Nonlocality
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In physics, nonlocality is a direct influence of one object on another, distant object, in violation of principle of locality. In classical physics, nonlocality in the form of action at a distance appeared in corpuscular theories and later disappeared in field theories. Action at a distance is incompatible with relativity. In quantum physics nonlocality re-appeared in the form of entanglement. Physical reality of entanglement has been demonstrated experimentally¹ together with the absence of local hidden variables. Entanglement is compatible with relativity; however, it prompts some of the more philosophically oriented discussions concerning quantum theory. More general nonlocality beyond quantum entanglement, but still compatible with relativity, is an active field of theoretical investigation but has yet to be observed.

(Non)locality and (in)determinism

Imagine two experimentalists, Alice and Bob, situated in their laboratories. Alice chooses and pushes one of two buttons, A0 and A1, on her apparatus. Bob observes on his apparatus one of two indicating lamps, b0 and b1, lighting. The four combinations are logically possible: (A0,b0), (A0,b1), (A1,b0) and (A1,b1).

It may happen that only two combinations, (A0,b0) and (A1,b1), occur in the experiment. Then one concludes that A has influence on B.

It may happen that all the four combinations occur with some (conditional) probabilities P(b0|A0), P(b1|A0) = 1 - P(b0|A0), P(b0|A1) and P(b1|A1) = 1 - P(b0|A1). If P(b0|A0) differs from P(b0|A1), one concludes that A has influence on B.

Here is a more complicated scenario: Alice pushes one of two buttons, A0 and A1; also Bob pushes one of two buttons, B0 and B1. Alice observes one of two outcomes, a0 and a1; also Bob observes one of two outcomes, b0 and b1. Logically, 16 combinations are possible:

where each of X,Y,x,y is 0 or 1. Imagine that only 8 combinations occur, with the following (conditional) probabilities:

That is, the two outcomes are perfectly anticorrelated (either (a0,b1) or (a1,b0), equiprobably) when (A1,B1) is chosen. In the three other cases ((A0,B0), (A0,B1), (A1,B0)), the two outcomes are perfectly correlated (either (a0,b0) or (a1,b1), equiprobably).

Does it imply that some influence exists (A on B, or B on A), or not?

The question is important, since the answer depends on our fundamental assumptions about nature.

On one hand, Alice cannot send a message to Bob, using her buttons A0, A1 and his indicators b0, b1. (Nor Bob to Alice.) In this sense the answer is negative (influence need not exist).

On the other hand, no one is able to design apparata that behave as specified, without using a kind of influence (A on B, or B on A). In this sense the answer is affirmative (some influence must exist).

Thorough logical analysis reveals that the affirmative answer follows from the assumptions of local realism and counterfactual definiteness. These fundamental assumptions, deeply rooted in our physical intuition, are incompatible with quantum theory. Different interpretations of quantum mechanics reject different parts of local realism and/or counterfactual definiteness (for detail, see Principle of locality). Thus, the definition of nonlocality, given in the beginning of this article, is tentative. Here is an elaborate definition.

A phenomenon is nonlocal if it implies a direct influence of one object on another, distant object, provided that local realism and counterfactual definiteness are taken for granted.

This subtlety explains why a nonlocal phenomenon is not necessarily a channel for direct signaling.

Nonlocality in quantum mechanics

Einstein, Podolsky and Rosen

Main article: EPR Paradox

In 1935, Einstein, Podolsky and Rosen published a thought experiment ² with which they hoped to expose the inadequacies of the Copenhagen interpretation of quantum mechanics in relation to the lack of determinism at the microscopic scale that it described. In particular, they hoped to demonstrate that the probabilistic nature of the results of measurements on particles could be described through the means of some ‘hidden’ variables that predetermine the result of a measurement, but to which an observer does not have access.

In physical terms, this experiment can be represented as a spin-zero particle decaying into two spin-half particles such that there is no interaction between the two particles after decay. Since spin is a conserved quantity, measurements of spin on the two particles must anti-correlate. The quantum state of the two particles prior to measurement can be written as³

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.

This state allows us to predict the probability of a particular result. Alice, for example, will measure her particle to be spin-up in an average of fifty percent of measurements. However, when Alice measures her particle it causes the state to collapse so that if Alice measures spin-up, Bob must measure spin-down and vice versa. Hence, either party is capable of setting the spin of the other’s particle instantaneously. Such behaviour is non-local because the measurement of one particle is able to influence the physical state of another independent of the distance between them, so that no information could travel between them.

