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In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin 's "Radon–Nikodym" theorem for completely positive maps.
Mathematically, a quantum operation is a linear map Φ between spaces of trace class operators on Hilbert spaces H and G such that. If S is a density operator, Tr (Φ (S)) ≤ 1. Φ is completely positive, that is for any natural number n, and any square matrix of size n whose entries are trace-class operators and which is non-negative, then is ...
Quantum channel. In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the general dynamics of a qubit. An example of classical information is a text document transmitted over the Internet.
The Choi-Jamiołkowski isomorphism is a mathematical concept that connects quantum gates or operations to quantum states called Choi states. It allows us to represent a gate's properties and behavior as a Choi state. In the generalised gate teleportation scheme, we can teleport a quantum gate from one location to another using entangled states ...
Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps. Every positive functional (in particular every state) is automatically completely positive. Given the algebras and of complex-valued continuous functions on compact Hausdorff spaces , every positive map is completely positive.
Quantum depolarizing channel. A quantum depolarizing channel is a model for quantum noise in quantum systems. The -dimensional depolarizing channel can be viewed as a completely positive trace-preserving map , depending on one parameter , which maps a state onto a linear combination of itself and the maximally mixed state, The condition of ...
These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2, C).
The Lorentz group is a subgroup of the Poincaré group —the group of all isometries of Minkowski spacetime. Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is the isotropy subgroup with respect to the origin of the isometry group of Minkowski spacetime.