Quantum steering

In the usual formulation of quantum steering, two distant parties, Alice and Bob, are considered, they share an unknown quantum state ${\displaystyle \rho }$ with induced states ${\displaystyle \rho _{A}}$ and ${\displaystyle \rho _{B}}$ for Alice and Bob respectively. Alice and Bob can both perform local measurements on their own subsystems, for instance, Alice and Bob measure ${\displaystyle x}$ and ${\displaystyle y}$ and obtain the outcome ${\displaystyle a}$ and ${\displaystyle b}$. After running the experiment many times, they will obtain measurement statistics ${\displaystyle p(a,b|x,y)}$, 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 having the measurement ${\displaystyle M=\{X_{1},\cdots ,X_{n}\}}$, where the elements ${\displaystyle X_{i}}$ make up a POVM and the set ${\displaystyle \{a_{1},\cdots ,a_{n}\}}$ are the corresponding outcomes. Then Bob's local state assemblage (a set of positive operators) corresponding to Alice's measurement ${\displaystyle M}$ is

${\displaystyle {\mathcal {S}}=\{\rho _{a_{1}|M},\cdots ,\rho _{a_{n}|M}\}}$ with ${\displaystyle \sum _{i=1}^{n}p(a_{i}|M)\rho _{a_{i}|M}=\rho _{B}}$ where the probability ${\displaystyle p(a_{i}|M)=\mathrm {Tr} (\rho _{a_{i}|M})}$.

Similar to the case of quantum entanglement, we define first un-steerable states. We introduce the local hidden state assemblage ${\displaystyle {\mathcal {A}}=\{\sigma _{\lambda }\}}$ for which ${\displaystyle \sum _{\lambda }p(\lambda )=\sum _{\lambda }\mathrm {Tr} (\sigma _{\lambda })=1}$ and ${\displaystyle \sum _{\lambda }p(\lambda )\sigma _{\lambda }=\rho _{B}}$. We say that a state is un-steerable if for an arbitrary POVM measurement ${\displaystyle M=\{X_{1},\cdots ,X_{n}\}}$ and state assemblage ${\displaystyle {\mathcal {S}}=\{\rho _{a_{1}|M},\cdots ,\rho _{a_{n}|M}\}}$, there exists a local hidden state assemblage ${\displaystyle {\mathcal {A}}=\{\sigma _{\lambda }\}}$ such that

${\displaystyle \rho _{a_{i}|M}=\sum _{\lambda }p(\lambda )p(a_{i}|M,\lambda )\sigma _{\lambda }}$ for all ${\displaystyle a_{i}}$.

A state is called a steering state if it is not un-steerable.

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