Multiplicativity of Completely Bounded p-Norms Implies a New Additivity Result

2005/06/30 by Igor Devetak, Marius Junge, Christopher King +2 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Mathematical Inequalities and Applications #Spectral Theory in Mathematical Physics #math.OA #quant-ph

paper · pdf · doi:10.1007/s00220-006-0034-0

published as Commun. Math. Phys. 266, 37-63 (2006) · Final version for Commun. Math. Physics. Section 5.2 of previous version deleted in view of the results in quant-ph/0601071 Other changes minor

arxiv created 2006/01/23 · openalex publication_date 2006/05/08 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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(gamma₁₂) - S(gamma₁) is restricted to states of the form gamma₁₂ = (I \ot Phi)(| psi >< psi |). We show that this follows from multiplicativity of the completely bounded norm of Phi considered as a map from L₁ -> 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 L₁ -> 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.

Citations

• Capacity of the noisy quantum channel
• Qubit Entanglement Breaking Channels
• Completely positive linear maps on complex matrices
• General Entanglement Breaking Channels

Cited by

• A matrix convexity approach to some celebrated quantum inequalities
• Completely Bounded Qusi-Norms, Their Mutiplicativity, and New Additivity Results of Quantum Channels