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 exhibits 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 $\rho$ with induced states $\rho _{A}$ and $\rho _{B}$ for Alice and Bob respectively. Alice and Bob can both perform local measurements on their own subsystems, for instance, Alice and Bob measure $x$ and $y$ and obtain the outcome $a$ and $b$. After running experiment many times, they will obtain a measurement statistics $p(a,b|x,y)$, 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 ${\mathcal {M}}=\{x_{1},\cdots ,x_{n}\}$ where each $x_{i}$ is a set of POVM, $\{a_{i}|x_{i}|a_{i}=1,\cdots ,k\}$, $a_{i}$'s are outcome of observable $x_{i}$. The Bob's states assemblage corresponding to Alice's measurement assemblage ${\mathcal {M}}=\{x_{1},\cdots ,x_{n}\}$ is ${\mathcal {S}}=\{\{\rho _{a_{1}|x_{1}}\},\cdots ,\{\rho _{a_{n}|x_{n}}\}\}$ where each $\rho _{a_{i}|x_{i}}$ is non-negative and $\sum _{a_{i}}\rho _{a_{i}|x_{i}}=\rho _{B}$ and $p(a_{i}|x_{i})=\mathrm {Tr} (\rho _{a_{i}|x_{i}})$. 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 ${\mathcal {A}}=\{p(\lambda ),\sigma _{\lambda }\}$ for which $\sum _{\lambda }p(\lambda )=1$, $\sigma _{\lambda }$'s are non-negative and $\sum _{\lambda }p(\lambda )\sigma _{\lambda }=\rho _{B}$. We say that a state is un-steerable if for arbitrary measurement assemblage ${\mathcal {M}}=\{x_{1},\cdots ,x_{n}\}$ and state assemblage ${\mathcal {S}}=\{\{\rho _{a_{1}|x_{1}}\},\cdots ,\{\rho _{a_{n}|x_{n}}\}\}$ there exist a local hidden state assemblage ${\mathcal {A}}=\{p(\lambda ),\sigma _{\lambda }\}$ such that $\rho _{a_{i}|x_{i}}=\sum _{\lambda }p(\lambda )p(a_{i}|x_{i},\lambda )\sigma _{\lambda }$ for all $a_{i}$ and $x_{i}$. A state is called steering state if it is no 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.

• local hidden variable model $p(a,b|x,y)=\sum _{\lambda }p(a|x,\lambda )p(b|y,\lambda )p(\lambda )$ ;
• local hidden state model $p(a,b|x,y)=\sum _{\lambda }p(a|x,\lambda )\mathrm {Tr} (F_{b|y}\sigma _{\lambda })p(\lambda )$;
• separable state model $p(a,b|x,y)=\sum _{\lambda }\mathrm {Tr} (E_{a|x}\chi _{\lambda })\mathrm {Tr} (F_{b|y}\sigma _{\lambda })p(\lambda )$.