Quantum Theory: Concepts and Methods

Last updated

Quantum Theory: Concepts and Methods is a 1993 quantum physics textbook by Israeli physicist Asher Peres. Well-regarded among the physics community, it is known for unconventional choices of topics to include.

Contents

Contents

In his preface, Peres summarized his goals as follows:

The purpose of this book is to clarify the conceptual meaning of quantum theory, and to explain some of the mathematical methods that it utilizes. This text is not concerned with specialized topics such as atomic structure, or strong or weak interactions, but with the very foundations of the theory. This is not, however, a book on the philosophy of science. The approach is pragmatic and strictly instrumentalist. This attitude will undoubtedly antagonize some readers, but it has its own logic: quantum phenomena do not occur in a Hilbert space, they occur in a laboratory. [a]

The book is divided into three parts. The first, "Gathering the Tools", introduces quantum mechanics as a theory of "preparations" and "tests", and it develops the mathematical formalism of Hilbert spaces, concluding with the spectral theory used to understand the quantum mechanics of continuous-valued observables. Part II, "Cryptodeterminism and Quantum Inseparability", focuses on Bell's theorem and other demonstrations that quantum mechanics is incompatible with local hidden-variable theories. (Within its substantial discussion of the failure of hidden variable theories, the book includes a FORTRAN program for testing whether a list of vectors forms a Kochen–Specker configuration. [b] ) Part III, "Quantum Dynamics and Information", covers the role of spacetime symmetry in quantum physics, the relation of quantum information to thermodynamics, semiclassical approximation methods, quantum chaos, and the treatment of measurement in quantum mechanics.

To generate the figures in his chapter on quantum chaos, including plots in phase space of chaotic motion, Peres wrote PostScript code that executed simulations in the printer itself. [c]

The book develops the methodology of mathematically representing quantum measurements by POVMs, [4] [5] and it provided the first pedagogical treatment of how to use a POVM for quantum key distribution. [6] Peres downplayed the importance of the uncertainty principle; that specific term only appears once in his index, and its entry points to that same page in the index. [7] The text itself does discuss the uncertainty principle, pointing out how an oversimplified "derivation" of it breaks down, and posing as a homework problem the task of finding three quantum-physics textbooks with a demonstrably incorrect uncertainty relation. [8]

Reception

Physicist Leslie E. Ballentine gave the textbook a positive review, declaring it a good introduction to quantum foundations and ongoing research therein. [9] John C. Baez also gave the book a positive assessment, calling it "clear-headed" and finding that it contained "a lot of gems that I hadn't seen", such as the Wigner–Araki–Yanase theorem. [10] Michael Nielsen wrote of the textbook, "Revelation! Suddenly, all the key results of 30 years of work (several of those results due to Asher) were distilled into beautiful and simple explanations." [11] Nielsen and Isaac Chuang said in their own influential textbook that Peres' was "superb", providing "an extremely clear exposition of elementary quantum mechanics" as well as an "extensive discussion of the Bell inequalities and related results". [12] Jochen Rau's introductory textbook on quantum physics described Peres' work as "an excellent place to start" learning about Bell inequalities and related topics like Gleason's theorem. [13]

N. David Mermin wrote that Peres had bridged the "textual gap" between conceptually-oriented books, aimed at understanding what quantum physics implies about the nature of the world, and more practical books intended to teach how to apply quantum mechanics. Mermin found the book praiseworthy, noting that he had "only a few complaints". He wrote:

Peres is careless in discriminating among the various kinds of assumptions one needs to prove the impossibility of a no-hidden-variables theory that reproduces the statistical predictions of quantum mechanics. I would guess that this is because even though he is a master practitioner of this particular art form, deep in his heart he is so firmly convinced that hidden variables cannot capture the essence of quantum mechanics, that he is simply not interested in precisely what you need to assume to prove that they cannot. [4]

Mermin called the book "a treasure trove of novel perspectives on quantum mechanics" and said that Peres' choice of topics is "a catalogue of common omissions" from other approaches. [4]

Meinhard E. Mayer declared that he would "recommend it to anyone teaching or studying quantum mechanics", finding Part II the most interesting of the book. While he noted some disappointment with Peres' selection of topics to include in the chapter on measurement, he reserved most of his negativity for the publisher, saying (as Ballentine also did [9] ) that they had priced the book beyond the reach of graduate students:

Such pricing practices are not justified when one considers that many publishers provide very little copyediting or typesetting any more, as is obvious from the "TeX"-ish look of most books published recently, this one included. [14]

Mermin, Mayer and Baez noted that Peres briefly dismissed the many-worlds interpretation of quantum mechanics. [4] [10] [14] Peres argued that all varieties of many-worlds interpretations merely shifted the arbitrariness or vagueness of the wavefunction collapse idea to the question of when "worlds" can be regarded as separate, and that no objective criterion for that separation can actually be formulated. [d] Moreover, Peres dismissed "spontaneous collapse" models like Ghirardi–Rimini–Weber theory in the same brief section, designating them "mutations" of quantum mechanics. [4] In a review that praised the book's thoroughness, Tony Sudbery noted that Peres disparaged the idea that human consciousness plays a special role in quantum mechanics. [15]

