Einstein-Podolsky-Rosen paradox

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The Einstein-Podolsky-Rosen paradox , also EPR paradox , or EPR effect , is a quantum mechanical phenomenon that was intensely discussed in the 20th century . The effect was named after Albert Einstein , Boris Podolsky and Nathan Rosen , who introduced this phenomenon as part of a thought experiment . Sometimes there is also talk of an EPR argument . It shows, for example, that quantum mechanics violates the assumption of locality , which is one of the basic assumptions of classical physics.

In this context, Einstein also spoke of a "spooky action at a distance" .

Basic problem

In the original formulation of their thought experiment, Einstein, Podolsky and Rosen wanted to prove that the quantum-mechanical description of physical reality, which is "brought to the point" in this paradox, must be incomplete. To put it even more simply: it is shown that quantum mechanics is not a classical theory .

There are several experimental arrangements which show the behavior characteristic of the EPR effect. Such an EPR-like experiment basically has the following characteristics:

  • A system of two particles (T1, T2) is considered, which initially interact directly with one another and then move far apart (e.g. diametrically diverging particles after a decay). Such a system is described by a single , special quantum mechanical state. This state is not a product state, that is, the two particles are in a special entangled state .
  • Two so-called complementary measured variables are considered on the spatially separated particles , e.g. B. Place and momentum, or two differently directed angular momentum components. The simultaneous exact determination of these measured quantities is impossible according to Heisenberg's uncertainty principle .
  • It is shown that the values ​​of these measurands for the two particles, despite the separation and despite the uncertainty relation, are strictly correlated: After the measurement, one of the two particles is in an eigenvalue of the first measurand, the other in the complementary value of the second Size. Which of the two particles results in the measured value 1 and which results in 2 must be "diced" according to the laws of probability.

The version of the EPR experiment revised by David Bohm is most frequently discussed today . Here two particles with spin (intrinsic angular momentum) are considered whose total spin (sum of the spins of the individual particles) is zero. With this reformulation, the experiment can also be carried out in practice. Einstein, Podolsky and Rosen originally chose the position and momentum of the particles as complementary observables .

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

The EPR thought experiment and its interpretation

EPR's original argument for the supposed incompleteness of quantum mechanics

Einstein, Podolsky and Rosen (EPR) published the article Can quantum-mechanical description of physical reality be considered complete? In the Physical Review in 1935 . You consider the position and momentum of two particles (T1, T2) as complementary observables. The momentum of particle 1 (T1) is measured. The observed entangled state changes in such a way that the output of a momentum measurement on particle 2 (T2) can now be predicted exactly with probability 1. T2 was certainly not disturbed by an uncontrolled interaction. Instead, the location of T1 could just as easily be determined, as a result of which, again without a disturbance, the location of T2 would now be exactly predictable. The following assumptions now lead to the conclusion that quantum mechanics is incomplete:

  • In a complete theory, every element of physical reality must have a correspondence.
  • A physical quantity whose value can be predicted with certainty without disturbing the system on which it is measured is an element of physical reality .

Since the decision as to whether the location of T2 or its impulse is determined by measuring the respective counterparts at T1 only needs to be made shortly before the measurement, it certainly cannot have a disruptive influence on elements of the reality of T2. EPR conclude from this that both quantities must be part of the same physical reality. Since, however, according to quantum mechanics, only one of the quantities can be predicted for each individual particle, quantum mechanics is incomplete.

In an article of the same name in 1935, Niels Bohr objected to this argument that the concept of interference-free measurement was not adequately defined if it was limited to a mechanical interaction in the final phase of the experiment. Such is indeed not the case, but the experimental set-up that leads to the precise prediction of the location of T2 concludes the complementary experiment to determine its momentum, which is why both quantities are not elements of the same reality, but elements of two complementary realities.

The EPR experiment as a paradox

Occasionally the EPR experiment is also referred to as a paradox. At first glance it seems paradoxical that two complementary observables of a particle can be determined at the same time - one directly by measuring at T1, the other indirectly by measuring at T2. This is apparently a contradiction to the well-known Heisenberg uncertainty principle . In the Copenhagen interpretation , the paradox is resolved with the reference to the fact that the indirect determination via the measurement at T2 is not a measurement of the property of T1 at all.

Local hidden variable and EPR correlation experiments

From his work on EPR (1935) until the end of his life (1955) Einstein stubbornly pursued the goal of completing quantum mechanics in the sense of EPR . His basic assumption remained that quantum mechanics, taken by itself, contradicts “common sense” ( “ God does not roll the dice ” ).

The flawedness of the EPR considerations was proven in two steps. In the 1960s, John Stewart Bell laid the theoretical basis for an empirical review that succeeded in 1982.

