Let us introduce the features of this paradox, well-known by
quantum physicists, for readers not familiar with the subject.

Let us call any 2 events (points of space-time) "independent" if they are separated by a space-like interval, which means that no imaginary moving point starting at one event and going "no faster than light in the void" can ever reach the other (this concept of "no faster than light in the void" is physically meaningful independently of a definition of speed, as it is measurable in principle by sending a beam of light in a tube of void that would be inserted in the same direction). Then:

- According to the geometry of space-time in Relativity, between any
two independent events, it makes no sense to say that one physically
precedes the other, because the laws of physics work the same when
both events are exchanged. Precisely, any relation
*R*between any events*a*and*b*, candidate to be called a "time order" relation, would have to be either symmetric between independent events (*aRb*⇔*bRa*), or only defined relatively to a "viewpoint" (any kind of observer or measurement system, by which, for example, the "times" of events might be defined and compared), while any 2 independent events can be switched (replacing*aRb*by*bRa*) by taking another "viewpoint" which would be equally valid for physics, in the sense that all laws of physics work the same for both viewpoints. - Thus, if a law of physics could ever provide any way for
(what happens at) an event
*a*to causally influence (send a signal to) an independent event*b*, then*b*(or an event coming just after it and thus able to use information from it) could also influence*a*as well by the same method (taking the relation: "*a*can causally influence*b*" in the role of time order). - But this logically impossilble as it would lead to time contradictions (like with the idea of trying to change the past).

From a quantum process (such as spatially separating 2 electrons from a pair) we can produce a pair of 2 crosses with the following property of pure quantum correlation. At first, both crosses are attached together, oriented like × and +, so that they form a star with 8 vertices. On each of these 8 vertices is a bulb.

One experimenter keeps the + on Earth; another one takes the × with him to Mars.

Each experimenter is free to wait any amount of time, then, at any time, he can select one of both axis of his cross; as soon as he selects one axis, one and only one of both bulbs at the ends of this axis lights on.

And, whatever axis is chosen, both of its ends will have the same 50% probability to light on; but anyway, whatever they may choose, there is 85% probability that both vertices that light on (one on the Earth, the other on Mars) had been neighbors before separation. (The exact maximum value of the probability that quantum theory permits in perfect experimental conditions is (1+√1/2)/2).

This remains true no matter if the experimenters follow any strategy or not regarding the time and choice of the axis they will select. It does not depend whether one experimenter selects of axis before or after the other, and before or after he could "see" (get information of) the other experimenter making the other selection of axis, or if they do it at exactly "the same time" (relatively to whatever frame of reference).

Assuming any cross to get measured first on one of its axis, it has 50% chance of giving either result. After this, the chances for the other cross (for either possible choice of direction) switch to 85% or 15% depending on the result of the first measurement. But we should point out that the same experiments with the same effective chances of results can be equally interpreted saying that crosses are measured in either order : both order assumptions give the same effective predictions, so that no experiment can distinguish between both.

We can compare this with the case of classical correlation : if both objects were prepared by giving at random one of a given list (with probabilities) of ways of preparing both objects together to react in a determistic manner. Then we can simply say that the observation of one system affects not the other system, but only what we know of the other system (thus the probabilities

In fact, as we made clear by the way we introduced quantum processes by a similar mathematical structure to that of classical probabilistic processes, the mathematical language of quantum physics does not see any "fundamental difference" between classical correlations and quantum correlations (entanglement). In other words, the effective change that "occurs" (the content of the "action at distance"), that is, the change of the quantum state, looks mathematically similar to the subjective change of expectation of future measurement results, for an observer learning the result of the first observation. Except that... it would be mathematically incoherent to consider this information, that induces a change of expectation, as the a discovery of a reality that existed before the measurement was actually done.

Namely, in the above scenario of measurements, for any process (method of preparation) produced by a classical correlation (i.e. if the pair of crosses was prepared by a random choice of specific combinations of prepared reactions to each selection, that observations simply discover), you cannot do better than a 75% probability of having neighbor vertices light on. This 75% is obtained by preparing the crosses so that the vertices that would light on if their axis is selected, were 4 consecutive ones - but the choice of the series of 4 consecutive vertices among the 8 possible series was taken at random.

But the 85% chance, which can be obtained from a quantum process, cannot be explained saying that each experimenter discovers the answer locally determined by some hidden reality (a local hidden cause that is ready to answer either choice of measurement in a determined way).

