The gist of their argument can be seen through an example due to David Bohm, who considered the state
$|\Psi\rangle = \dfrac{|00\rangle + |11\rangle }{\sqrt{2}} = \dfrac{|++\rangle + |--\rangle }{\sqrt{2}}, $
also known as an EPR pair, which here we expressed in both the standard basis $\{ |0\rangle, |1\rangle \}$ and Hadamard basis $\{ |+\rangle, |-\rangle \}$.
Suppose Alice gets the first qubit and Bob gets the second qubit. If Alice measures her qubit in the standard basis, she immediately projects Bob's qubit onto either the state $|0\rangle$ or $|1\rangle$. She produces a similar kind of projection of Bob's system if she measures in the Hadamard basis.
This seems problematic because it suggests that Alice can affect Bob's qubit even though their qubits no longer interact, in which case you would expect that nothing that Alice does on her qubit should influence Bob's.
Einstein, Podolsky, and Rosen thought (incorrectly as it turns out) that this nonlocal effect implies that the quantum state provides an incomplete description of Alice and Bob's physical systems. They attempt to support this intuition by arguing that one might be able to explain what Bob obtains if he measures his qubit through local hidden variables (LHV) that are missing from quantum theory.
Like EPR, Erwin Schrödinger was bothered by the idea of nonlocality. However, unlike EPR, Schrodinger believed that a quantum state provided a complete description of a local, isolated system.
Thus, his response, Schrödinger introduced the notions of an entangled state for describing states like $|\Psi\rangle$, and steering for describing Alice's ability to influence Bob's system with her measurement choice. Schrödinger rejected the EPR argument for LHVs, instead, he argued (wrongly) that where quantum theory failed was in describing delocalized entangled systems, i.e., Bob's system is in some definite state, a local hidden state (LHS), which renders steering an unobservable effect in practice.
It turns out that steering is a form of quantum correlation that is distinct from entanglement and nonlocality. In fact, it lies in between the two, since not all entangled states can be used to demonstrate steering, but not all steerable states can be used to violate a Bell inequality, which would imply that a quantum correlation can not be explained by a LHV model (i.e. it is nonlocal).
Some intuition for distinguishing the three quantum-correlation-types can be obtained from the quantum information perspective if we think of the following tasks for three parties Alice, Bob, and Charlie.
Suppose that Alice and Bob share an entangled state and their goal is to convince Charlie of this fact. In this scenario, Alice and Bob are not allowed to communicate but they are allowed to share some common random bits. They each can communicate directly to Charlie.
(i) If Charlie trusts both Alice and Bob, then all they have to do is perform a tomographic measurement on their state. (A tomographic measurement is one whose results contain enough information to reconstruct the quantum state if it were unknown.) They give their results to Charlie, who can use the data to verify that the state is entangled.
(ii) If Charlie trusts Bob but not Alice, then Alice and Bob have to demonstrate steering, since it would be enough to convince Charlie if there is no LHS model for the trusted party.
(iii) If Charlie does not trust both, then Alice and Bob have to show that they can violate a Bell inequality, since this is the way Charlie can be guaranteed that the correlations that Alice and Bob can produce with their state is non-classical.
 |
| Nonlocality, steering, and entanglement from a task. Alice and Bob aim to convince Charlie that they share entangled qubits. Alice and Bob cannot communicate with each other but they are allowed to have some shared classical random bits. Here the white and black boxes are seen from Charlie's perspective. The white boxes denote a qubit basis measurement, labelled by a Bloch vector $\vec{n}$, while the black boxes refer to some unknown operation with input $x (y)$ and output $a (b)$. Describing from the top to bottom: If Charlie trusts both Alice and Bob then they can just perform some tomographically complete measurement to verify entanglement. If Charlie trusts only Bob, then Alice and Bob have to show that the state they share can be steered. If Charlie trusts neither, then they have to exhibit a Bell-inequality violation, by say, playing the CHSH game and winning sufficiently many games. |
Like Bell nonlocality, steering can be demonstrated through simple tests, e.g. for qubits it is sufficient to consider two measurements for Alice, which prepares a collection of four states for Bob. This means steering can be certified using violation of steering inequalities, analogous to Bell inequality violations for exhibiting nonlocality.
