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New multiplicativity results for qubit maps. (English) Zbl 1111.81036

Summary: Let \(\Phi\) be a trace-preserving, positivity-preserving (but not necessarily completely positive) linear map on the algebra of complex \(2\times 2\) matrices, and let \(\Omega\) be any finite-dimensional completely positive map. For \(p=2\) and \(p\geq 4\), we prove that the maximal \(p\)-norm of the product map \(\Phi\otimes\Omega\) is the product of the maximal \(p\)-norms of \(\Phi\) and \(\Omega\). Restricting \(\Phi\) to the class of completely positive maps, this settles the multiplicativity question for all qubit channels in the range of values \(p\geq 4\).

MSC:

81P68 Quantum computation
15A90 Applications of matrix theory to physics (MSC2000)
81P15 Quantum measurement theory, state operations, state preparations
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