New multiplicativity results for qubit maps

Abstract

Let $\Phi$ be a trace-preserving, positivity-preserving (but not necessarily completely positive) linear map on the algebra of complex $2\times 2$ matrices, and let $\Omega$ be any finite-dimensional completely positive map. For $p=2$ and $p\ge 4$, we prove that the maximal $p$-norm of the product map $Φ\ot Ω$ is the product of the maximal $p$-norms of $\Phi$ and $\Omega$. Restricting $\Phi$ to the class of completely positive maps, this settles the multiplicativity question for all qubit channels in the range of values $p\ge 4$.

14 pages; original proof simplified by using Gorini and Sudarshan's classification of extreme affine maps on R^3

References in corpus (7)

• Additivity for a class of unital qubit channels
• Additivity of the Classical Capacity of Entanglement-Breaking Quantum Channels
• Counterexample to an additivity conjecture for output purity of quantum channels
• On some additivity problems in quantum information theory
• Maximization of capacity and p-norms for some product channels
• On the multiplicativity conjecture for quantum channels
• Extending additivity from symmetric to asymmetric channels

Cited by in corpus (4)

• Qubit quantum channels: A characteristic function approach
• Exploring Non-Multiplicativity in the Geometric Measure of Entanglement
• Notes on multiplicativity of maximal output purity for completely positive qubit maps
• An application of decomposable maps in proving multiplicativity of low dimensional maps