Note that if Bob's system can be described by a LHS, Bob's probabilities for measurement outcomes must be compatible with a quantum state. This suggests the use of uncertainty relations for deriving some steering criteria.
One key difference between steering inequalities and Bell inequalities is that the latter is constructed without defining any particular measurement for Alice or Bob. For steering, the inequality does depend on Bob's measurement choice since we require outcome probabilities consistent with a quantum state.
AS EPR and Schrödinger have noted, steering can be demonstrated using pure entangled states. And from what we know about the EPR pair $|\Psi\rangle$, it can be used to show nonlocality through violation of the CHSH inequality, a Bell inequality for qubits. So to again show that the ideas are distinct, we need to consider mixtures, quantum states that are necessarily described using density operators.
The example that we will take here is the isotropic state for qubits
$\rho = p|\Psi\rangle\langle\Psi| + (1-p)\dfrac{I}{4}$,
which can be seen as a mixture of $|\Psi\rangle$ and the maximally mixed state for two qubits. Note that this is actually a family of quantum states with $0 \le p \le 1$. In standard matrix form, we would write it as
$\rho = \dfrac{p}{2}
\begin{pmatrix}
1 \\ 0 \\ 0 \\ 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0 & 1
\end{pmatrix}
+ \dfrac{1-p}{4}
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
=
\dfrac{1}{4}
\begin{pmatrix}
1+p & 0 & 0 & 2p \\
0 & 1-p & 0 & 0 \\
0 & 0 & 1-p & 0 \\
2p & 0 & 0 & 1+p
\end{pmatrix}$.
We can determine for which values $p$ the state $\rho$ is entangled by looking at the eigenvalues of the partial transpose:
$\rho^{T_B} =
\begin{pmatrix}
1+p & 0 & 0 & 0 \\
0 & 1-p & 2p & 0 \\
0 & 2p & 1-p & 0 \\
0 & 0 & 0 & 1+p
\end{pmatrix}
$
The Peres-Horodecki criterion says that when $\rho^{T_B}$ has a negative eigenvalue then $\rho$ is entangled. Thus, solving for the eigenvalues of $\rho^{T_B}$
$\mathrm{Det}(\rho^{T_B} - \lambda I)
= (1 + p - 4\lambda) ^2 [ (1- p - 4\lambda)^2 - 4p^2 ] = 0$
so the four eigenvalues are $\lambda = (1+p)/4$ counted thrice, and $\lambda = (1-3p)/4$. Thus, the last one gives the smallest eigenvalue and we get $\lambda < 0$ when $ p > 1/3$.
To establish Bell nonlocality for $\rho$, we consider the CHSH inequality given by
$ | \langle Z X' \rangle + \langle Z Z' \rangle + \langle X X' \rangle - \langle X Z' \rangle |\le 2$,
where $\langle A B \rangle = \mathrm{Tr}(\sigma A\otimes B)$ is the expectation value of observable $A \otimes B$ given the quantum state $\sigma$. In this case, the relevant observables (i.e. they give rise to the optimal violation) are
$
X =
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix},
\quad
Z =
\begin{pmatrix}
1 &0 \\
0 & -1
\end{pmatrix},
\quad
X' = \dfrac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix},
\quad
Z' = \dfrac{1}{\sqrt{2}}
\begin{pmatrix}
-1 & -1 \\
-1 & 1
\end{pmatrix}
$.
