EPR paradox

The Einstein - Podolsky-Rosen paradox, also EPR paradox, or EPR effect, is an intensely debated in the 20th century quantum mechanical phenomenon. The effect was named after Albert Einstein, Boris Podolsky and Nathan Rosen, who introduced this phenomenon in the context of a thought experiment. It is sometimes spoken of an EPR argument. It exemplifies that quantum mechanics violates the assumption of locality that is one of the basic assumptions of classical physics.

3.1 Applications of non-locality

4.1 Spinorraum
4.2 the wave function collapse

Basic problem

In the original formulation of their thought experiment it was Einstein, Podolsky and Rosen therefore, demonstrate that the quantum mechanical description of physical reality, which is " to the point " in this effect, must be incomplete. In even simpler: It is shown that quantum mechanics is not a classical theory.

There are several experimental arrangements show the characteristic behavior of the EPR effect. In principle, such an EPR -type experiment the following characteristics:

It is a system of two particles (T1, T2 ) is considered, the initially directly interact with each other and it far away from one another (eg diametrically diverging particles after a breakup ). Such a system is described by a single, specific quantum mechanical state. This state is not a product state, ie the two particles are in a special entangled state.
At the spatially separated particles called two complementary metrics are considered, such as position and momentum, or two different - looking angular momentum components. The simultaneous accurate determination of these measures is impossible according to the Heisenberg uncertainty relation.
It is shown that the values of these measurements for the two particles, in spite of the separation and in spite of the uncertainty are strictly correlated: One of the particles is located after the measurement in a net value of the first measured variable, the other on the complementary value of the second size. Which of the two particles 1 and the measured value which yields 2, which must follows the rules of probability theory are " erwürfelt ".

Most commonly, revised by David Bohm version of the EPR experiment is discussed today. Here two particles with spin ( angular momentum ) are considered, the total spin is (sum of the spins of the individual particles ) is zero. In this reformulation, the experiment is also practicable. Einstein, Podolski and Rosen chose initially position and momentum of the particles as complementary observables.

The following are the Bohmian variant is presented. First, the result of the EPR experiment is summarized and described its significance for the interpretation of quantum mechanics. Then the quantum mechanical explanation of the experiment and the necessary for their understanding properties of quantum mechanics are briefly presented.

The EPR thought experiment and its interpretation

The original EPR argument for the incompleteness of quantum mechanics

Einstein, Podolsky and Rosen ( EPR) consider in their 1935 published in Physical Review Article Can quantum -mechanical description of physical reality be Considered complete? Position and momentum of two particles (T1, T2) as complementary observables. Is the pulse of particles 1 (T1) is measured. Thus, the observed entangled state changes so that now the output of a pulse measurement on particle 2 (T2 ) can be predicted accurately with probability 1. This T2 was certainly not disturbed by uncontrolled interaction. It could just as well instead of the place are determined by T1, which, again without an error, the location of T2 would now be accurately predicted.

To conclude that quantum mechanics is incomplete, then perform the following assumptions:

In a complete theory, each element of physical reality must have a counterpart.
A physical quantity whose value is predicted with certainty, without disturbing the system in which it is measured, is an element of physical reality.

Since the decision whether the location of T2 or his pulse is determined by measuring the respective counterparts at T1, needs to be made until shortly before the measurement, it can certainly have no adverse effect on elements of the reality of T2. This close EPR that both variables would have to be part of the same physical reality. Since, however, according to the quantum mechanics for each individual particle, only one of the variables can be predicted, the quantum mechanics is incomplete.

Niels Bohr turned in the same year 1935 in a same article against this argument, that the concept of interference-free measurement is not adequately defined, when he was confined to a mechanical interaction in the final phase of the experiment. Such products does not, in fact, still close to the experimental setup, which would lead to the accurate prediction of the location of T2, just the complementary experiment to determine its momentum, which is why both sizes are not members of the same reality, but elements of two complementary realities are.

The EPR experiment as a paradox

Occasionally, also from the EPR experiment as a paradox of the question. This seems paradoxical at first glance, that two complementary observables of a particle can be determined simultaneously - the one about directly by measurement of T1, the other indirectly by measurement of T2. This is seemingly a contradiction to the well-known Heisenberg uncertainty relation. In the Copenhagen interpretation, the paradox is resolved by pointing out that the indirect determination of the measurement of T2 is just no measurement of the property of the T1.

Local hidden variable and EPR correlation experiments

Since the EPR paper (1935 ) Einstein pursued until his death (1955 ) a persistent goal to complete the quantum mechanics in the sense of EPR. His basic assumption was that quantum mechanics on its own "common sense" is contrary to ("God does not play dice ").

The inaccuracy of the EPR considerations, it was proved in two steps. John Stewart Bell laid in the 1960s, the theoretical basis for an empirical verification, which succeeded in 1982.

