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Genericcoder
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Can you guys give me a concrete example of a completely positive map from M_m → M_n?
Genericcoder said:Can you guys give me a concrete example of a completely positive map from M_m → M_n?
A completely positive map is a linear transformation between two matrix spaces which preserves positive semidefinite matrices. This means that if the input matrix is positive semidefinite, then the output matrix will also be positive semidefinite.
An example of a completely positive map from Mn to Mm is the transpose map, which takes an n x n matrix and produces an n x n matrix with its rows and columns switched. This map preserves positive semidefinite matrices because the transpose of a positive semidefinite matrix is also positive semidefinite.
A completely positive map is a stronger condition than a positive map. While a positive map only requires that the output matrix is positive semidefinite, a completely positive map also requires that the map preserves positivity under tensor products. In other words, if we apply the map to a tensor product of matrices, the resulting matrix should still be positive semidefinite.
Yes, a completely positive map can be extended to a larger matrix space. This is because the map preserves positive semidefinite matrices, so any new matrices added to the larger space will still be mapped to positive semidefinite matrices.
In quantum mechanics, completely positive maps are used to describe the evolution of density matrices, which represent the state of a quantum system. These maps are also used to describe the process of quantum measurement, as they preserve the positivity of the measured quantity.