Quantum steering

In physics, in the area of quantum information theory and quantum computation, quantum steering is a special kind of nonlocal correlations, which is intermediate between Bell nonlocality and quantum entanglement. A state exhibiting Bell nonlocality must also exhibit quantum steering, a state exhibiting quantum steering must also exhibit quantum entanglement. But for mixed quantum states, there exist examples which lie between these different quantum correlation sets. The notion was initially proposed by Schrödinger, and later made popular by Howard M. Wiseman, S. J. Jones, and A. C. Doherty.
Definition.
In the usual formulation of quantum steering, two distant parties, Alice and Bob, are considered, they share an unknown quantum state formula_1 with induced states formula_2 and formula_3 for Alice and Bob respectively. Alice and Bob can both perform local measurements on their own subsystems, for instance, Alice and Bob measure formula_4 and formula_5 and obtain the outcome formula_6 and formula_7. After running the experiment many times, they will obtain measurement statistics formula_8, this is just the symmetric scenario for nonlocal correlation. Quantum steering introduces some asymmetry between two parties, viz., Bob's measurement devices are trusted, he knows what measurement his device carried out, meanwhile, Alice's devices are untrusted. Bob's goal is to determine if Alice influences his states in a quantum mechanical way or just using some of her prior knowledge of his partial states and by some classical means. The classical way of Alice is known as the local hidden states model which is an extension of the local variable model for Bell nonlocality and also a restriction for separable states model for quantum entanglement.
Mathematically, consider Alice has a measurement assemblage formula_9 where each formula_10 is a set of POVM, formula_11, formula_12's are outcome of observable formula_10. The Bob's states assemblage corresponding to Alice's measurement assemblage formula_9 is formula_15 where each formula_16 is non-negative and formula_17 and formula_18. Similar as in the case of quantum entanglement, to define entanglement states, we must define the unentangled states (separable states), here we need to introduce the local hidden states assemblage formula_19 for which formula_20, formula_21's are non-negative and formula_22. We say that a state is un-steerable if for arbitrary measurement assemblage formula_9 and state assemblage formula_15 there exists a local hidden state assemblage formula_19 such that formula_26 for all formula_12 and formula_10. A state is called a steering state if it is not un-steerable.
Local hidden state model.
Let us do some comparison among Bell nonlocality, quantum steering, and quantum entanglement. By definition, a Bell nonlocal which does not admit a local hidden variable model for some measurement setting, a quantum steering state is a state which does not admit a local hidden state model for some measurement assemblage and state assemblage, and quantum entangled state is a state which is not separable. They share a great similarity.