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

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

2005-06-23

arxiv.org/abs/quant-ph/0506196v2

Commun. Math. Phys. 266, 37-63 (2006)

Physics

Quantum Physics

Final version for Commun. Math. Physics. Section 5.2 of previous version deleted in view of the results in quant-ph/0601071 Ot

Scientific paper

10.1007/s00220-006-0034-0

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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