Principle of Locality

Principle of Locality

In physics, the principle of locality is that distant objects cannot have direct influence on one another: an object is influenced directly only by its immediate surroundings. This was stated as follows by Albert Einstein in his article "Quantum Mechanics and Reality" ("Quanten-Mechanik und Wirklichkeit", Dialectica 2:320-324, 1948):

The following idea characterises the relative independence of objects far apart in space (A and B): external influence on A has no direct influence on B; this is known as the Principle of Local Action, which is used consistently only in field theory. If this axiom were to be completely abolished, the idea of the existence of quasienclosed systems, and thereby the postulation of laws which can be checked empirically in the accepted sense, would become impossible.

Local realism is the combination of the principle of locality with the assumption that all objects must objectively have their properties already before these properties are observed. Einstein liked to say that the Moon is "out there" even when no one is observing it.

Local realism is a significant feature of classical general relativity and classical Maxwell's theory, but quantum mechanics rejects this principle. Every theory that, like quantum mechanics, is compatible with violations of Bell's inequalities must abandon local realism. (Most physicists believe that experiments have demonstrated such violations, but some local realists dispute the claim, in view of the recognised loopholes in the tests.) Different interpretations of quantum mechanics reject different parts of local realism.

In most of the conventional interpretations, such as the version of the Copenhagen interpretation in which the wavefunction is not real, the many-worlds interpretation, and the interpretation based on Consistent Histories, it is realism that is rejected. The actual definite properties of a physical system "do not exist" prior to the measurement and the wavefunction is only interpreted as a mathematical tool used to calculate the probabilities of the outcome of the experiments, which is, in agreement with positivism in philosophy, the only topic that science should discuss.

In the version of the Copenhagen interpretation where the wavefunction is real, it is the principle of locality that is violated. The wavefunction is a real object that exists prior to the measurement, but the measurement causes the wavefunction collapse which is a non-local process.

The Bohm interpretation always wants to preserve realism, and it needs to violate the principle of locality to achieve the required correlations. In fact, it needs to violate not only locality, but also causality which seems to imply a real conflict with the special theory of relativity because real, superluminal signals would have to be propagated.

Because the differences between the different interpretations are mostly philosophical ones (except for the Bohm interpretation), the physicists usually use the language in which the important statements are independent of the interpretation we choose. In this framework, only the measurable action at a distance - a superluminal propagation of real, physical information - would be usually considered to be a violation of locality by the physicists. Such phenomena have never been seen, and they are not predicted by the current theories (with the possible exception of the Bohm theory). This is why relativistic quantum field theory is usually considered to be a local theory.

The above is only about one sense of locality. In another use of the term, if we have two observables, each localized within two distinct spacetime regions which happen to be at a spacelike separation from each other, both observables would commute and we have locality. This interpretation of the word "locality" is closely related to the relativistic version of causality in physics.

Nonlocality

Nonlocality in quantum mechanics, refers to the property of entangled quantum states in which both the entangled states "collapse" simultaneously upon measurement of one of their entangled components, regardless of the spatial separation of the two states. This "spooky action at a distance" is the content of Bell's theorem and the EPR paradox.

Actions obtained from nonlocal Lagrangians are called nonlocal actions. The actions appearing in the fundamental theories of physics, such as the Standard Model, are local actions - nonlocal actions play a part in theories which attempt to go beyond the Standard Model, and also appear in some effective field theories. Nonlocalization of a local action is also an essential aspect of some regularization procedures.

Action at a Distance

In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. This term was used most often with early theories of gravity and electromagnetism to describe how an object could "know" the mass (in the case of gravity) or charge (in electromagnetism) of another distant object.

According to Albert Einstein's theory of special relativity, instantaneous action-at-a-distance was seen to violate the relativistic upper limit on speed of propagation of information. If one of the interacting objects were suddenly displaced from its position, the other object would feel its influence instantaneously, meaning information had been transmitted faster than the speed of light.

Electricity

Coulomb's law in electrostatics appears to be a theory with action-at-a-distance - Coulomb's law deals with charges which have always been static. Efforts to develop a theory of interaction between moving charges, electrodynamics, led to the necessity to introduce the concept of a field with physical properties. In the theory of electrodynamics as formulated in Maxwell's equations, interactions between moving charges are mediated by propagating deformations of an electromagnetic field. These deformations propagate with the speed of light and therefore do not violate special relativity. The deformations of the field can carry momentum independently, thus facilitating conservation of angular momentum.

Gravity

Newton's theory of gravity offered no prospect of identifying any mediator of gravitational interaction. His theory assumed that gravitation acts instantaneously, regardless of distance. Newton had shown mathematically that if the gravitational interaction is not instantaneous, angular momentum is not conserved, and Kepler's observations gave strong evidence that in planetary motion angular momentum is conserved. (The mathematical proof is only valid in the case of a Euclidean geometry)

A related question, raised by Ernst Mach, was how rotating bodies know how much to bulge at the equator. How do they know their rate of rotation? This, it seems, requires an action-at-a-distance from distant matter, informing the rotating object about the state of the universe. Einstein coined the term Mach's principle for this question.

Einstein: One of the conditions that a relativistic theory of gravitation must meet is to be mediated with a speed that does not exceed lightspeed. It could be seen from the previous success of electrodynamics that the relativistic theory of gravitation would have to use the concept of a field or something similar.

