Contractivity of positive and trace-preserving maps under Lp norms

2006/01/30 by David Pérez-Garcı́a, David Perez-Garcia, Michael M. Wolf +3 · 141 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebra over a field #Combinatorics #Domain (mathematical analysis) #Eigenvalues and eigenvectors #Hermitian matrix #Mathematical analysis #Mathematics #Matrix Theory and Algorithms #Physics #Positive-definite matrix #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #Spectral Theory in Mathematical Physics #Subspace topology #TRACE (psycholinguistics) #Unital #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1063/1.2218675

published in Journal of Mathematical Physics 47(8) (American Institute of Physics)

arxiv created 2006/01/30 · openalex publication_date 2006/08/01 · arxiv updated 2015/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

We provide a complete picture of contractivity of trace preserving positive maps with respect to p-norms. We show that for p>1 contractivity holds in general if and only if the map is unital. When the domain is restricted to the traceless subspace of Hermitian matrices, then contractivity is shown to hold in the case of qubits for arbitrary p⩾1 and in the case of qutrits if and only if p=1, ∞. In all noncontractive cases best possible bounds on the p-norms are derived.

Citations

• Multiplicativity of Completely Bounded p-Norms Implies a New Additivity Result
• Entropy production rates of bistochastic strictly contractive quantum channels on a matrix algebra
• Quantifying Entanglement

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