In 1964, Bell set up Bell's inequality , which is now named after him , and showed that it is valid for every classical theory . He showed that the basic assumptions of EPR would enforce the validity of the inequality: Exactly if EPR were right with their criticism, Bell's inequality would have to be fulfilled in the experiment. In other words: The validity of Bell's inequality would be incompatible with quantum mechanics. In particular, it is true that the quantum mechanical theory violates the inequality strongly enough that an empirical decision about the validity of the EPR assumptions is possible .

Thus, Bell's inequality made it possible to decide between quantum mechanics and Einstein's assumptions in specific experiments (“either-or” decision), i. H. to falsify either theories.

Bell's inequality implicitly and in accordance with EPR introduces “hidden local variables” for the empirical examination, which precisely fill the role of the possibly “incomplete” description of the reality of quantum mechanics. So it can be proven empirically that Bell's inequality

  • (A) is verifiably violated in at least one case , the existence of local hidden variables can be excluded. The EPR effect then does not provide a starting point for considering quantum mechanics to be incomplete. In particular, it must also be admitted that the (naive) realism of the EPR argument that the world can be described completely “classically” does not apply;
  • (B) is always adhered to , the incompleteness of quantum mechanics would then be proven via the then presumed existence of “local hidden variables”. The perception of reality and locality represented by EPR would be strengthened.

The experimental decision between these two alternatives (by Alain Aspect among others ) confirms the quantum mechanical predictions and refutes the EPR assumption of locality and realism , i.e. H. at least one of these two assumptions is incorrect. The experiments show a (quantum mechanically required) correlation between the measurement results of a spin experiment, which is significantly greater than in a classical theory, i.e. H. according to Bell's inequality, would be conceivable. This “non-locality” is shown in the quantum mechanical system description by the fact that the state of the system is determined at any point in time by a single abstract state vector at all points (x, y, z) .

Applications of non-locality

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

However, it is not possible to use the EPR effect to communicate at faster speeds than light : the individual measurement - regardless of whether the other particle has already been measured - always gives an unpredictable result in and of itself. Only when the result of the other measurement is known - through classic, sub-light communication - can the correlation be determined or exploited.

Quantum theoretical foundations of the EPR experiment

Spinor space

The quantum mechanics of the spin 1/2 degree of freedom of a particle takes place in a particularly simple Hilbert space, the 2-dimensional spinor space for a single particle. In addition, only very simple properties of this space play a role for the EPR experiment.

  • The first is that the eigenvectors of two non-commuting operators form two different bases of the same subspace. We can illustrate this with the 2-dimensional spinor space corresponding to one of the two particles (T1, T2), as shown in the adjacent figure.
    Illustration of the 2-dimensional spinor space
    The figure shows the x and y components of the spin as complementary observables in the form of coordinate systems rotated by 45 degrees against each other (each corresponding to the eigenvector bases of the operators belonging to the observables). If the perpendicular from the state vector ψ falls on the coordinate axis belonging to a measured value (the eigenvector of the eigenvalue), then one obtains the quantum theoretical probability of finding this measured value during a measurement. The fact that this probability can obviously only be equal to one for exactly one value of one of the observables explains precisely the Heisenberg uncertainty relation.
  • The second is the fact that the quantum mechanical state space of a multi-particle system results as the direct product of the state spaces of its constituent parts, i.e. a 4-dimensional, linear vector space is obtained as the state vector of a 2-particle spin 1/2 system, that of all ordered pairs of vectors of the 2-dimensional spinor spaces is generated. As a result, the collapse of the wave function by measuring one particle generally 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 better be called the projection of the state vector in our picture . It is postulated in quantum mechanics in order to describe the preparation of a system or the quantum mechanical measuring process . In the usual interpretations of quantum mechanics ( Copenhagen interpretation and related approaches) the projection of the state vector is introduced as an independent postulate: If an observable is measured on a system, its state vector suddenly changes into the projection of the previous state vector onto the eigenspace for the measured eigenvalue. In the case of an entangled state, this means that the state also changes with regard to the probabilities of measurement results on the other system. Let, for example, be the initial state (except for normalization) , where the eigenvector for measuring a positive spin in a certain direction (“x-direction”) on system 1 is. By measuring z. B. of a negative spin in the x-direction on system 1, all components of the initial state that contain the eigenvector for positive spin at T1 disappear. The state therefore changes into , i.e. H. At T2 a further measurement of the spin in the x-direction will definitely result in a positive spin. If you write the collapsed wave function in the eigenvector base of the complementary observable (spin in y- or x-direction, the rotated coordinate system in the picture), you can see that both values ​​are again equally likely in one of these directions. If an observer of T2 could make exact copies of its quantum state, he could actually determine which observables the observer of the first particle measured, and a (faster than light) flow of information from observer 1 to observer 2 would be possible. However , there are no such “ quantum amplifiers ”.