This problem has been resolved by Einstein's theory of general relativity in which gravitational interaction is mediated by deformation of space-time geometry. Matter warps the geometry of space-time and these effects are, as with electric and magnetic fields, propagated at the speed of light. Thus, in the presence of matter, space-time becomes non-Euclidean, resolving the apparent conflict between Newton's proof of the conservation of angular momentum and Einstein's theory of special relativity. Mach's question is resolved because local space-time geometry is informing a rotating body about the rest of the universe. In Newton's theory of motion, space acts on objects, but is not acted upon. In Einstein's theory of motion, matter acts upon space-time geometry, deforming it, and space-time geometry acts upon matter.

Quantum Mechanics

Current physical theories incorporate the upper limit on propagation of interaction as one of their basic building blocks, hence ruling out instantaneous action-at-a-distance. At the same time however, instantaneous action at a distance appears to be an essential feature of some very fundamental quantum mechanical effects like entanglement and quantum nonlocality. For these implications of quantum mechanics, Einstein coined the term "spooky action at a distance".

The question of whether this 'spooky action' at a distance constitutes a violation of the relativistic upper limit on the propagation of interaction is not straightforward. According to the laws of quantum mechanics, entanglement cannot be employed for relaying information from one place to another.

Quantum Teleportation

Quantum teleportation is a technique discussed in quantum information science to transfer a quantum state to an arbitrarily distant location using an entangled state and the transmission of some classical information.

Indistinguishability

Let's say that Alice has a rubidium atom (the element physicists in this field like to use for their experiments), which is in its ground state, and Bob also has such an atom, as well in its ground state. It is important to see that these two atoms are indistinguishable; that means that there really is no difference between them.

If Alice and Bob had, say, two glass balls, which exactly look alike, and they exchanged them, then something would change. If you had a powerful microscope, you could certainly find some difference between the two balls. For atoms of the same kind and in the same quantum state, however, there really is no difference at all. The physical situation with Alice having the first atom and Bob the second is exactly the same as vice versa.1 In a certain sense, it is even wrong to say that the two atoms have any individuality or identity. It would be more appropriate to say that the two locations in space both have the property that the fundamental quantum fields have those values which define the ground state of the rubidium atom.

Quantum Teleportation: The Result

Now, imagine Alice's atom being in some complicated (excited) quantum state. Assume that we do not know this quantum state — and of course, we cannot find out by inspection (measurement). But what we can do is to teleport the quantum state to Bob's rubidium atom. After this operation, Bob's atom is exactly in the state that Alice's atom was before.

Now note that Bob's atom afterwards is indistinguishable from Alice's atom before. In a way, the two are the same — because it does not make sense to claim that two atoms are different only because they are at different locations. If Alice had gone to Bob and given him the atom we would have exactly the same physical situation.

But Alice and Bob were not required to meet. They only needed to share entanglement. The method: As a prerequisite, Bob has produced two particles (or atoms, to stay with the concrete example) called I and II, which are maximally entangled, e.g. in the Bell state.

This means that the particles I and II are neither in the state |0> nor in |1> but rather in both simultaneously, and if you measure one of them you will find it to be |0> or |1> each with probability 1/2. But the other is then always in the same state, and this connects them by some "spooky action at a distance", as Einstein has put it.

Bob passed particle I to Alice (by moving it there through a quantum channel) and kept particle II. From now on, Alice can send a quantum state to Bob, whenever she wants (or Bob to Alice).

If Alice now has a particle which she wants to teleport to Bob, she does a so-called Bell measurement on the particle to be sent and on particle I. The Bell measurement (for details see the article on it) projects Alice's two particle into one of the four Bell states. The two measured particles are afterwards in a known entangled state, specified by the measurement result, and the particle to be sent has lost its former state. That state, however, is not gone: it was teleported to Bob's particle II due to the previously existing entanglement.

One problem remains: The state of Bob's particle might be "rotated". This depends on the outcome of Alice's measurement, and Alice has to tell Bob this outcome by using some classical (i.e. ordinary) communication channel and Bob will then apply a unitary operation (a so-called Pauli operator) onto his particle.2 Afterwards, he really has the desired state.

Star Trek beaming: That the term teleportation reminds of the beaming process in the Star Trek TV series might have been intentional. After all, the physicists coining the term might have thought, it is as if the particle would be brought there. But this analogue is prone to miss the point: Only the information about the quantum state is brought there, the particle to take up the state must already be present.

Entanglement swapping: If a state being teleported is itself entangled with another state, the entanglement is teleported with it. To illustrate: If Alice has a particle which is entangled with a particle owned by Carol, and she teleports it to Bob, then afterwards, Bob's particle is entangled with Carol's.

A more symmetric way to describe the situation is the following: Alice has one particle, Bob two, and Carol one. Alice's particle and Bob's first particle are entangled, and so are Bob's second and Carol's particle:

Alice-:-:-:-:-Bob1 Bob2-:-:-:-:-Carol

Now, Bob performs a Bell measurement on his two particles, which projects them into a Bell state, i.e. they are now entangled. But, more spectacularly, Alice's and Carol's particles are now entangled as well, although the two never met:

Alice......Bob1-:-Bob2......Carol
|............................................|
\-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-/

This effect allows, at least in theory, to build a quantum repeater. Breaking news: [[1]] demonstrates one node of a quantum repeater.