Quantum Entanglement

1. The collapse approach

The Copenhagen interpretation, or collapse approach to quantum mechanics, was devised by German physicist Werner Heisenberg in the 1920s[1], and modified by Italian physicists Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber[2][3], and British physicist Roger Penrose[4][5].

The collapse approach states that the measurement of a quantum system invokes a 'collapse' of the quantum wave function, from a superpositional state, into a state that can be described classically, in accordance with Born's rule[6].

At first glance, the collapse approach appears to contradict German-Swiss-American physicist Albert Einstein's theory of special relativity, which states that nothing can travel faster than the speed of light[7]. This is because of an affect known as entanglement, a term coined by Austrian physicist Erwin Schrödinger in 1935[8].

2. Entanglement

Schrödinger stated that:

"If two separated bodies, about which, individually, we have maximal knowledge, come into a situation in which they influence one another and then again separate themselves, then there regularly arises that which I just called entanglement [Verschränkung] of our knowledge of the two bodies...Our knowledge remains maximal, but at the end, if the bodies have again separated themselves, that knowledge does not again decompose into a logical sum of knowledge of the individual bodies"[9].

Entanglement can be illustrated with examples of any observable property, such as position, momentum, or spin. Two entangled electrons, for example, must possess spins of opposite signs.

Spin can be measured at any angle but is usually described as being 'up' or 'down', or 'left' or 'right', when measured in horizontal or vertical planes.

This means that measuring the spin of one member of an entangled pair of electrons instantaneously determines the spin of the other, even if it's very far away.

Schrödinger showed that there's no equation that describes the state of a single entangled electron, and the overall spin-state cannot be equated with any combination of the individual states. This means that entangled electrons cannot really be said to be individuals.

Picture of linked circles representing entangled photons.

A symbolic representation of a pair of entangled photons. Image credit: Neolexx/CC-SA.

2.1 The EPR paper

Einstein did not like the collapse approach because it suggests that instantaneous action at a distance occurs when the wave function collapses. Einstein, American physicist Boris Podolsky, and American-Israeli physicist Nathan Rosen presented what became known as the EPR paper, in 1935[10].

The EPR paper states that quantum mechanics is incomplete. There must be hidden variables that explain why there's no need for instantaneous travel, something Einstein famously referred to as "spooky actions at a distance"[11].

Einstein tried to think of a way to ascribe observable properties to a system without measuring it directly. He realised that if the position of one electron in an entangled pair was measured, then he could also determine its momentum by measuring that of the second electron.

This would contradict Heisenberg's statement that an electron's position and momentum cannot be known simultaneously. Einstein hoped that the effects of entanglement could be explained if the motion of the photons were somehow guided by the electromagnetic field.

In 1964, British physicist John Stewart Bell devised a way to theoretically test for a hidden variable theory, such as Einstein's[12]. American mathematician Simon Kochen and Swiss mathematician Ernst Specker showed that Einstein's hidden variable theory could not be correct in 1967[13].

American physicists Stuart Freedman and John Clauser performed the first experimental test in 1972[14]. Freedman and Clauser showed that Einstein was wrong; the information does appear to be sent instantaneously.

This was verified by French physicist Alain Aspect in 1982[15][16]. Aspect showed that if information is sent though spacetime, then it must travel faster than the speed of light. An experiment in 2008 showed that it must travel at least 10,000 times this speed[17].

Quantum entanglement does not allow any meaningful information to be sent faster than the speed of light, however, because we cannot control what information is sent.

2.2 Quantum holism

Quantum 'action at a distance' is similar to Newtonian action at a distance, where the force of gravity was thought to affect objects instantaneously across great distances, but it differs in two respects:

  • Firstly, quantum action at a distance does not have the symmetry that the gravitational force has. In quantum mechanics, the first measurement always determines the outcome of the second; they are not of mutual influence.

  • Secondly, in quantum mechanics, the effects are irrespective of distance, whereas in the Newtonian model the gravitational force decreases proportionally to the square of the distance between objects.

A better interpretation may be quantum holism[18]. Holism refers to the idea that aspects of a state are not determined by its constituent parts, but by the state as a whole.

2.3 Quantum teleportation

In 1993, physicist Charles Bennett and a team of researchers at IBM showed that the effects of quantum entanglement allow for teleportation as long as the object travels at the speed of light and the original copy is destroyed[19].

This was first demonstrated in 1998 by physicists in Europe and the United States who teleported a photon about one metre across a room[20]. Photons have since been teleported over 140 km[21], and macroscopic objects were first teleported in 2012[22].

3. Other approaches

3.1 The Bohm approach

In 1952, American physicist David Bohm suggested that there is no need for instantaneous action at a distance because the collapse approach is incorrect, and there is no collapse of the wave function[23][24].

