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

@article{Devetak2006MultiplicativityOC,
title={Multiplicativity of Completely Bounded p-Norms Implies a New Additivity Result},
author={Igor Devetak and Marius Junge and Christopher King and Mary Beth Ruskai},
journal={Communications in Mathematical Physics},
year={2006},
volume={266},
pages={37-63}
}
• I. Devetak, +1 author M. Ruskai
• Published 2006
• Mathematics, Physics
• Communications in Mathematical Physics
AbstractWe 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 $$(\mathcal{I} \otimes \Phi)\left( | \psi \rangle \langle \psi | \right)$$. We show that this follows from multiplicativity of the completely bounded norm of Φ considered as a map from L1 → Lp for Lp spaces defined by the Schatten p-norm on matrices, and give… Expand
A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity
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Minimal entropy of states emerging from noisy quantum channels
• Mathematics, Computer Science
• IEEE Trans. Inf. Theory
• 2001
It is proved that for a tensor product of two unital stochastic maps on qubit states, using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states. Expand
Non-commutative vector valued Lp-spaces and completely p-summing maps
Let $E$ be an operator space in the sense of the theory recently developed by Blecher-Paulsen and Effros-Ruan. We introduce a notion of $E$-valued non commutative $L_p$-space for $1 \leq p < \infty$Expand
A Note on the p->q norms of Completely Positive Maps
King and Ruskai asked whether the $p\to q$ norm of a completely positive map $\Phi$, acting between Schatten $p$ and $q$ classes of self-adjoint operators, $||\Phi||_{p\to q} = \max_{A=A^*}Expand On Some Additivity Problems in Quantum Information Theory • Physics, Mathematics • 2000 A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantumExpand Inequalities for the Moments of the Eigenvalues of the Schrodinger Hamiltonian and Their Relation to Sobolev Inequalities • Physics • 2002 Estimates for the number of bound states and their energies, ej ≤ 0, are of obvious importance for the investigation of quantum mechanical Hamiltonians. If the latter are of the single particle formExpand A relation between completely bounded norms and conjugate channels We show a relation between a quantum channel$\Phi$and its conjugate$\Phi^C$, which implies that the$p\to p$Schatten norm of the channel is the same as the$1\to p\$ completely bounded norm of theExpand
Counterexample to an additivity conjecture for output purity of quantum channels
• Mathematics, Physics
• 2002
A conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel, as measuredExpand