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Quantum Theory
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
The quantum confusability graph V(E) is a vector space of operators on a Hilbert space. There is no obvious connection to the vertices and edges of classical graphs we introduced earlier, so we will rely on a few theorems to motivate this definition. We should point out some key properties of the quantum confusability graph. First, it is closed under the dagger operation. Indeed, this follows from the fact that (Ei†Ej)†=Ej†Ei∈V(E). Also, V(E) contains the identity operator I since ∑kEk†Ek=I. In general, any subspace V of the operators on a Hilbert space that contains I and is closed under the dagger operation is called a quantum graph4. Like classical graphs, every quantum graph can be realized as the quantum confusability graph of some quantum channel, though many different quantum channels can have the same quantum confusability graph.
Numerical estimates of the essential spectra of quantum graphs with delta-interactions at vertices
Published in Applicable Analysis, 2019
Víctor Barrera-Figueroa, Vladimir S. Rabinovich, Miguel Maldonado Rosas
In the modern terminology, a quantum graph is a mathematical object that comprises a metric graph , differential or pseudo-differential operators on the edges of the graph , and conditions at its vertices (see, e.g. [1]). Vertex conditions are usually chosen in order for the associated unbounded operator with appropriate domain to be self-adjoint in .
Parameter dependent differential operators on graphs and their applications
Published in Applicable Analysis, 2021
A quantum graph is a metric graph endowed with differential or, more generally, pseudodifferential operators on the edges and general connection conditions at the vertices. They have a big mathematical interest and arise as simplified models in many important processes in physics, chemistry, and engineering. We mention, in particular, the free-electron theory of conjugated molecules, photonic crystals, carbon nano structures, thin quantum and electromagnetic waveguides, electric and neuronal networks, etc. We refer to the following reviews, books, and papers on this topic [1–12].