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Terminologically, quantum channels are completely positive (CP) trace-preserving maps between spaces of operators. In other words, a quantum channel is just a quantum operation viewed not merely as the reduced dynamics of a system but as a pipeline intended to carry quantum information. (Some authors use the term "quantum operation" to include ...
In order that a quantum operation preserve the set of density matrices, we need the additional assumption that it is trace-preserving. In the context of quantum information, the quantum operations defined here, i.e. completely positive maps that do not increase the trace, are also called quantum channels or stochastic maps. The formulation here ...
Choi's technique can be used to obtain a similar result for a more general class of maps. Φ is said to be Hermitian-preserving if A is Hermitian implies Φ(A) is also Hermitian. One can show Φ is Hermitian-preserving if and only if it is of the form = =
Every *-homomorphism is completely positive.[1]For every linear operator : between Hilbert spaces, the map () (), is completely positive. [2] Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
Let denote the space of Hermitian matrices, + denote the set consisting of positive semi-definite Hermitian matrices and + + denote the set of positive definite Hermitian matrices. For operators on an infinite dimensional Hilbert space we require that they be trace class and self-adjoint , in which case similar definitions apply, but we discuss ...
The Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs. [4] The Hermitian-adjacency matrix of a mixed graph is another important concept, as it is a Hermitian matrix that plays a role in studying the energies of mixed graphs. [5]