Measurement problem

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In quantum mechanics, the measurement problem is the problem of definite outcomes: quantum state have superpositions but quantum measurements only give one definite result. [1] [2]

Contents

The wave function in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different states. However, actual measurements always find the physical system in a definite state. Any future evolution of the wave function is based on the state the system was discovered to be in when the measurement was made, meaning that the measurement "did something" to the system that is not obviously a consequence of Schrödinger evolution. The measurement problem is describing what that "something" is, how a superposition of many possible values becomes a single measured value.

To express matters differently (paraphrasing Steven Weinberg), [3] [4] the Schrödinger equation determines the wave function at any later time. If observers and their measuring apparatus are themselves described by a deterministic wave function, why can we not predict precise results for measurements, but only probabilities? As a general question: How can one establish a correspondence between quantum reality and classical reality? [5]

Schrödinger's cat

A thought experiment called Schrödinger's cat illustrates the measurement problem. A mechanism is arranged to kill a cat if a quantum event, such as the decay of a radioactive atom, occurs. The mechanism and the cat are enclosed in a chamber so the fate of the cat is unknown until the chamber is opened. Prior to observation, according to quantum mechanics, the atom is in a quantum superposition, a linear combination of decayed and intact states. Also according to quantum mechanics, the atom-mechanism-cat composite system is described by superpositions of compound states. Therefore, the cat would be described as in a superposition, a linear combination of two states an "intact atom-alive cat" and a "decayed atom-dead cat". However, when the chamber is opened the cat is either alive or it is dead: there is no superposition observed. After the measurement the cat is definitively alive or dead. [6] : 154 

The cat scenario illustrates the measurement problem: how can an indefinite superposition yield a single definite outcome? It also illustrates other issues in quantum measurement, [7] : 585  including when does a measurement occur? Was it when the cat was observed? How is a measurement apparatus defined? The mechanism for detecting radioactive decay? The cat? The chamber? What is the role of the observer?

Interpretations

The views often grouped together as the Copenhagen interpretation are the oldest and, collectively, probably still the most widely held attitude about quantum mechanics. [8] [9] N. David Mermin coined the phrase "Shut up and calculate!" to summarize Copenhagen-type views, a saying often misattributed to Richard Feynman and which Mermin later found insufficiently nuanced. [10] [11]

Generally, views in the Copenhagen tradition posit something in the act of observation which results in the collapse of the wave function. This concept, though often attributed to Niels Bohr, was due to Werner Heisenberg, whose later writings obscured many disagreements he and Bohr had during their collaboration and that the two never resolved. [12] [13] In these schools of thought, wave functions may be regarded as statistical information about a quantum system, and wave function collapse is the updating of that information in response to new data. [14] [15] Exactly how to understand this process remains a topic of dispute. [16]

Bohr discussed his views in a 1947 letter to Pauli. [17] Bohr points out that the measurement processes such as cloud chambers or photographic plates involve enormous amplification requiring energies far in excess of the quantum effects being studied and he notes that these processes are irreversible. [18] He considered a consistent account of this issue to be an unsolved problem.

Hugh Everett's many-worlds interpretation attempts to solve the problem by suggesting that there is only one wave function, the superposition of the entire universe, and it never collapses—so there is no measurement problem. Instead, the act of measurement is simply an interaction between quantum entities, e.g. observer, measuring instrument, electron/positron etc., which entangle to form a single larger entity, for instance living cat/happy scientist. Everett also attempted to demonstrate how the probabilistic nature of quantum mechanics would appear in measurements, a work later extended by Bryce DeWitt. However, proponents of the Everettian program have not yet reached a consensus regarding the correct way to justify the use of the Born rule to calculate probabilities. [19] [20]