That is, Alice's observables are just the Pauli operators $X$ and $Z$, while Bob's operators are rotated versions $X'$ and $Z'$. As we have seen before, these observables correspond to Alice measuring her qubit in the standard or Hadamard basis, with Bob measuring his qubit in the basis $\{ \cos\theta |0\rangle + \sin\theta |1\rangle, \sin\theta |0\rangle - \cos\theta |1\rangle \}$, for $\theta = \pm\pi/8$. More specifically, a measurement in the qubit basis
$\left\{
\begin{pmatrix}
a \\
b
\end{pmatrix},
\begin{pmatrix}
b^* \\
-a^*
\end{pmatrix}
\right\}$
yields observable
$O =
\begin{pmatrix}
a \\
b
\end{pmatrix}
\begin{pmatrix}
a^* & b^*
\end{pmatrix}
-
\begin{pmatrix}
b^* \\
-a^*
\end{pmatrix}
\begin{pmatrix}
b & -a
\end{pmatrix}
=
\begin{pmatrix}
|a|^2 - |b|^2 & 2 a b^* \\
2 b a^* & |b|^2 - |a|^2
\end{pmatrix}
$.
Let $\beta = Z\otimes Z' + Z \otimes X' + X \otimes X' - X \otimes Z'$ be the CHSH operator. Then
$\beta = \sqrt{2}
\begin{pmatrix}
1 & 0 & 0 & 1 \\
0 & -1 & 1 & 0 \\
1 & 1 & -1& 0 \\
1 & 0 & 0 & 1
\end{pmatrix}$,
and so we can easily compute $\mathrm{Tr}(\rho \beta) = \dfrac{1}{2\sqrt{2}} (2+ 6p +2p - 2) =2\sqrt{2} p$. Hence, we observe a CHSH inequality violation when $ 2 - |\mathrm{Tr}(\rho \beta)| < 0$, which for the isotropic state happens when $ p > 1/\sqrt{2}$.
Finally, we consider the following steering inequality:
$\langle X X \rangle + \langle Y Y \rangle + \langle Z Z \rangle \le \sqrt{3}$,
where $Y$ is the other Pauli operator $ Y =
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}$.
Here we have the steering operator $\Sigma = X \otimes X + Y \otimes Y + Z \otimes Z$. A direct calculation gives
$\Sigma = \begin{pmatrix}
1 & 0 & 0 & 2 \\
0 &-1 & 0 & 0 \\
0 & 0 & - 1 & 0 \\
2 & 0 & 0 & 1
\end{pmatrix}$,
which means that $\mathrm{Tr}(\rho \Sigma) = \dfrac{1}{4} (2 + 10p + 2p - 2) = 3p$.
Thus, the steering is observed when $\sqrt{3} - |\mathrm{Tr}(\rho \Sigma) | < 0$, which happens when $p > 1\sqrt{3}$. (Actually this result is conservative since the steering limit is $p > 1/2$ if we consider an infinite number of measurements, but that is obviously impractical to test.)
 |
| Nonlocality, steering, and entanglement for the family of qubit isotropic states $\rho(p)$. This red line plots the smallest eigenvalue of the partial transpose of $\rho$. The green line plots $\sqrt{3} - |\mathrm{Tr}(\rho \Sigma)|$, which comes from a steering inequality. The blue line plots $2 - \mathrm{Tr}(\rho \beta) |$, which comes from the CHSH inequality. The value of $p$ where the red, green, and blue lines become negative indicate the respective bounds for entangled, steerable, and Bell nonlocal isotropic states. |
So to recap, for the family of qubit isotropic states, we have seen that a particular $\rho$ is entangled only if $p > 1/3$, it is steerable for Pauli measurements only if $p > 1/\sqrt{3}$, and it exhibits Bell nonlocality only if $ p> 1/\sqrt{2}$.
This simple example at least illustrates that these notions are indeed different for mixed states. It continues to be a pretty challenging task to quantify these various kinds of quantum correlations in higher dimensions.
References:
H. M. Wiseman, S. J. Jones, and A. C. Doherty, Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen paradox, Physical Review Letters, 98 (2007) 140402.
E.G. Cavalcanti, S.J. Jones, H.M. Wiseman and M.D. Reid, Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox, Physical Review A 80 (2009) 032112.
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