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A note on the \(p \rightarrow q\) norms of 2-positive maps. (English) Zbl 1163.47015

This paper is concerned with a mathematical question raised by C.King and M.B.Ruskai [Quantum Inf.Comput.4, No.6–7, 500–512 (2004; Zbl 1162.47307)], namely:
Is the \(p \to q\) norm of a completely positive map \(\Phi\) acting between Schatten \(p\) and \(q\) classes of Hermitian operators, \(\|\Phi\|_{p \to q}=\sup_{A=A^*}\frac{\|\Phi (A)\|_q}{\|A\|_p}\), equal to the \(p \to q\) norm of the map when acting between Schatten classes of general, not necessarily Hermitian, operators?
A general proof has been given by J.Watrous [Quantum Inf.Comput.5, No.1, 57–67 (2005; Zbl 1162.47308)] for any completely positive map and for all values \(p,q\geq 1\). In this paper, the author presents an alternative proof, which shows that the statement holds more generally for all 2-positive maps.

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47L90 Applications of operator algebras to the sciences
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
82B10 Quantum equilibrium statistical mechanics (general)
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References:

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[4] King, C.; Ruskai, M. B., Comments on multiplicativity of maximal \(p\)-norms when \(p = 2\), Quant. Inform. Comput., 4, 500-512 (2004) · Zbl 1162.47307
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[7] R. Schrader, Perron-Frobenius theory for positive maps on trace ideals, in: Roberto Longo (Ed.), Mathematical Physics in Mathematics and Physics: Quantum and Operator Algebraic Aspects, Fields Inst. Commun., vol. 30, Amer. Math. Soc., Providence, RI, 2001 (see also arXiv eprint math-ph/0007020).; R. Schrader, Perron-Frobenius theory for positive maps on trace ideals, in: Roberto Longo (Ed.), Mathematical Physics in Mathematics and Physics: Quantum and Operator Algebraic Aspects, Fields Inst. Commun., vol. 30, Amer. Math. Soc., Providence, RI, 2001 (see also arXiv eprint math-ph/0007020).
[8] Watrous, J., Notes on super-operator norms induced by Schatten norms, Quant. Inform. Comput., 5, 58-68 (2005) · Zbl 1162.47308
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