Nonlocality

In classical physics, nonlocality (action at a distance) is a direct influence of one object on another distant object, in violation of the principle of locality. In quantum physics and quantum field theories, the term nonlocal means that correlations cannot be described by any local hidden variable theory. Such a form of nonlocality does not allow superluminal communication. Many states which possess entanglement produce such correlations when measured, as demonstrated by Bell's theorem. This has been verified experimentally implying the absence of local hidden variables underlying quantum mechanics.

Entanglement is compatible with relativity, though it prompts some of the more fundamentally oriented discussions concerning quantum theory. A more general nonlocality beyond quantum entanglement — retaining compatibility with relativity — is an active field of theoretical investigation and has yet to be observed.

Different notions of nonlocality in physics

One needs to distinguish between different notions of nonlocality appearing in physics:

1. Aharonov-Bohm-like interactions.
2. Non-local correlations arising from quantum entanglement. For example Einstein-Podolsky-Rosen Paradox (EPR) correlations.
3. The impossibility to localize a particle, that is, wavefunctions with finite spatial extent are not eigenfunctions of a position operator.
4. The field equation contains an infinite order of derivatives. An example is the square root of the Klein–Gordon equation. See also nonlocal Lagrangian.
5. Coupling of a field to derivatives of potentials, e.g., Darwin-like terms.
6. Locality conditions in quantized theories, that is, the commutator of fields vanishes for spacelike distances.

Example

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:

$\textstyle (AX,BY,ax,by)$

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

$P({ax,by}{|}{AX,BY})={\begin{cases}{\frac {1}{2}},&{\mbox{if }}x\oplus y=XY\\0,&{\mbox{otherwise}}\end{cases}}$

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 there is no influence of A on B, or of B on A, since it is easily checked that P(bx|A0) = P(bx|A1) for both x = 0 and x = 1 in the above example. That is to say, this particular set of probabilities is non-signalling.

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 with which they hoped to expose the incompleteness of the Copenhagen interpretation of quantum mechanics in relation to the violation of realism and local causality at the microscopic scale that it described. This was later formalised by the suggestion of the existence of ‘hidden’ variables that determine the statistics of measurement results, but to which an observer does not have access.

Bohm later modified the original EPR thought experiment, simplifying the mathematics and highlighting assumptions like locality (which Einstein et al had tacitly assumed). In his version of the experiment, a spin-zero particle decays 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 as

$\left|\psi _{AB}\right\rangle ={\frac {1}{\sqrt {2}}}{\bigg (}\left|\uparrow \right\rangle _{A}\left|\downarrow \right\rangle _{B}-\left|\downarrow \right\rangle _{A}\left|\uparrow \right\rangle _{B}{\bigg )}$

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 on each particle. Alice, for example, will measure her particle to be spin-up in an average of fifty percent of measurements. However, according to the Copenhagen interpretation, Alice's measurement causes the state of the two particles to collapse, so that if Alice measures spin-up in some direction n, the quantum state after measurement is the corresponding eigenstate

$\left|\psi _{AB}\right\rangle =\left|\uparrow _{\mathbf {n} }\right\rangle _{A}\left|\downarrow _{\mathbf {n} }\right\rangle _{B}$

implying that if Bob also measures spin in direction n, he must get a spin-down result. Hence, spin measurements in the same direction are always anti-correlated, regardless of spatial separation. The orthodox quantum description of such behaviour is non-local because the measurement of one particle instantaneously influences the physical state of another, independent of the distance between them.

Einstein, Podolsky and Rosen saw this as evidence of the incompleteness of the Copenhagen interpretation of Quantum Theory: after Alice's measurement in any direction, Bob's particle has a definite physical spin in that direction. If Alice's measurement has no effect on the physical description of Bob's particle, then the outcome of spin measurements on Bob's particle must be a definite physical property wihch is unaccounted for by the quantum mechanical state description. Based on this perspective, their paper concludes:

While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.

Demonstration

See also: Bell test experiments

In 1964, John Bell showed that such local hidden variables could never reproduce the statistical outcomes of individual measurements, as predicted by Quantum Theory. Starting from the presumption of hidden variables, Bell derived inequalities based on expected correlations between measurements that would have to be true if hidden variable models were correct. If the Bell inequalities are violated, then reality cannot be described by such local hidden variables and the mystery of quantum non-local causation remains. According to Bell:

This [grossly nonlocal structure] is characteristic... of any such theory which reproduces exactly the quantum mechanical predictions.

