression of quantum algorithms using high-level constructs. The point of quantum languages is not so much to provide a tool for programmers, but to provide tools for researchers to understand better how quantum...
, as they characterise mappings between density matrices
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
.
The use of the super- prefix here is in no way related to its other use in mathematical physics. That is to say superoperators have no connection to supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
and superalgebra
Superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading....
which are extensions of the usual mathematical concepts defined by extending the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of numbers to include Grassmann number
Grassmann number
In mathematical physics, a Grassmann number, named after Hermann Grassmann, is a mathematical construction which allows a path integral representation for Fermionic fields...
s. Since superoperators are themselves operators the use of the super- prefix is used to distinguish them from the operators upon which they act.
Example von Neumann Equation
In quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
the Schrodinger Equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
,
expresses the time evolution of the state vector by the action of the Hamiltonian which is an operator mapping state vectors to state vectors.In the more general formulation of John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
, statistical states and ensembles are expressed by density operators rather than state vectors.
In this context the time evolution of the density operator is expressed via the von Neumann equation in which density operator is acted upon by a superoperator
mapping operators to operators. It is defined by taking the commutatorCommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
with respect to the Hamiltonian operator:
where
As commutator brackets are used extensively in QM this explicit superoperator presentation of the Hamiltonian's action is typically omitted.
Example Derivatives of Functions on the Space of Operators
When considering an operator valued function of operators as for example when we define the quantum mechanical Hamiltonian of a particle as a function of the position and momentum operators, we may (for whatever reason) define an “Operator Derivative” as a superoperator mapping an operator to an operator.
For example if
then its operator derivative is the superoperator defined by:This “operator derivative” is simply the Jacobian matrix of the function (of operators) where one simply treats the operator input and output as vectors and expands the space of operators in some basis. The Jacobian matrix is then an operator (at one higher level of abstraction) acting on that vector space (of operators).