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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.
Moreover, the eigenvalues of this matrix are 1,1,1 and −1. (This matrix happens to be the Choi matrix of T , in fact.) Incidentally, a map Φ is said to be co-positive if the composition Φ ∘ {\displaystyle \circ } T is positive.
While the convex-sum decomposition, which is a sort of generalised eigenvalue decomposition since a gen-extreme Choi state can be mapped to a pure state, is difficult to solve for large-dimensional channels, it shall be comparable with the eigen-decomposition of the Choi state to find the set of Kraus operators. Both of the decompositions are ...
Choi Kyung-Ju (Korean: 최경주; born 19 May 1970), commonly known as K. J. Choi, is a South Korean professional golfer who currently plays on the PGA Tour Champions. Since turning pro in 1994, he has won more than twenty professional golf tournaments worldwide, including eight on the PGA Tour .
Overnight leader K.J. Choi won the Senior British Open by two shots over Australia’s Richard Green after carding a 2-under 70 on Sunday. The South Korean golfer, who took a one-shot lead into ...
There also exists an infinite-dimensional algebraic generalization of Choi's theorem, known as "Belavkin's Radon-Nikodym theorem for completely positive maps", which defines a density operator as a "Radon–Nikodym derivative" of a quantum channel with respect to a dominating completely positive map (reference channel). It is used for defining ...
In quantum information theory, the channel-state duality refers to the correspondence between quantum channels and quantum states (described by density matrices).Phrased differently, the duality is the isomorphism between completely positive maps (channels) from A to C n×n, where A is a C*-algebra and C n×n denotes the n×n complex entries, and positive linear functionals on the tensor product
The current proof of Choi's theorem has some problems (though it is essentially correct). The notation for the Choi matrix, (Φ(E ij)) ij is a bit confusing.It needs to mean (in matrix-of-matrix notation) the matrix whose ijth entry is Φ(E ij), but I'm not convinced this is the most obvious interpretation of (Φ(E ij)) ij.
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