Clauser, Horne, Shimony and Holt (CHSH) reformulated these inequalities for a scenario closest to the one given above. 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.

The variable λ here is commonly referred to as a hidden variable, but it is equivalent (and more intuitive) to think of them as local "strategies" that each occur with some probability p($\lambda$) when an entangled pair of states is created. EPR's criteria for local causation then stipulates that each local strategy defines independent distributions for the outcome probabilities if Alice inputs measurement A and Bob inputs measurement B:

$P\left({a,b}{|}{A,B,\lambda }\right)=P\left({a}{|}{A,\lambda }\right)P\left({b}{|}{B,\lambda }\right)$

where, for instance, $P\left({a}{|}{A,\lambda }\right)$ denotes the probability of Alice getting the outcome a given that she input the measurement A.

If each strategy $\lambda _{i},1\leq i\leq k$ has an associated probability ρ($\lambda _{i}$) of being selected (such that the probabilities sum to unity) we can average these probabilities over the distribution on the $\lambda _{i}$:

$P\left({a,b}{|}{A,B}\right)=\sum _{i=1}^{k}P\left({a}{|}{A,\lambda _{i}}\right)P\left({b}{|}{B,\lambda _{i}}\right)\rho \left(\lambda _{i}\right)$

For each measurement A and B, the correlator E(A, B) is then defined as:

$E\left({A,B}\right)=\sum _{a,b}{ab}P\left({a,b}{|}{A,B}\right)$

In the case that Alice and Bob are measuring spin-1/2 particles, a and b only take on values 1 or -1. Hence the product ab is equal to 1 if Alice and Bob get the same outcome, and -1 if they get different outcomes. E(A,B) can therefore be seen as the expectation that Alice's and Bob's outcomes are correlated.

In the case that Alice chooses from one of two measurements $A_{1}$ or $A_{2}$, and Bob chooses from $B_{1}$ or $B_{2}$, the CHSH inequality uses a correlation function defined as

$S_{CHSH}=E\left({A_{1},B_{1}}\right)+E\left({A_{1},B_{2}}\right)+E\left({A_{2},B_{1}}\right)-E\left({A_{2},B_{2}}\right)$

Using only local strategies, it can be shown that the correlation function always obeys the following CHSH inequality:

$-2\leq S_{CHSH}\leq 2$

However, if instead of local hidden variables we adopt the rules of Quantum Theory, it is possible to construct an entangled pair of particles (one each for Alice and Bob) and a set of measurements $A_{1},A_{2},B_{1},B_{2}$ such that $S_{CHSH}=2{\sqrt {2}}$. Experimentalists such as 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 non-local in the EPR sense.

Non-locality and entanglement

See also: Quantum entanglement

In the media and popular science, quantum non-locality is often portrayed as being equivalent to entanglement. While it is true that a bipartite quantum state must be entangled in order for it to produce non-local correlations, there exist entangled states which do not produce such correlations. A well-known example of this is the Werner state that is entangled for certain values of $p_{sym}$, but can always be described using local hidden variables. On the other hand, reasonably simple examples of Bell inequalities have been found for which the quantum state giving the largest violation is never a maximally entangled state, showing that entanglement is, in some sense, not even proportional to non-locality.

In short, entanglement of a two-party state is necessary but not sufficient for that state to be non-local. It is important to recognise that entanglement is more commonly viewed as an algebraic concept, noted for being a precedent to non-locality as well as quantum teleportation and superdense coding, whereas nonlocality is interpreted according to experimental statistics and is much more involved with the foundations and interpretations of Quantum Mechanics.

