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| <ref>[http://www.theory.caltech.edu/~preskill/ph229]</ref>{{Refimprove|date=October 2009}}
| | Oscar is how he's known as and he totally loves this name. Minnesota is where he's been residing for many years. Doing ceramics is what her family and her appreciate. She is a librarian but she's always needed her personal company.<br><br>Here is my site - [http://www.animecontent.com/blog/349119 over the counter std test] |
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| In [[physics]], a '''superoperator''' is a [[linear operator]] acting on a [[vector space]] of [[Linear map|linear operators]].
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| Sometimes the term refers more specially to a [[completely positive map]] which does not increase or preserves the [[trace (linear algebra)|trace]] of its [[Parameter|argument]].
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| This specialized meaning is used extensively in the field of [[quantum computing]], especially [[quantum programming]], as they characterise mappings between [[density matrix|density matrices]].
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| 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_(mathematics)|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.
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| ==Example von Neumann Equation==
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| In [[quantum mechanics]] the [[Schrödinger Equation]], <math>i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi</math> expresses the time evolution of the state vector <math>\psi</math> by the action of the Hamiltonian <math>\hat{H}</math> which is an operator mapping state vectors to state vectors.
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| In the more general formulation of [[John von Neumann]], statistical states and ensembles are expressed by [[density operator]]s rather than state vectors.
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| 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''' <math>\mathcal{H}</math> mapping operators to operators. It is defined by taking the [[commutator]] with respect to the Hamiltonian operator:
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| <math>i \hbar \frac{\partial}{\partial t}\rho = \mathcal{H}[\rho]</math>
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| where
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| <math>\mathcal{H}[\rho] = [\hat{H},\rho] \equiv \hat{H}\rho - \rho\hat{H}</math>
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| As commutator brackets are used extensively in QM this explicit superoperator presentation of the Hamiltonian's action is typically omitted.
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| ==Example Derivatives of Functions on the Space of Operators==
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| When considering an operator valued function of operators <math>\hat{H} = \hat{H}(\hat{P})</math> 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” <math> \frac{\Delta \hat{H}}{\Delta \hat{P}} </math>
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| as a '''superoperator''' mapping an operator to an operator.
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| For example if <math> H(P) = P^3 = PPP</math> then its operator derivative is the superoperator defined by:
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| <math> \frac{\Delta H}{\Delta P}[X] = X P^2 + PXP + P^2X</math>
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| 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).
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| ==See also==
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| [[Lindblad superoperator]]
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| ==References==
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| {{Reflist}}
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| [[Category:Quantum information theory]]
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Oscar is how he's known as and he totally loves this name. Minnesota is where he's been residing for many years. Doing ceramics is what her family and her appreciate. She is a librarian but she's always needed her personal company.
Here is my site - over the counter std test