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 also preserves or does not increase 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 numbers. Since superoperators are themselves operators the use of the super- prefix is used to distinguish them from the operators upon which they act.
Left/right multiplication Fix a choice of basis for the underlying Hilbert space { | i ⟩ } i {\displaystyle \{|i\rangle \}_{i}} . Defining the left and right multiplication superoperators by L ( A ) [ ρ ] = A ρ {\displaystyle {\mathcal {L}}(A)[\rho ]=A\rho } and R ( A ) [ ρ ] = ρ A {\displaystyle {\mathcal {R}}(A)[\rho ]=\rho A} respectively one can express the commutator as
[ A , ρ ] = L ( A ) [ ρ ] − R ( A ) [ ρ ] . {\displaystyle [A,\rho ]={\mathcal {L}}(A)[\rho ]-{\mathcal {R}}(A)[\rho ].}
Next we vectorize the matrix ρ {\displaystyle \rho } which is the mapping
ρ = ∑ i , j ρ i j | i ⟩ ⟨ j | ↦ | ρ ⟩ ⟩ = ∑ i , j ρ i j | i ⟩ ⊗ | j ⟩ , {\displaystyle \rho =\sum _{i,j}\rho _{ij}|i\rangle \langle j|\mapsto |\rho \rangle \!\rangle =\sum _{i,j}\rho _{ij}|i\rangle \otimes |j\rangle ,}
where | ⋅ ⟩ ⟩ {\displaystyle |\cdot \rangle \!\rangle } denotes a vector in the Fock-Liouville space. The matrix representation of L ( A ) {\displaystyle {\mathcal {L}}(A)} is then calculated by using the same mapping
A ρ = ∑ i , j ρ i j A | i ⟩ ⟨ j | ↦ ∑ i , j ρ i j ( A | i ⟩ ) ⊗ | j ⟩ = ∑ i , j ρ i j ( A ⊗ I ) ( | i ⟩ ⊗ | j ⟩ ) = ( A ⊗ I ) | ρ ⟩ ⟩ = L ( A ) [ ρ ] , {\displaystyle A\rho =\sum _{i,j}\rho _{ij}A|i\rangle \langle j|\mapsto \sum _{i,j}\rho _{ij}(A|i\rangle )\otimes |j\rangle =\sum _{i,j}\rho _{ij}(A\otimes I)(|i\rangle \otimes |j\rangle )=(A\otimes I)|\rho \rangle \!\rangle ={\mathcal {L}}(A)[\rho ],}
indicating that L ( A ) = A ⊗ I {\displaystyle {\mathcal {L}}(A)=A\otimes I} . Similarly one can show that R ( A ) = ( I ⊗ A T ) {\displaystyle {\mathcal {R}}(A)=(I\otimes A^{T})} . These representations allows us to calculate things like eigenvalues associated to superoperators. These eigenvalues are particularly useful in the field of open quantum systems, where the real parts of the Lindblad superoperator's eigenvalues will indicate whether a quantum system will relax or not.
Examples
Von Neumann's equation In quantum mechanics the Schrödinger equation,
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