User:Boris Tsirelson/Sandbox1

=Entanglement (physics)=

There are three interrelated meanings of the word "entanglement" in physics. They are listed below and then discussed, both separately and in relation to each other.


 * A hypothetical combination of empirical facts incompatible with local causality and counterfactual definiteness (below called "empirical entanglement", which is not a standard terminology).
 * A prediction of the quantum theory stating that the empirical entanglement must occur in appropriate physical experiments (below called "quantum entanglement").
 * In quantum theory there is a technical notion of "entangled state".

Empirical entanglement
Some people understand it easily, others find it difficult and confusing.

It is easy, since no physical or mathematical prerequisites are needed. Nothing like Newton laws, Schrodinger equation, conservation laws, nor even particles or waves. Nothing like differentiation or integration, nor even linear equations.

It is difficult and confusing for the very same reason! It is highly abstract. Many people feel uncomfortable in such a vacuum of concepts and rush to return to the particles and waves.

The framework, and local causality
The following concepts are essential here. Otherwise the apparata are not restricted; they may use all kinds of physical phenomena. In particular, they may receive any kind of information that reaches them. We treat each apparatus as a black box: the switch position is its input, the light flashed is its output; we need not ask about its internal structure.
 * A physical apparatus that has a switch and several lights. The switch can be set to one of several possible positions. A little after that the apparatus flashes one of its lights.
 * "Local causality": widely separated apparata are incapable of signaling to each other.

However, not knowing what is inside the black boxes, can we know that they do not signal to each other? There are two approaches, "loose" and "strict".

The loose approach: we open the black boxes, look, see nothing like mobile phones and rely on our knowledge and intuition.

The strict approach: we do not open the black boxes. Rather, we place them, say, 1,000,000 km apart and restrict the experiment to a time interval of, say, 1 sec. Relativity theory states that they cannot signal to each other, for a good reason: a faster-than-light communication in one inertial reference frame would be a backwards-in-time communication in another inertial reference frame!

Counterfactual definiteness
In this section a single apparatus is considered.

An experiment performed on the apparatus is described by a pair (x,y) where x is the input (the switch position) and y is the output (the light flashed). Is y a function of x? We may repeat the experiment with the same x and get a different y (especially if the apparatus tosses a coin). We can set the switch to x again, but we cannot set all molecules to the same microstate. Still, we may try to imagine the past changed, asking a counterfactual question: It is meant that only the input x is changed in the past, nothing else. The question may seem futile, since an answer cannot be verified empirically. Strangely enough, the question will appear to be very useful in the next section.
 * Which outcome the experimenter would have received if he/she did set the switch to another position?

Classical physics can interpret the question as a change of external forces acting on a mechanical system of a large number of microscopic particles. It is unfeasible to calculate the answer, but anyway, the question makes sense, and the answer exists in principle:
 * $$ y = f(x) $$

for some function $$ f : X \to Y, $$ where X is the finite set of all possible inputs, and Y is the finite set of all possible outputs. Existence of this function f is called "counterfactual definiteness".

Repeating the experiment we get
 * $$ y_i = f_i(x_i) $$

for $$ i=1,2,\dots $$ Each time a new function fi appears; thus xi=xj does not imply yi=yj. In the case of a single apparatus, counterfactual definiteness is not falsifiable, that is, has no observable implications. Surprisingly, for two (and more) apparata the situation changes dramatically.

Local causality plus counterfactual definiteness
For two apparata, A and B, an experiment is described by two pairs, (xA,yA) and (xB,yB) or, equivalently, by a combined pair ((xA,xB), (yA,yB)). Counterfactual definiteness alone (without local causality) takes the form
 * $$ (y_{\rm A},y_{\rm B}) = f_{\rm A,B} (x_{\rm A},x_{\rm B}) $$

or, equivalently,
 * $$ y_{\rm A} = f_{\rm A} (x_{\rm A},x_{\rm B}), y_{\rm B} = f_{\rm B} (x_{\rm A},x_{\rm B}). $$

Assume in addition that A and B are widely separated and the local causality applies. Then xA cannot influence yB, and xB cannot influence yA, therefore
 * $$ y_{\rm A} = f_{\rm A} (x_{\rm A}), \quad y_{\rm B} = f_{\rm B} (x_{\rm B}). $$

Wikipedia
Counterfactual definiteness -- Quantum entanglement is probably the most mysterious of all the phenomena discovered by physics.

Some striking features of quantum physics, explained below, are prerequisites.

Classical physics considers first of all closed (autonomous) physical systems. For quantum physics, systems open to external influence are of great importance.

Measurement and influence
In classical physics an ideal measurement exerts no influence on the object. It only reveals some properties of the object to the experimenter. The experimenter is able to choose an observable (a variable to be measured), or to measure all possible observables at once, and still, the object may be treated as a closed system.

Quantum physics is strikingly different. The influence of a measurement on the object is almost inevitable. If a macroscopic measuring device together with some environment is treated as a part of the quantum system (the object), and the experimenter only observes the reading of the device, then in some sense the experimenter does not influence the object. This is a subtle point related to quantum decoherence. Typically, macroscopic devices are not included into the quantum system, which implies that every measurement inevitably exerts a substantial influence on the object.

In general, two measuring devices cannot be applied simultaneously to the same object. Thus, in general, two observables are incompatible. For example, the coordinate qx and the momentum px of a particle are incompatible (but qx and py are compatible). The experimenter may choose one of the two observables, coordinate or momentum, and measure it, thus exerting a substantial influence on the other observable.

Local causality and influence
Local causality negates action on a distance. Basically it states that if two objects A, B are far apart in space then any external influence on A has no direct influence on B.

A strict relativistic interpretation states that a signal cannot propagate faster than light. More exactly, let A, B be two domains in space-time. (For example, A may be a given space ship during a given one-second time interval, according to its local clock, and B another space ship, 1,000,000 km apart, during its time interval.) Assume that a light ray emitted from A cannot reach B. (For example, because it can travel only 300,000 km during the given second.) Then any external influence exerted within A is of no consequence within B. (For example, a sudden explosion on the first space ship during its time interval cannot cause anxiety on the second space ship before the end of its time interval.)

Local causality does not contradict the evident existence of objects extended in space (solid bodies, sea waves and many others). For example, the hull of a space ship being a single solid body, it may seem that its parts move in ideal, non-delayed coordination; but this is an illusion. If one part was hit by a meteorite 30 nanoseconds before, this fact cannot have yet any consequence for another part 20 m apart.

An extended object is a manifestation of correlations (rather than nonlocality). Slowly propagating signals (for example, newspapers delivered by surface mail) routinely create strong correlations between remote objects, possibly not interacting with each other.