What is the difference between nonlocality and entanglement?

Question

I'm a bit confused about the difference and relation between (quantum) nonlocality and entanglement.

To give some context about my confusion, I was reading this paper: Brunner, Nicolas, et al. "Bell nonlocality." Reviews of Modern Physics 86.2 (2014): 419.
Section 6.B.1 states:

The first inequality for detecting genuine multipartite nonlocality was introduced by Svetlichny (1987). Focusing on a tripartite system, Svetlichny derived a Bell-type inequality which holds for any distribution of Eq. (72). Thus a violation of such inequality implies the presence of genuine tripartite nonlocality. It should be noted that this in turn implies the presence of genuine tripartite entanglement

This seems to say that presence of genuine tripartite nonlocality $\implies$ genuine tripartite entanglement, in other words, all nonlocal states are entangled states? I imagine this as saying that entanglement is more general than nonlocality.

But then on the wiki page for Quantum Nonlocality under the heading Entanglement and Nonlocality, it mentions that quantum nonlocality and entanglement are not the same things. Of course I recognise wikipedia is not the most reliable source of information but I'm confused nonetheless.

It seems that entanglement is only a phenomenon of the quantum mechanics formulation, but nonlocality is independent of the model. I also know that there are forms of nonlocal correlations that are stronger than quantum mechanics (Ver Steeg, Greg. Foundational aspects of nonlocality. California Institute of Technology, 2009). But since entanglement is a quantum mechanical phenomenon this seems to imply that nonlocality is more general than entanglement? How could it be possible to have stronger forms of nonlocality if any nonlocal system is entangled according to "Bell Nonlocality"?

I think perhaps I am not comparing equivalent things and maybe that is why I'm getting confused. I do notice that "Bell Nonlocality" was only talking about genuinely multipartite nonlocality, so perhaps only specific systems violating the Svetlichny inequality are both (genuinely multipartite) nonlocal and entangled while there are nonlocal but not entangled systems that do not violate the Svetlichny inequality. Or perhaps, there are so many definitions of nonlocality that a general statement can't be made?

Answer 1

Entanglement is a simple concept in quantum mechanics: when a multipartite system is represented by a single quantum state, the system parts are said to be entangled.

Nonlocality (at least in the quantum context) is the interpretational theorisation of the implications of entanglement. This explains why it may be so complicated to define it properly, and so difficult to have people agree about what it entails.

So entanglement appears as a mathematical statement in a model, while nonlocality appears within the interpretation(s) of the model.

In that sense one cannot say which among nonlocality and entanglement is more general - they just do not belong to the same conceptual categories. Discussions about nonlocality may be very nebulous at times, while entanglement is a clear, technical, notion that everybody agrees about.

Answer 2

Entanglement/separability is a property of a quantum state (with respect to a bipartition of the underlying space, or more generally with respect to some multipartite structure of the space).

Nonlocality is a property of a conditional probability distribution describing how different measurement choices result in different measurement outcomes. These are also referred to as behaviours, for example in the review you cited. In the simplest case, these are vectors of the form $\{p(ab|xy)\}_{a,b,x,y}$, with $a,b$ labelling possible measurement outcomes, and $x,y$ labelling possible measurement choices. Such a behaviour is said to be nonlocal if it cannot be decomposed as $p(ab|xy)=\sum_\lambda p_\lambda p(a|x,\lambda)p(b|y,\lambda)$ for some set of probability distributions $\lambda\mapsto p_\lambda$, $a\mapsto p(a|x,\lambda)$, $b\mapsto p(b|y,\lambda)$.

Within quantum mechanics, nonlocal states are entangled. However, you can consider probability distributions (behaviours, in the terminology above) which are not only nonlocal, but also cannot be achieved by any quantum mechanical system at all, and so talking about "entanglement" doesn't even make sense for these. You can find an explicit example here. See also this other question on qc.SE for more details.

Answer 3

To have an entangled quatum state you need at least two states or particles. Nonlocality is a characteristic of quantum mechanics that manifests even for individual states, so they are not the same. Note that quantum nonlocality is a controversial issue and is subjected to interpretation/definition.