TY - JOUR

T1 - Multiplicativity of completely bounded p-norms implies a new additivity result

AU - Devetak, Igor

AU - Junge, Marius

AU - King, Christoper

AU - Ruskai, Mary Beth

N1 - Copyright:
Copyright 2006 Elsevier B.V., All rights reserved.

PY - 2006/8

Y1 - 2006/8

N2 - We prove additivity of the minimal conditional entropy associated with a quantum channel Φ, represented by a completely positive (CP), trace-preserving map, when the infimum of S(γ12) - S(γ1) is restricted to states of the form (I ⊗ Φ)(|ψ » « ψ|). We show that this follows from multiplicativity of the completely bounded norm of Φ considered as a map from L 1 → L p for L p spaces defined by the Schatten p-norm on matrices, and give another 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.

AB - We prove additivity of the minimal conditional entropy associated with a quantum channel Φ, represented by a completely positive (CP), trace-preserving map, when the infimum of S(γ12) - S(γ1) is restricted to states of the form (I ⊗ Φ)(|ψ » « ψ|). We show that this follows from multiplicativity of the completely bounded norm of Φ considered as a map from L 1 → L p for L p spaces defined by the Schatten p-norm on matrices, and give another 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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U2 - 10.1007/s00220-006-0034-0

DO - 10.1007/s00220-006-0034-0

M3 - Article

AN - SCOPUS:33745606948

VL - 266

SP - 37

EP - 63

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -