In theoretical physics, quantum nonlocality refers to the phenomenon by which the measurement statistics of a multipartite quantum system do not allow an interpretation with local hidden variables. Quantum nonlocality has been experimentally verified under a variety of physical assumptions, with a notable exception being the many-worlds interpretation which violates an assumption of Bell's theorem.
Quantum nonlocality does not allow for faster-than-light communication, and hence is compatible with special relativity and its universal speed limit of objects. Thus, quantum theory is local in the strict sense defined by special relativity and, as such, the term "quantum nonlocality" is sometimes considered a misnomer. Still, it prompts many of the foundational discussions concerning quantum theory.
History
Einstein, Podolsky and Rosen
In the 1935 EPR paper, Albert Einstein, Boris Podolsky and Nathan Rosen described "two spatially separated particles which have both perfectly correlated positions and momenta" as a direct consequence of quantum theory. They intended to use the classical principle of locality to challenge the idea that the quantum wavefunction was a complete description of reality, but instead they sparked a debate on the nature of reality.Clauser, John F., and Abner Shimony. "Bell's theorem. Experimental tests and implications." Reports on Progress in Physics 41.12 (1978): 1881. Afterwards, Einstein presented a variant of these ideas in a letter to Erwin Schrödinger, which is the version that is presented here. The state and notation used here are more modern, and akin to David Bohm's take on EPR. The quantum state of the two particles prior to measurement can be written as
where .
Here, subscripts "A" and "B" distinguish the two particles, though it is more convenient and usual to refer to these particles as being in the possession of two experimentalists called Alice and Bob. The rules of quantum theory give predictions for the outcomes of measurements performed by the experimentalists. Alice, for example, will measure her particle to be spin-up in an average of fifty percent of measurements. However, according to the textbook quantum mechanics, Alice's measurement causes the state of the two particles to collapse, so that if Alice performs a measurement of spin in the z-direction, that is with respect to the basis , then Bob's system will be left in one of the states . Likewise, if Alice performs a measurement of spin in the x-direction, that is, with respect to the basis , then Bob's system will be left in one of the states . Schrödinger referred to this phenomenon as "steering". This steering occurs in such a way that no signal can be sent by performing such a state update; quantum nonlocality cannot be used to send messages instantaneously and is therefore not in direct conflict with causality concerns in special relativity.
In the orthodox view of this experiment, Alice's measurement—and particularly her measurement choice—has a direct effect on Bob's state. However, under the assumption of locality, actions on Alice's system do not affect the "true", or "ontic" state of Bob's system. We see that the ontic state of Bob's system must be compatible with one of the quantum states or , since Alice can make a measurement that concludes with one of those states being the quantum description of his system. At the same time, it must also be compatible with one of the quantum states or for the same reason. Therefore, the ontic state of Bob's system must be compatible with at least two quantum states; the quantum state is therefore not a complete descriptor of his system. Einstein, Podolsky and Rosen saw this as evidence of the incompleteness of quantum theory, since the wavefunction is explicitly not a complete description of a quantum system under this assumption of locality. Their paper concludes:
Although various authors (most notably Niels Bohr) criticised the ambiguous terminology of the EPR paper, the thought experiment nevertheless generated a great deal of interest. Their notion of a "complete description" was later formalised by the suggestion of hidden variables that determine the statistics of measurement results, but to which an observer does not have access.von Neumann, J. (1932/1955). In Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, translated into English by Beyer, R.T., Princeton University Press, Princeton, cited by Baggott, J. (2004) Beyond Measure: Modern physics, philosophy, and the meaning of quantum theory, Oxford University Press, Oxford, , pages 144–145. Bohmian mechanics provides such a completion of quantum mechanics, with the introduction of hidden variables; however the theory is explicitly nonlocal. The interpretation therefore does not give an answer to Einstein's question, which was whether or not a complete description of quantum mechanics could be given in terms of local hidden variables in keeping with the "Principle of Local Action".
Bell inequality
In 1964 John Bell answered Einstein's question by showing that such local hidden variables can never reproduce the full range of statistical outcomes predicted by quantum theory. Bell showed that a local hidden variable hypothesis leads to restrictions on the strength of correlations of measurement results. If the Bell inequalities are violated experimentally as predicted by quantum mechanics, then reality cannot be described by local hidden variables and the mystery of quantum nonlocal causation remains. However, Bell notes that the non-local hidden variable model of Bohm are different:
Clauser, Horne, Shimony and Holt (CHSH) reformulated these inequalities in a manner that was more conducive to experimental testing (see CHSH inequality).
In the scenario proposed by Bell (a Bell scenario), two experimentalists, Alice and Bob, conduct experiments in separate labs. At each run, Alice (Bob) conducts an experiment in her (his) lab, obtaining outcome . If Alice and Bob repeat their experiments several times, then they can estimate the probabilities , namely, the probability that Alice and Bob respectively observe the results when they respectively conduct the experiments x,y. In the following, each such set of probabilities will be denoted by just . The form is termed a "box".
Bell formalized the idea of a hidden variable by introducing the parameter to locally characterize measurement results on each system: "It is a matter of indifference ... whether λ denotes a single variable or a set ... and whether the variables are discrete or continuous". However, it is equivalent (and more intuitive) to think of as a local "strategy" or "message" that occurs with some probability when Alice and Bob reboot their experimental setup. Bell's assumption of local causality then stipulates that each local strategy defines the distributions of independent outcomes if Alice conducts experiment x and Bob conducts experiment
Here () denotes the probability that Alice (Bob) obtains the result when she (he) conducts experiment and the local variable describing her (his) experiment has value ().
Suppose that can take values from some set . If each pair of values has an associated probability of being selected (shared randomness is allowed, i.e., can be correlated), then one can average over this distribution to obtain a formula for the joint probability of each measurement result:
A box admitting such a decomposition is called a Bell local or a classical box. Fixing the number of possible values which can each take, one can represent each box as a finite vector with entries . In that representation, the set of all classical boxes forms a convex polytope. In the Bell scenario studied by CHSH, where can take values within , any Bell local box must satisfy the CHSH inequality:
where
The above considerations apply to model a quantum experiment. Consider two parties conducting local polarization measurements on a bipartite photonic state. The measurement result for the polarization of a photon can take one of two values (informally, whether the photon is polarized in that direction, or in the orthogonal direction). If each party is allowed to choose between just two different polarization directions, the experiment fits within the CHSH scenario. As noted by CHSH, there exist a quantum state and polarization directions which generate a box with equal to . This demonstrates an explicit way in which a theory with ontological states that are local, with local measurements and only local actions cannot match the probabilistic predictions of quantum theory, disproving Einstein's hypothesis. Experimentalists such as Alain Aspect have verified the quantum violation of the CHSH inequality as well as other formulations of Bell's inequality, to invalidate the local hidden variables hypothesis and confirm that reality is indeed nonlocal in the EPR sense.
Possibilistic nonlocality
Bell's demonstration is probabilistic in the sense that it shows that the precise probabilities predicted by quantum mechanics for some entangled scenarios cannot be met by a local hidden variable theory. (For brevity, here and henceforth "local theory" means "local hidden variables theory".) However, quantum mechanics permits an even stronger violation of local theories: a possibilistic one, in which local theories cannot even agree with quantum mechanics on which events are possible or impossible in an entangled scenario. The first proof of this kind was due to Daniel Greenberger, Michael Horne, and Anton Zeilinger in 1993. The state involved is often called the GHZ state.
In 1993, Lucien Hardy demonstrated a logical proof of quantum nonlocality that, like the GHZ proof is a possibilistic proof. It starts with the observation that the state defined below can be written in a few suggestive ways:
where, as above, .
The experiment consists of this entangled state being shared between two experimenters, each of whom has the ability to measure either with respect to the basis or . We see that if they each measure with respect to , then they never see the outcome . If one measures with respect to and the other , they never see the outcomes However, sometimes they see the outcome when measuring with respect to , since
This leads to the paradox: having the outcome we conclude that if one of the experimenters had measured with respect to the basis instead, the outcome must have been or , since and are impossible. But then, if they had both measured with respect to the basis, by locality the result must have been , which is also impossible.
Nonlocal hidden variable models with a finite propagation speed The work of Bancal et al. studies whether influences that propagate at a finite speed (possibly superluminal) can explain quantum correlations. In the case of a bipartite experiment, even if no signalling is allowed, the correlations can always be explained by a finite, but superluminal, hidden influence. In this scenario, any bipartite experiment revealing Bell nonlocality can just provide lower bounds on the hidden influence's propagation speed. However, certain experiments with three or more parties will either result in signalling (i.e. the influences cannot be hidden anymore) or rule out all finite speed causal influences explaining quantum correlations.
Analogs of Bell's theorem in more complicated causal structures
The random variables measured in a general experiment can depend on each other in complicated ways. In the field of causal inference, such dependencies are represented via Bayesian networks: directed acyclic graphs where each node represents a variable and an edge from a variable to another signifies that the former influences the latter and not otherwise, see the figure.