John Stewart Bell presented in 1964 to the named today after him Bell's inequality. He showed that the basic assumptions of EPR the validity of the inequality force - if EPR were correct in their criticism. In other words, the validity of Bell 's inequality is incompatible with quantum mechanics. In particular, the quantum mechanical theory violates the inequality strong enough that an empirical rule on the validity of the EPR assumptions is possible.

Thus, the Bell's inequality resulted in the ability to decide in concrete experiments between quantum mechanics and Einstein's assumptions ( " either - or " decision ), that is, to falsify one of the theories.

The Bell's inequality implicitly and corresponding EPR for empirical verification "hidden local variables ", which exactly fill the role of possibly " incomplete " description of reality in quantum mechanics. So can be demonstrated empirically that the Bell's inequality

(A) is violated verifiable at least one case, the existence of local hidden variable can be ruled out. The EPR effect then provides no point to keep the quantum mechanics is incomplete. In particular, is also admitting that the (naive ) realism of the EPR argument does not apply, leave the world completely " classic " to describe;

(B ) is always met, the incompleteness of quantum mechanics would be the then presumable existence of "local hidden variables " proven. The forward by EPR of reality and locality finds that there would be strengthened.

The experimental decision between these two alternatives ( among others by Alain Aspect ) confirmed the quantum mechanical predictions and showed that the EPR adoption of locality and realism (ie, at least one of these two assumptions is not true ). The experiments show a ( quantum mechanically required ) correlation between the measurement results of a spinning experiment, which is significantly greater than this, i.e., according to the Bell inequality would be conceivable in a classical theory. This " non-locality " is reflected in the quantum mechanical description system by the fact that the state of the system is determined at each time by a single abstract state vector at the same time at all points (x, y, z).

Applications of non-locality

A first practical application of proven non-locality of quantum physical reality is quantum cryptography. In this context, the so-called Aharonov - Bohm effect deserves attention.

However, it is not possible to communicate by means of the EPR effect than light speed: The single measurement results - regardless of whether the other particle has been measured - always in itself unpredictable result. Only if the result of the other measurement - by classical, under light- fast communication - is known, one can determine or exploit the correlation.

Quantum theoretical foundations of the EPR experiment

Spinorraum

The quantum mechanics of a particle Spin-1/2-Freiheitsgrades takes place in a particularly simple Hilbert space, the 2-dimensional Spinorraum for a single particle. Moreover, only very simple properties of this space for the EPR experiment play a role.

The first is that the eigenvectors of two non- commuting operators form two different bases of the same subspace. This we can illustrate where each of the two particles (T1, T2) corresponding 2 -dimensional Spinorraums as in the illustration. Illustration of the two -dimensional Spinorraumes The figure shows an observable complementary x and y component of the spins in the form of 45 degrees to each other rotated coordinate systems (which respectively correspond to the bases of the eigenvector associated with the observables operators). If you draw a perpendicular from the state vector ψ on that pertains to the measurement coordinate axis ( the eigenvector corresponding to the eigenvalue ), we obtain the quantum mechanical probability to find precisely this value in a measurement. The fact that this probability can be only one value for one of the observables equal to one obviously explains exactly the Heisenberg Uncertainty Principle.
The second is the fact that the quantum-mechanical state of a space Mehrteilchensystems obtained as the direct product of the state space of its components, so as to obtain a state vector of a 2-Teilchen-Spin-1/2-Systems 4-dimensional linear subspace, for the all ordered pairs of basis vectors of the two -dimensional Spinorräume exists. The result is that the collapse of the wave function by the measurement of a particle in general also changes the state of the other particle ( see next section).

Collapse of the wave function

The so-called collapse of the wave function should be called in our image better projection of the state vector. It is postulated in the quantum mechanics to describe the preparation of a system or the quantum measurement. In the usual interpretation of quantum mechanics ( the Copenhagen interpretation and related approaches ) is the projection of the state vector is introduced as an independent postulate: If an observable measured on a system that is going to be state vector by leaps and bounds in the projection of the current state vector to the eigenvector corresponding to the measured eigenvalue over. In a folded state, this means that the state changes so that also with respect to the probabilities for the measurement results on the respective other system. Is about the initial state ( up to normalization ), the eigenvector for the measurement of a positive spins in a particular direction ("X - direction " ) of system 1 is. By measuring, for example, a negative spin in the x-direction of system 1 is now disappearing all the components of the initial state containing the eigenvector to positive spin at T1. Thus, the state transitions to, i.e. at T2 is a further measurement of the spins in the x direction give with certainty a positive spin. If we write the collapsed wave function in the eigenvector basis of complementary observables (spin in the y- or x - direction, the rotated coordinate system in the picture) out so you can see that both values are equally likely again in one of these directions. So could an observer of T2 exact make copies of the quantum state, he could actually determine what observable, the observer of the first particle is measured, and it would be a ( faster than light ) flow of information from one observer to observer 2 possible. Such " quantum repeater " but there is no.

Aharonov–Bohm effect No-Cloning-Theorem Digital Object Identifier Stanford Encyclopedia of Philosophy Internet Encyclopedia of Philosophy Karl Popper

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