Bohm devised a different type of hidden variable theory known as Bohmian mechanics, or the Bohm approach. This suggests that quantum objects follow paths that are determined by a guiding equation, an idea that was first devised by French physicist Louis de Broglie in 1927[25], and was supported by Bell[26][27].

3.2 The Everett approach

In 1957, American physicist Hugh Everett III suggested that Bohm is right, there is no collapse of the wave function, but he interpreted this very differently, devising the many worlds, or Everett approach[28][29].

There's still no consensus over which of these explanations, if either, is correct.

4. References

  1. Faye, J., 'Copenhagen Interpretation of Quantum Mechanics', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  2. Ghirardi, G. C., Rimini, A. and Weber, T., 1985, 'A model for a unified quantum description of macroscopic and microscopic systems' in 'Quantum Probability and Applications II', Springer Berlin Heidelberg.

  3. Ghirardi, G. C., Rimini, A. and Weber, T., 1986, 'Unified dynamics for microscopic and macroscopic systems', Physical Review D, 34, pp.470-491.

  4. Penrose, R., 1999 (1989), 'The Emperor's New Mind', Oxford University Press.

  5. Ghirardi, G. C., 'Collapse Theories', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  6. Born, M., 1926, 'Quantenmechanik der stoßvorgänge' ('On the quantum mechanics of collision processes'), Zeitschrift für Physik, 38, pp.803-827.

  7. Einstein, A., 1905, 'On the electrodynamics of moving bodies', Annalen der Physik, 17, pp.891-921, reprinted in 'The principle of relativity; original papers', 1920, The University of Calcutta.

  8. Schrödinger, E., 1935, 'Discussion of probability relations between separated systems', Mathematical Proceedings of the Cambridge Philosophical Society, 31, pp.555-563.

  9. Schrödinger, E., 1935, 'Die gegenwärtige Situation in der Quantenmechanik' ('The present situation in quantum mechanics'), Naturwissenschaften, 23, pp.823-828. Translation by Trimmer, J. D., 1980, Proceedings of the American Philosophical Society, 23, pp.323-338.

  10. Einstein, A., Podolsky, B. and Rosen, N., 1935, 'Can quantum-mechanical description of physical reality be considered complete?', Physical review, 47, pp.777-780.

  11. Einstein, A., 1971 (1947), 'Letter to Born, 3 March 1947', Letter 84 in 'The Born Einstein Letters', Macmillan.

  12. Bell, J. S., 1964, 'On the Einstein Podolsky Rosen paradox', Physics, 1, pp.195-200.

  13. Kochen, S. and Specker, E. P., 1967, 'The problem of hidden variables in quantum mechanics', Journal of Mathematics and Mechanics, 17, pp.59-87.

  14. Freedman, S. J. and Clauser, J. F., 1972, 'Experimental test of local hidden-variable theories', Physical Review Letters, 28, pp.938.

  15. Aspect, A., Grangier, P., and Roger, G., 1982, 'Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: a new violation of Bell's inequalities', Physical review letters, 49, pp.91-94.

  16. Aspect, A., Dalibard, J., and Roger, G., 1982, 'Experimental test of Bell's inequalities using time-varying analyzers', Physical review letters, 49, pp.1804-1807.

  17. Salart, D., Baas, A., Branciard, C., Gisin, N., and Zbinden, H., 2008, 'Testing the speed of ‘spooky action at a distance’', Nature, 454, pp.861-864.

  18. Healey, R., 'Holism and Nonseparability in Physics', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  19. Bennett, C. H., et al, 1993, 'Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels', Physical review letters, 70, pp.1895.

  20. Furusawa, A., et al, 1998, 'Unconditional quantum teleportation', Science, 282, pp.706-709.

  21. Ma, X. S., et al, 2012, 'Quantum teleportation over 143 kilometres using active feed-forward', Nature, 489, pp.269-273.

  22. Bao, X. H., et al, 2012, 'Quantum teleportation between remote atomic-ensemble quantum memories', Proceedings of the National Academy of Sciences, 109, pp.20347-20351.

  23. Bohm, D., 1952, 'A suggested interpretation of the quantum theory in terms of 'hidden' variables, I', Physical Review, 85, pp.166-179.

  24. Bohm, D., 1952, 'A suggested interpretation of the quantum theory in terms of 'hidden' variables, II', Physical Review, 85, pp.180.

  25. De Broglie, L., 1927, 'La mécanique ondulatoire et la structure atomique de la matière et du rayonnement' ('Wave mechanics and atomic structure of matter and radiation'), J. Phys. Radium, 8, pp.225-241.

  26. Bell, J. S., 2004 (1987), 'Speakable and Unspeakable in Quantum Mechanics', Cambridge University Press.

  27. Goldstein, S., 'Bohmian Mechanics', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  28. Everett, H., III, 1957, 'The Theory of the Universal Wavefunction', PhD thesis.

  29. Vaidman, L., 'Many-Worlds Interpretation of Quantum Mechanics', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

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