Annihilating and breaking Lorentz cone entanglement

Abstract

Linear maps between finite-dimensional ordered vector spaces with orders induced by proper cones $C_A$ and $C_B$ are called entanglement breaking if their partial application sends the maximal tensor product $K\otimes_{\max} C_A$ into the minimal tensor product $K\otimes_{\min} C_B$ for any proper cone $K$. We study the larger class of Lorentz-entanglement breaking maps where $K$ is restricted to be a Lorentz cone of any dimension, i.e., any cone over a Euclidean ball. This class of maps appeared recently in the study of asymptotic entanglement annihilation and it is dual to the linear maps factoring through Lorentz cones. Our main results establish connections between these classes of maps and operator ideals studied in the theory of Banach spaces. For operators $u:X\rightarrow Y$ between finite-dimensional normed spaces $X$ and $Y$ we consider so-called central maps which are positive with respect to the cones $C_A=C_X$ and $C_B=C_Y$. We show how to characterize when such a map factors through a Lorentz cone and when it is Lorentz-entanglement breaking by using the Hilbert-space factorization norm $\gamma_2$ and its dual $\gamma^*_2$. We also study the class of Lorentz-entanglement annihilating maps whose local application sends the Lorentzian tensor product $C_A\otimes_{L} C_A$ into the minimal tensor product $C_B\otimes_{\min} C_B$. When $C_A$ is a cone over a finite-dimensional normed space and $C_B$ is a Lorentz cone itself, the central maps of this kind can be characterized by the $2$-summing norm $\pi_2$. Finally, we prove interesting connections between these classes of maps for general cones, and we identify examples with particular properties, e.g., cones with an analogue of the $2$-summing property.