In physics, in the area of quantum information theory and quantum computation, quantum steering is a special kind of nonlocal correlation, 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 Erwin Schrödinger,[1] [2] and later made popular by Howard M. Wiseman, S. J. Jones, and A. C. Doherty.[3]
In the usual formulation of quantum steering, two distant parties, Alice and Bob, are considered, they share an unknown quantum state
\rho
\rhoA
\rhoB
x
y
a
b
p(a,b|x,y)
Mathematically, consider Alice having the measurement
M=\{X1, … ,Xn\}
Xi
\{a1, … ,an\}
M
l{S}=\{\rho | |
a1|M |
, … ,\rho | |
an|M |
\}
n | |
\sum | |
i=1 |
p(ai|M)\rho
ai|M |
=\rhoB
p(ai|M)=Tr(\rho
ai|M |
)
Similar to the case of quantum entanglement, we define first un-steerable states. We introduce the local hidden state assemblage
l{A}=\{\sigmaλ\}
\sumλp(λ)=\sumλTr(\sigmaλ)=1
\sumλp(λ)\sigmaλ=\rhoB
M=\{X1, … ,Xn\}
l{S}=\{\rho | |
a1|M |
, … ,\rho | |
an|M |
\}
l{A}=\{\sigmaλ\}
\rho | |
ai|M |
=\sumλp(λ)p(ai|M,λ)\sigmaλ
ai
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.
p(a,b|x,y)=\sumλp(a|x,λ)p(b|y,λ)p(λ)
p(a,b|x,y)=\sumλp(a|x,λ)Tr(Fb|y\sigmaλ)p(λ)
p(a,b|x,y)=\sumλTr(Ea|x\chiλ)Tr(Fb|y\sigmaλ)p(λ)