Operator space approach to steering inequality

In Junge and Palazuelos (2011 Commun. Math. Phys. 306 695-746) and Junge et al (2010 Commun. Math. Phys. 300 715-39) the operator space theory was applied to study bipartite Bell inequalities. The aim of the paper is to follow this line of research and use the operator space technique to analyze the steering scenario. We obtain a bipartite steering functional with unbounded largest violation of steering inequality, as well as constructing all ingredients explicitly. It turns out that the unbounded largest violation is obtained by a non maximally entangled state. Moreover, we focus on the bipartite dichotomic case where we construct a steering functional with unbounded largest violation of steering inequality. This phenomenon is different to the Bell scenario where only the bounded largest violation can be obtained by any bipartite dichotomic Bell functional.