To Einstein, Podolsky and Rosen, this implied an effect being transmitted at superluminal speeds and in doing so, violating the laws of special relativity². Their position was that this suggested the presence of hidden variables that predetermined the value of the measurement at the time the particles were entangled, which would restore determinism to physics.

Demonstration of nonlocality

See also: Bell test experiments

In 1964, John Bell proposed a test to discover if such hidden variables existed ⁴. Starting from the presumption of hidden variables, Bell derived inequalities based on expected correlations between measurements that would have to be true if hidden variables existed. If the Bell inequalities were violated, then quantum mechanics would be right, and hidden variables would not be an appropriate physical model.

Clauser, Horne, Shimony and Holt reformulated these inequalities for a scenario closest to the one given above.⁵ They proposed a scheme whereby Alice and Bob can make measurements of particle spin along two arbitrary axes. The statistical correlation of the results of these measurements was assumed to result from non-quantum mechanical information carried locally by the two particles as part of a collection of hidden variables denoted by λ.

With the further restriction, imposed by the physical reality, that measurement along the same axis by Alice and Bob must lead to anti-correlated results, a table of possible local plans λ can be produced by assigning spin-up and spin-down with numerical values ±1.

Each of these plans has an associated probability ρ(λ_i) of being selected, such that the four probabilities sum to unity. If Alice and Bob have outcomes labelled a and b then we can write the probability of obtaining a particular combination of outcomes given the set of inputs. For a particular measurement by Alice denoted A and a particular measurement B by Bob:

where the quantity in the sum is the probability of Alice obtaining result a having chosen measurement A, and Bob obtaining result b having chosen measurement B₁ given the local plan λ_i, multiplied by the probability that this plan is used. A correlation function can then be defined as:

The CHSH inequality, which must be true if only hidden variables determine the correlations, uses a correlation function defined as

Using suitable local plans and the correlation function, it can be shown that this obeys the following inequality, which is the CHSH inequality³:

Experimentalists such as Aspect used this inequality¹, as well as other formulations of Bell's inequality, to invalidate the hidden variables hypothesis and confirm the existence of nonlocality in quantum mechanics, implying that the lack of determinism in the Copenhagen interpretation was justified.

Generalising nonlocality

The upper boundary for the CHSH inequality was found by Cirel'son to be ⁶, but further to this work, Sandu Popescu and Daniel Rohrlich postulated a correlation function for a particular set of measurements that would allow an upper boundary of 4.⁷ This demonstrated that it was possible for nature to be more non-local than quantum mechanics allows. From this, it was possible to abstract physical measurements of nonlocality into a non-local box.⁸

In the case of measurements by Alice and Bob, such a box would take an input X from Alice and an input Y from Bob, and output two values a and b for Alice and Bob respectively and separately, where a, b, X and Y could only take the values zero or one. The box itself determines the joint probability for an output pair given the particular pair of inputs received. This probability, denoted p_{ab | XY} has the properties

and

These arise from the normal probabilistic conditions of positivity and normalisation.

The definition of non-local boxes described only encapsulates non-signalling boxes⁸, which means that neither Alice nor Bob can signal their choice of input to one another. This is a physically logical proposition, since when a party sets their input it is physically analogous to making a measurement, which should effectively provide a result immediately. Since there is no restriction on the separation of parties, signalling to Bob would potentially require considerable time to elapse between measurement and result, which is a physically unrealistic scenario.

For a box to be non-signalling does not, however, require that it involves nonlocality since two non-communicating parties could simply have shared information by discussing their choice of inputs in advance. There are therefore both local and non-local boxes.

The non-signalling requirement imposes further conditions on the joint probability, in that the probability of a particular output a or b should depend only on its associated input. This is formalised by the conditions:

and

The constraints above are all linear, and so define a convex polytope representing the set of all non-signalling boxes with a given number of inputs and outputs. The polytope is convex because two boxes that exist in the polytope can be mixed in a probabilistic manner to produce another box that also exists within the polytope.

Since this polytope contains all possible non-signalling boxes of a given number of inputs and outputs, it has as subsets both local boxes and those boxes which can violate Cirel’son’s bound by relying on quantum mechanical correlations.

Popescu and Rohrlich’s maximum algebraic violation of the CHSH inequality can be reached by a box, referred to as a standard PR box after these authors, with joint probability given by:

where denotes addition modulo two.⁹

References