The de Broglie–Bohm theory tries to solve the measurement problem very differently: the information describing the system contains not only the wave function, but also supplementary data (a trajectory) giving the position of the particle(s). The role of the wave function is to generate the velocity field for the particles. These velocities are such that the probability distribution for the particle remains consistent with the predictions of the orthodox quantum mechanics. According to the de Broglie–Bohm theory, interaction with the environment during a measurement procedure separates the wave packets in configuration space, which is where apparent wave function collapse comes from, even though there is no actual collapse. [21]

A fourth approach is given by objective-collapse models. In such models, the Schrödinger equation is modified and obtains nonlinear terms. These nonlinear modifications are of stochastic nature and lead to behaviour that for microscopic quantum objects, e.g. electrons or atoms, is unmeasurably close to that given by the usual Schrödinger equation. For macroscopic objects, however, the nonlinear modification becomes important and induces the collapse of the wave function. Objective-collapse models are effective theories. The stochastic modification is thought to stem from some external non-quantum field, but the nature of this field is unknown. One possible candidate is the gravitational interaction as in the models of Diósi and Penrose. The main difference of objective-collapse models compared to the other approaches is that they make falsifiable predictions that differ from standard quantum mechanics. Experiments are already getting close to the parameter regime where these predictions can be tested. [22]

The Ghirardi–Rimini–Weber (GRW) theory proposes that wave function collapse happens spontaneously as part of the dynamics. Particles have a non-zero probability of undergoing a "hit", or spontaneous collapse of the wave function, on the order of once every hundred million years. [23] Though collapse is extremely rare, the sheer number of particles in a measurement system means that the probability of a collapse occurring somewhere in the system is high. Since the entire measurement system is entangled (by quantum entanglement), the collapse of a single particle initiates the collapse of the entire measurement apparatus. Because the GRW theory makes different predictions from orthodox quantum mechanics in some conditions, it is not an interpretation of quantum mechanics in a strict sense.

The role of decoherence

Erich Joos and Heinz-Dieter Zeh claim that the phenomenon of quantum decoherence, which was put on firm ground in the 1980s, resolves the problem. [24] The idea is that the environment causes the classical appearance of macroscopic objects. Zeh further claims that decoherence makes it possible to identify the fuzzy boundary between the quantum microworld and the world where the classical intuition is applicable. [25] [26] Quantum decoherence becomes an important part of some modern updates of the Copenhagen interpretation based on consistent histories. [27] [28] Quantum decoherence does not describe the actual collapse of the wave function, but it explains the conversion of the quantum probabilities (that exhibit interference effects) to the ordinary classical probabilities. See, for example, Zurek, [5] Zeh [25] and Schlosshauer. [29]

The present situation is slowly clarifying, described in a 2006 article by Schlosshauer as follows: [30]

Several decoherence-unrelated proposals have been put forward in the past to elucidate the meaning of probabilities and arrive at the Born rule ... It is fair to say that no decisive conclusion appears to have been reached as to the success of these derivations. ...

As it is well known, [many papers by Bohr insist upon] the fundamental role of classical concepts. The experimental evidence for superpositions of macroscopically distinct states on increasingly large length scales counters such a dictum. Superpositions appear to be novel and individually existing states, often without any classical counterparts. Only the physical interactions between systems then determine a particular decomposition into classical states from the view of each particular system. Thus classical concepts are to be understood as locally emergent in a relative-state sense and should no longer claim a fundamental role in the physical theory.

See also

For a more technical treatment of the mathematics involved in the topic, see Measurement in quantum mechanics.