Generalising nonlocality

Whilst the CHSH inequality gives restrictions on the CHSH value attainable by local hidden variable theories, the rules of Quantum Theory do not allow us to violate Tsirelson's bound of $2{\sqrt {2}}$, even if we exploit measurements of entangled particles. The question remained whether this was the maximum CHSH value one can attain without explicitly allowing instantaneous signaling. In 1994 two physicists, Sandu Popescu and Daniel Rohrlich, postulated a particular set of non-signalling correlated measurements that give $S_{CHSH}=4$: the algebraic maximum. This demonstrated that there were apparently reasonable theories of parts of Nature that drastically violated the predictions of Quantum Theory. The attempt to understand what uniquely identified Quantum Theory from such general theories motivated an abstraction from physical measurements of non-locality, to the study of non-local boxes.

In the case of measurements by Alice and Bob, such a box would take an input A from Alice and an input B from Bob, and output two values a and b for Alice and Bob respectively and separately, where a, b, A and B 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 is denoted $P\left({{a,b}{|}{A,B}}\right)$ and has the properties

$P\left({{a,b}{|}{A,B}}\right)\geq 0\quad \forall {a,b,A,B}$

and

$\sum _{a,b}P\left({{a,b}{|}{A,B}}\right)=1\quad \forall {A,B}$

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

A box is local, or admits a local hidden variable model, if its output probabilities can be characterized in the following way:

$P\left({{a,b}{|}{A,B}}\right)=\sum _{\lambda }p(\lambda )\;P\left({{a}{|}{A,\lambda }}\right)\;P\left({{b}{|}{B,\lambda }}\right)$

where $P\left({{a}{|}{A,\lambda }}\right)$ and $P\left({{b}{|}{B,\lambda }}\right)$ describe single input/output probabilities at Alice's or Bob's system alone, and the value of $\lambda$ is chosen at random according to some fixed probability distribution given by $p(\lambda )$. Intuitively, $\lambda$ corresponds to a hidden variable, or to a shared randomness between Alice and Bob. If a box violates this condition, it is explicitly non-local. However, the study of non-local boxes often also encapsulates local boxes.

The set of non-local boxes most commonly studied are the so-called non-signalling boxes, for which neither Alice nor Bob can signal their choice of input to the other. Physically, this is a reasonable restriction: setting the input is physically analogous to making a measurement, which should effectively provide a result immediately. Since there may be a large spatial separation between the parties, signalling to Bob would potentially require considerable time to elapse between measurement and result, which is a physically unrealistic scenario.

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 allows for the notion of a reduced or marginal probability on both Alice and Bob's measurements, and is formalised by the conditions:

$\sum _{b}P\left({a,b}{|}{A,B}\right)=\sum _{b}P\left({a,b}{|}{A,B'}\right)\equiv P\left({a}{|}{A}\right)\quad \forall {a,X,B,B'}$

and

$\sum _{a}P\left({a,b}{|}{A,B}\right)=\sum _{a}P\left({a,b}{|}{A'B}\right)\equiv P\left({b}{|}{B}\right)\quad \forall {b,B,A,A'}$

The constraints above are all linear, and so define a polytope representing the set of all non-signalling boxes with a given number of inputs and outputs. Moreover, the polytope is convex because any two boxes that exist in the polytope can be mixed (as above, according to some variable $\lambda$ with probabilities $p(\lambda )$) to produce another box that also exists within the polytope.

Local boxes are clearly non-signalling, however non-local boxes may or may not be non-signalling. 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 achieve Tsirelson’s bound in accord with quantum mechanical correlations. Indeed, the set of local boxes form a convex sub-polytope of the non-signalling polytope.

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

$P\left({a,b}{|}{A,B}\right)={\begin{cases}{\frac {1}{2}},&{\mbox{if }}a\oplus b=AB\\0,&{\mbox{otherwise}}\end{cases}}$

where $\oplus$ denotes addition modulo two.

Various attempts have been made to explain why Nature does not allow for stronger nonlocality than Quantum Theory permits. For example, in a recent publication it was found that quantum mechanics cannot be more non-local without violating the Heisenberg uncertainty principle. Strikingly, it has been discovered that if PR boxes did exist, any distributed computation could be performed with only one bit of communication. An even stronger result is that for any non-local box theory which violates Tsirelson's bound, there cannot be a sensible measure of mutual information between pairs of systems. This suggests a deep link between nonlocality and the information-theoretic properties of Quantum Mechanics.

See also

• Quantum pseudo-telepathy
• Wheeler-Feynman absorber theory

External links

• Citizendium: Entanglement