literature

  • A. Einstein, B. Podolsky, N. Rosen: Can quantum-mechanical description of physical reality be considered complete? , Phys. Rev. 47 (1935), pp. 777-780 doi : 10.1103 / PhysRev.47.777 .
  • N. Bohr, Can Quantum-Mechanical Description of Physical Reality be Considered Complete? , (= Reply), in: Physical Review, 48 (1935), p. 700. doi : 10.1103 / PhysRev.48.696 .
  • C. Kiefer (eds.), Albert Einstein, Boris Podolsky, Nathan Rosen: Can the quantum mechanical description of physical reality be regarded as complete? Reprint of the original work in German translation with detailed commentary. doi : 10.1007 / 978-3-642-41999-7 .
  • D. Bohm, Y. Aharonov: Discussion of Experimental Proof for the Paradox of Einstein, Rosen and Podolsky , Phys. Rev. 108 (1957), pp. 1070-1076 doi : 10.1103 / PhysRev.108.1070
  • Alexander Afriat, Franco Selleri: The Einstein, Podolsky, and Rosen paradox in atomic, nuclear, and particle physics. Plenum Press, New York 1999, ISBN 0-306-45893-4 .
  • Max Jammer: The Philosophy of Quantum Mechanics , John Wiley & Sons, New York 1974, pp. 159-251, ISBN 0-471-43958-4 .
  • John Stewart Bell : Quantum Mechanics, Six Possible Worlds and Further Articles , de Gruyter, Berlin 2015, especially the introduction of Alain Aspect, Article 2 (translation of the article cited below), 10 and 21, ISBN 978-3-11-044790-3 .

Web links

Commons : EPR paradox  - collection of images, videos and audio files

References and footnotes

  1. ↑ Einstein uses this term in a letter to Max Born on March 3, 1947. Quotation: "... But I cannot seriously believe in it because the theory is incompatible with the principle that physics is a reality in time and space should represent, without ghostly long-range effects. " To be read in Albert Einstein Max Born, Briefwechsel 1916-1955 , published by 'Langen Müller in the FA Herbig Verlagsbuchhandlung GmbH, Munich' (3rd edition, 2005) on pages 254ff.
  2. ↑ For quantum entanglement see also state (quantum mechanics) #Another example .
  3. ↑ Albert Einstein's mistake consisted in the fact that he took these assumptions - biased on this point - for granted that he demanded their validity for quantum mechanics as well. Long after Einstein's death, this was falsified by experiments in connection with Bell's inequality .
  4. ↑ Cf. N. Bohr: Can quantum-mechanical description of physical reality be considered complete? , in: Phys. Rev. 48 (1935), p. 700.
  5. ↑ On the subject of "flawedness of the EPR work", it should be emphasized again that it is not a matter of errors in logic, wrong conclusions or the like, but that the philosophical basic premises of the considerations of EPR have been refuted by experiments. To put it simply, the errors consisted in the basic assumption that quantum mechanics must meet all the essential properties of a classical theory.
  6. ↑ JS Bell: On the Einstein-Podolsky-Rosen paradox . In: Physics . tape 1 , no. 3 , 1964, pp. 195-200 .
  7. ↑ This is an explicit example of the epistemological view of Karl Popper , according to which a theory cannot be verified , but only falsified .
  8. ↑ Comment: Here you can see very clearly that in physics it is ultimately the experiment that counts.
  9. ↑ Alain Aspect, Philippe Grangier, and Gérard Roger: Experimental Realization of Einstein-Podolsky-Rosen-Bohm Thought Experiment : A New Violation of Bell's Inequalities . In: Phys. Rev. Lett. tape 49 , 1982, pp. 91-94 , doi : 10.1103 / PhysRevLett.49.91 .
  10. ↑ If one follows the Copenhagen interpretation of quantum mechanics (Niels Bohr), one must assume that both assumptions (locality and reality) are violated, the latter because, according to Niels Bohr, quantum mechanical measurements do not simply determine states , but rather produce (prepare) them .
  11. ↑ The non-locality of the non-relativistic state vector does not contradict the locality postulate of a relativistic quantum field theory or the corresponding C * -algebra . Because this postulate applies to the observables , not to the states. See Klaus Fredenhagen and Katarzyna Rejzner: QFT on curved spacetimes: axiomatic framework and examples. 2014, arxiv : 1412.5125 .