References and notes

  1. ↑ Schlosshauer, Maximilian (2005-02-23). "Decoherence, the measurement problem, and interpretations of quantum mechanics". Reviews of Modern Physics. 76 (4): 1267–1305. arXiv: quant-ph/0312059 . doi:10.1103/RevModPhys.76.1267. ISSN   0034-6861.
  2. ↑ Leggett, A. J. (2005-02-11). "The Quantum Measurement Problem". Science. 307 (5711): 871–872. doi:10.1126/science.1109541. ISSN   0036-8075.
  3. ↑ Weinberg, Steven (1998). "The Great Reduction: Physics in the Twentieth Century". In Michael Howard & William Roger Louis (eds.). The Oxford History of the Twentieth Century . Oxford University Press. p.  26. ISBN   0-19-820428-0.
  4. ↑ Weinberg, Steven (November 2005). "Einstein's Mistakes". Physics Today . 58 (11): 31–35. Bibcode:2005PhT....58k..31W. doi: 10.1063/1.2155755 . S2CID   120594692.
  5. 1 2 Zurek, Wojciech Hubert (22 May 2003). "Decoherence, einselection, and the quantum origins of the classical". Reviews of Modern Physics . 75 (3): 715–775. arXiv: quant-ph/0105127 . Bibcode:2003RvMP...75..715Z. doi:10.1103/RevModPhys.75.715. S2CID   14759237.
  6. ↑ Baggott, J. E. (2013). The quantum story: a history in 40 moments (Impression: 3 ed.). Oxford: Oxford Univ. Press. ISBN   978-0-19-965597-7.
  7. ↑ Griffiths, David J.; Schroeter, Darrell F. (2018-08-16). Introduction to Quantum Mechanics. Cambridge University Press. doi:10.1017/9781316995433. ISBN   978-1-316-99543-3.
  8. ↑ Schlosshauer, Maximilian; Kofler, Johannes; Zeilinger, Anton (August 2013). "A snapshot of foundational attitudes toward quantum mechanics". Studies in History and Philosophy of Science Part B . 44 (3): 222–230. arXiv: 1301.1069 . Bibcode:2013SHPMP..44..222S. doi:10.1016/j.shpsb.2013.04.004. S2CID   55537196.
  9. ↑ Ball, Philip (2013). "Experts still split about what quantum theory means". Nature . doi:10.1038/nature.2013.12198. S2CID   124012568.
  10. ↑ Mermin, N. David (1989). "What's Wrong with this Pillow?". Physics Today . 42 (4): 9. Bibcode:1989PhT....42d...9D. doi:10.1063/1.2810963.
  11. ↑ Mermin, N. David (2004). "Could Feynman have said this?". Physics Today . 57 (5): 10–11. Bibcode:2004PhT....57e..10M. doi: 10.1063/1.1768652 .
  12. ↑ Howard, Don (December 2004). "Who Invented the "Copenhagen Interpretation"? A Study in Mythology". Philosophy of Science . 71 (5): 669–682. doi:10.1086/425941. ISSN   0031-8248. S2CID   9454552.
  13. ↑ Camilleri, Kristian (May 2009). "Constructing the Myth of the Copenhagen Interpretation". Perspectives on Science. 17 (1): 26–57. doi:10.1162/posc.2009.17.1.26. ISSN   1063-6145. S2CID   57559199.
  14. ↑ Englert, Berthold-Georg (2013-11-22). "On quantum theory". The European Physical Journal D . 67 (11): 238. arXiv: 1308.5290 . Bibcode:2013EPJD...67..238E. doi:10.1140/epjd/e2013-40486-5. ISSN   1434-6079. S2CID   119293245.
  15. ↑ Peierls, Rudolf (1991). "In defence of "measurement"". Physics World . 4 (1): 19–21. doi:10.1088/2058-7058/4/1/19. ISSN   2058-7058.
  16. ↑ Healey, Richard (2016). "Quantum-Bayesian and Pragmatist Views of Quantum Theory". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy . Metaphysics Research Lab, Stanford University.
  17. ↑ Bohr, Niels (1985) [May 16, 1947], Jørgen Kalckar (ed.), Niels Bohr: Collected Works, vol. 6: Foundations of Quantum Physics I (1926–1932), pp. 451–454
