### Abstract

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(γ_{12}) - S(γ_{1}) 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.

Original language | English (US) |
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Pages (from-to) | 37-63 |

Number of pages | 27 |

Journal | Communications in Mathematical Physics |

Volume | 266 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1 2006 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications in Mathematical Physics*,

*266*(1), 37-63. https://doi.org/10.1007/s00220-006-0034-0