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Multiplicativity of completely bounded p-norms implies a new additivity result

Igor Devetak·Marius Junge·Christopher King·Mary Beth Ruskai
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Abstract

We prove additivity of the minimal conditional entropy associated with a quantum channel Phi, represented by a completely positive (CP), trace-preserving map, when the infimum of S(gamma_{12}) - S(gamma_1) is restricted to states of the form gamma_{12} = (I \ot Phi)(| psi >< psi |). We show that this follows from multiplicativity of the completely bounded norm of Phi considered as a map from L_1 -> L_p for L_p spaces defined by the Schatten p-norm on matrices; we also give an independent proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L_1 -> L_p norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.

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Igor Devetak, Marius Junge, Christopher King, Mary Beth Ruskai. 2006-01-23. Multiplicativity of completely bounded p-norms implies a new additivity result. https://doi.org/10.1007/s00220-006-0034-0

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