  18. ↑ Stenholm, Stig (1983), "To fathom space and time", in Meystre, Pierre (ed.), Quantum Optics, Experimental Gravitation, and Measurement Theory, Plenum Press, p. 121, The role of irreversibility in the theory of measurement has been emphasized by many. Only this way can a permanent record be obtained. The fact that separate pointer positions must be of the asymptotic nature usually associated with irreversibility has been utilized in the measurement theory of Daneri, Loinger and Prosperi (1962). It has been accepted as a formal representation of Bohr's ideas by Rosenfeld (1966).
  19. ↑ Kent, Adrian (2010). "One world versus many: the inadequacy of Everettian accounts of evolution, probability, and scientific confirmation". Many Worlds?. Oxford University Press. pp. 307–354. arXiv: 0905.0624 . ISBN   9780199560561. OCLC   696602007.
  20. ↑ Barrett, Jeffrey (2018). "Everett's Relative-State Formulation of Quantum Mechanics". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy . Metaphysics Research Lab, Stanford University.
  21. ↑ Sheldon, Goldstein (2017). "Bohmian Mechanics". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy . Metaphysics Research Lab, Stanford University.
  22. ↑ Bassi, Angelo; Lochan, Kinjalk; Satin, Seema; Singh, Tejinder P.; Ulbricht, Hendrik (2013). "Models of wave-function collapse, underlying theories, and experimental tests". Reviews of Modern Physics . 85 (2): 471–527. arXiv: 1204.4325 . Bibcode:2013RvMP...85..471B. doi:10.1103/RevModPhys.85.471. S2CID   119261020.
  23. ↑ Bell, J. S. (2004). "Are there quantum jumps?". Speakable and Unspeakable in Quantum Mechanics. pp. 201–212.
  24. ↑ Joos, E.; Zeh, H. D. (June 1985). "The emergence of classical properties through interaction with the environment". Zeitschrift für Physik B . 59 (2): 223–243. Bibcode:1985ZPhyB..59..223J. doi:10.1007/BF01725541. S2CID   123425824.
  25. 1 2 Zeh, H. D. (2003). "Chapter 2: Basic Concepts and Their Interpretation". In Joos, E. (ed.). Decoherence and the Appearance of a Classical World in Quantum Theory (2nd ed.). Springer-Verlag. p. 7. arXiv: quant-ph/9506020 . Bibcode:2003dacw.conf....7Z. ISBN   3-540-00390-8.
  26. ↑ Jaeger, Gregg (September 2014). "What in the (quantum) world is macroscopic?". American Journal of Physics . 82 (9): 896–905. Bibcode:2014AmJPh..82..896J. doi:10.1119/1.4878358.
  27. ↑ Belavkin, V. P. (1994). "Nondemolition principle of quantum measurement theory". Foundations of Physics . 24 (5): 685–714. arXiv: quant-ph/0512188 . Bibcode:1994FoPh...24..685B. doi:10.1007/BF02054669. S2CID   2278990.
  28. ↑ Belavkin, V. P. (2001). "Quantum noise, bits and jumps: uncertainties, decoherence, measurements and filtering". Progress in Quantum Electronics. 25 (1): 1–53. arXiv: quant-ph/0512208 . Bibcode:2001PQE....25....1B. doi:10.1016/S0079-6727(00)00011-2.
  29. ↑ Maximilian Schlosshauer (2005). "Decoherence, the measurement problem, and interpretations of quantum mechanics". Reviews of Modern Physics. 76 (4): 1267–1305. arXiv: quant-ph/0312059 . Bibcode:2004RvMP...76.1267S. doi:10.1103/RevModPhys.76.1267. S2CID   7295619.
  30. ↑ Schlosshauer, Maximilian (January 2006). "Experimental motivation and empirical consistency in minimal no-collapse quantum mechanics". Annals of Physics . 321 (1): 112–149. arXiv: quant-ph/0506199 . Bibcode:2006AnPhy.321..112S. doi:10.1016/j.aop.2005.10.004. S2CID   55561902.

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