Other examples of similar experiments, elsewhere on the web, with the advantage of working with some 100% certainties (instead of arguing on precise values of probabilities of uncertain results):

- With 2 observers, each choosing between 3 questions : described here, "a puzzle".
- With 3 observers, each choosing between 2 questions : GHZ experiment, also described there.

See also : A
simple proof of Bell's Non-locality Theorem , discussed
here

CHSH
inequality

However, despite its strict absence of any effective means to send an information faster than light, this phenomenon has some paradoxical properties : a sort of conspiracy, or play of hide and seek, with the question whether or not any hidden signal goes faster than light, even while this cannot be used to effectively send a signal in practice.

If we consider things from the viewpoint that there is a hidden definite relation of simultaneity (a division of space-time into space-like slices, telling among any two events which event happens before which, even if they are separated by a space-like interval), then the EPR paradox can be interpreted as follows:

The first measurement done produces its random result with probabilities given by the state of the local system as it "globally" was (in the above example, it is probability 0.5 each), but, immediately (in the sense of this absolute simultaneity) this result affects the states of any other systems (probabilities of results if measured) that were correlated to the first.How does it affect the other system ? Assume the first experimenter chose the vertical axis. Then the second one, no matter which diagonal he chooses, has only 15% chance for its result to differ in terms of the up/down description: but 15% chance to differ if choosing one diagonal, plus 15% сhance to differ if choosing the other, gives maximum 30% chances for the result to differ from one choice of diagonal to the other. But if the first experimenter chose the horizontal axis then the same argument goes in terms of right/left, which means minimum 70% chances to differ in terms of up/down, in contradiction with the previous case. As if the first experimenter actually acted at distance, on ... the correlation between what the second measurement would be for one choice of direction, and what it would be for the other. Then, what makes this "not really an action at distance", is the fact that the first observer could not choose the result of his observation, which was instead given "at random" strictly following its probability law, so that the probablities of result for the second measurement remain unchanged subjectively to an observer ignoring the result of the first measurement. Namely, as long as the result of the measurement is not specified, the "effect" on the states of other systems at distance, is neither the effect coming from one possible result, nor the one coming from the other possible result, but the average between them (barycenter weighted by probabilities), which is, in fact, the preservation of the state they had before the measurement.

As for the "effect" that is "transmitted at distance", i.e. "the chances for the result to differ from one choice of diagonal to the other", it is not really an effect because it is not measurable (as only one diagonal can be measured). So, the quantum correlation, while suggesting a sort of interconnectedness between distant places, does not provide any means to transmit information faster than light, because the formalism of quantum physics expresses it in the same language (as if it had the same nature) as classical correlation.

*Counterfactual definiteness* is the idea that there would
be a (hidden) reality of what would have been the result of a
measurement other than (and physically no more doable after) the
one actually made.

The EPR paradox shows that the probabilities of behavior of
quantum-correlated systems cannot be explained as a hidden (random
choice of) predetermination of all measurement results that was
fixed once for all, prior to both measurements. At least, if we
admit the "no-conspiracy" hypothesis, that is: this determination
was not prepared by an entity that could either control or predict
the choices made by experimenters. We shall admit this
no-conspiracy hypothesis for the rest of this discussion. Thus,
the idea of counterfactual definiteness (predetermination of
results) cannot be valid in a global manner (independently of
choices and of time order between them).

Also, it is remarkable that the mathematical formalism of quantum
theory does not contain itself any trace of counterfactual
definiteness (as its predictions are probabilistic, where
probabilities are computed as numbers).

Now we are going to enter some more logical details : is it still
possible to defend (find a logical possibility for) a partial
applicability of the concept of counterfactual definiteness, such
as, not as a definite but as a probabilistic information;
relatively to the time order between choices, or relatively to the
choice by the other experimenter, or relatively to either
measurement result ?

First thought experiment : imagine that the × is measured first
along one axis, but that the counterfactual measurement (what
would have been found along the other axis) was made too; and that
both results are "top". Knowing these 2 data, what probabilities
of expectation can we give to the measurement results on the other
system ?

(to be continued)

(Wikipedia : Counterfactual
definiteness)

Related pages:

Introduction to quantum physics: states, correlations and measurements (describes the mathematical structure of quantum physics, and how it coherently provides such predictions)

Next pages:

The double-slit experiment

Quantum decoherence

Interpretations of quantum physics