# Einstein–Podolsky–Rosen (EPR) Paradox

October 18, 2016 2:12 pm

In their famous paper entitled ”Can quantum-mechanical description of physical reality be considered complete” 1, Albert Einstein and his co-authors Boris Podolsky and Nathan Rosen consider quantum mechanics to be incomplete. They argue that a complete theory must represent both the position and the momentum of a particle at any time with arbitrary accuracy. Quantum mechanics indeed does not – hence it is incomplete.

Their key argument, as summarized in Niels Bohr’s reply 2  is the following: Two particles that had interacted in the past, are now far apart and no more interaction is present. So, according to the authors, a measurement of one particle does not affect the other, spacelike separated particle. Using an apparatus as shown below left, one can measure the height of one particle relative to the other. The experiment has to be repeated, until both particles pass through the two slots, which are a distance L apart. The result is denoted as  Now the height of one particle (e.g. ) is measured after passing the board. With this setup the height of one particle is measured directly and the height of the other particle is measured indirectly, without affecting it.

Apparatus for Einstein-Podolsky-Rosen Gedankenexperiment

In a next step the position measurement is combined with a momentum measurement, which is depicted above on the right side. Initially  , since the initial momenta are horizontal. Since the total momentum of board and particles is unchanged during the passage, this measurement yields the total momentum of the particles denoted as . Actually the vertical recoil of the board measures the change in , since  and  may change by knocking against the slots as the particles pass through. When the particles pass through the slots the vertical recoil is measured yielding . Again after passing the slots one can infer   from a direct measurement of  . However measuring   may affect  If the board does not recoil up or down the apparatus measures   and . Next a direct measurement is of   and either  or   is performed. Since there is no interaction between the two particles the result of the   measurement cannot depend on weather   or    is measured. If the result is   the result    follows from  . But if a   measurement is carried out   cannot be measured and quantum mechanics does not predict the result of the   measurement. But the results of the  measurement cannot depend on weather   or  is measured, by assumption. But still   is given by , a predetermined value independent of what is measured on the other particle. Since quantum mechanics does not predict this result quantum theory is incomplete. This is the EPR claim.

To be more precise the EPR paradox refers to a Gedankenexperiment challenging the dogma that the description of reality, given by the wave function, is complete: ”In a complete theory there is an element corresponding to each element of reality”, or as states in the paper: ”every element of the physical reality must have a counterpart in the physical theory.” The authors thereby claim that the outcome of a measurement, which already exist before the actual measurement takes place, is determined by an element of reality, a part of the real physical world: ”If without in any way disturbing the system, we can predict with certainty the value of a physical quantity, the there exists an element of reality corresponding to this physical quantity” Furthermore the systems properties are independent of which interventions are carried out on spatially separated systems: ”…since at the time of the measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system”  1 . So the elements of reality are local, in the sense that they can only be influenced by an event which is located in the past light cone of the elements point in spacetime. These are the two main assumptions in the EPR claim, namely locality and realism, which are often referred to just as local realism.

In quantum mechanics the behavior of a particle is described by the concept of states. It is completely characterized by a wave function  , which is a function of variables such as x and p to describe the particles behavior. Corresponding to each physical observable quantity A there is an operator (may be assigned the same letter, but hatted). If is an eigenstate of , which is if where  is a real number, then the physical quantity A has with certainty the value a whenever the particle is in state . Since ) a precise measurement of p will result in an equally distributed probability of x. The authors concluded from that: ”… when the momentum of a particle is known, its coordinate has no physical reality… From this follows either (1) the quantum- mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality.” This is the authors starting point used to derive a contradiction with quantum mechanics Showing that it is possible to assign two different wave functions to the same reality (the second system after the interaction with the first): The system is characterized by a two particle wave function denoted as . For the momentum operator we get  , where its eigenfunction, corresponding to the eigenvalue is given by . Consequently the two particle wave function from before can be written as , where  is the eigenfunction of the operator corresponding to the eigenvalue  of the momentum of the second particle. On the other hand if the measured observable is the coordinate of the first particle, with eigenfunction corresponding to the eigenvalue, the two particle wave function becomes  , with . Here is eigenfunction of the operator . Since  it is possible for   and   to be eigenfunctions of two noncommuting observables, corresponding to physical quantities. So in the first case   is considered to be an element of reality, whereas in in the second case   is considered to be an element of reality. But both wavefunctions   and   belong to the same reality.  So depending on which measurement is performed on the first system, the second system is left in different wavefunctions, this mechanism is commonly referred to as collapse of wavefunction. On the other hand since the particles no longer interact, no real change can take place in the second system (same reality). So it is stated explicitly in the paper ”Previously we proved that either (1) the quantum- mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality.” The consequence of (1) would be that quantum mechanics is local and there must be some unknown underlying mechanism acting on these variables to give rise to the observed effects of ”noncommuting quantum observables”, i.e. the Heisenberg uncertainty principle. Such a theory is called a hidden variable theory. If quantum mechanics were complete (first option failed) then the second option would hold, that is, incompatible quantities (operators corresponding to two physical quantities do not commute) cannot have real values simultaneously. But if quantum mechanics were complete, then incompatible quantities could indeed have simultaneous, real values. Thus the negation of (1) leads to the negation of the only alternative (2), concluding that the quantum-mechanical description using wave functions is incomplete. The problem within the EPR claim is that the two main assumptions are incompatible with quantum mechanics. So basically EPR shows that quantum mechanics is not a classical theory.

If position and momentum in the EPR Gedankenexperiment are replaced by spin or polarization measurements it becomes experimentally testable, which is schematically illustrated above for spin measurements along arbitrary directions and . In 1951 David J. Bohm reformulated the EPR argument for the spin of two spatially separated entangled particles to illuminate the essential features of the EPR paradox 3 .

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