Manuel Bächtold analyzed Peres' textbook from a standpoint of philosophical pragmatism. [16] John Conway and Simon Kochen used a Kochen–Specker configuration from the book in order to prove their free will theorem. [17] Peres' insistence in his textbook that the classical analogue of a quantum state is a Liouville density function was influential in the development of QBism. [18]

John Watrous places Peres' textbook among the "indispensable references", along with Nielsen and Chuang's Quantum Computation and Quantum Information and Mark Wilde's Quantum Information Theory. [19] In their obituary for Peres, William Wootters, Charles Bennett and coauthors call Quantum Theory: Concepts and Methods the "modern successor" to John von Neumann's 1955 Mathematical Foundations of Quantum Mechanics. [7]

Editions

Notes

  1. ↑ Preface, p. xi. The last remark is often quoted, [1] for example by Brukner and Zeilinger [2] and by Czartowski and Życzkowski. [3]
  2. ↑ Section 7-5, "Appendix: Computer test for Kochen–Specker contradiction", p. 209
  3. ↑ Section 11-7, "Appendix: PostScript code for a map", p. 370
  4. ↑ Section 12-1, "The ambivalent observer", p. 374

Related Research Articles

The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics, stemming from the work of Niels Bohr, Werner Heisenberg, Max Born, and others. While "Copenhagen" refers to the Danish city, the use as an "interpretation" was apparently coined by Heisenberg during the 1950s to refer to ideas developed in the 1925–1927 period, glossing over his disagreements with Bohr. Consequently, there is no definitive historical statement of what the interpretation entails.

In quantum mechanics, counterfactual definiteness (CFD) is the ability to speak "meaningfully" of the definiteness of the results of measurements that have not been performed. The term "counterfactual definiteness" is used in discussions of physics calculations, especially those related to the phenomenon called quantum entanglement and those related to the Bell inequalities. In such discussions "meaningfully" means the ability to treat these unmeasured results on an equal footing with measured results in statistical calculations. It is this aspect of counterfactual definiteness that is of direct relevance to physics and mathematical models of physical systems and not philosophical concerns regarding the meaning of unmeasured results.

An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Quantum mechanics has held up to rigorous and extremely precise tests in an extraordinarily broad range of experiments. However, there exist a number of contending schools of thought over their interpretation. These views on interpretation differ on such fundamental questions as whether quantum mechanics is deterministic or stochastic, local or non-local, which elements of quantum mechanics can be considered real, and what the nature of measurement is, among other matters.

Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measurement. "Local" here refers to the principle of locality, the idea that a particle can only be influenced by its immediate surroundings, and that interactions mediated by physical fields cannot propagate faster than the speed of light. "Hidden variables" are supposed properties of quantum particles that are not included in quantum theory but nevertheless affect the outcome of experiments. In the words of physicist John Stewart Bell, for whom this family of results is named, "If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

<span class="mw-page-title-main">Wigner's friend</span> Thought experiment in theoretical quantum physics

Wigner's friend is a thought experiment in theoretical quantum physics, first published by the Hungarian-American physicist Eugene Wigner in 1961, and further developed by David Deutsch in 1985. The scenario involves an indirect observation of a quantum measurement: An observer observes another observer who performs a quantum measurement on a physical system. The two observers then formulate a statement about the physical system's state after the measurement according to the laws of quantum theory. In the Copenhagen interpretation, the resulting statements of the two observers contradict each other. This reflects a seeming incompatibility of two laws in the Copenhagen interpretation: the deterministic and continuous time evolution of the state of a closed system and the nondeterministic, discontinuous collapse of the state of a system upon measurement. Wigner's friend is therefore directly linked to the measurement problem in quantum mechanics with its famous Schrödinger's cat paradox.

In physics, a hidden-variable theory is a deterministic physical model which seeks to explain the probabilistic nature of quantum mechanics by introducing additional variables.

In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic. The procedure for finding a probability involves combining a quantum state, which mathematically describes a quantum system, with a mathematical representation of the measurement to be performed on that system. The formula for this calculation is known as the Born rule. For example, a quantum particle like an electron can be described by a quantum state that associates to each point in space a complex number called a probability amplitude. Applying the Born rule to these amplitudes gives the probabilities that the electron will be found in one region or another when an experiment is performed to locate it. This is the best the theory can do; it cannot say for certain where the electron will be found. The same quantum state can also be used to make a prediction of how the electron will be moving, if an experiment is performed to measure its momentum instead of its position. The uncertainty principle implies that, whatever the quantum state, the range of predictions for the electron's position and the range of predictions for its momentum cannot both be narrow. Some quantum states imply a near-certain prediction of the result of a position measurement, but the result of a momentum measurement will be highly unpredictable, and vice versa. Furthermore, the fact that nature violates the statistical conditions known as Bell inequalities indicates that the unpredictability of quantum measurement results cannot be explained away as due to ignorance about "local hidden variables" within quantum systems.

In quantum mechanics, the measurement problem is the problem of definite outcomes: quantum systems have superpositions but quantum measurements only give one definite result.

In the interpretation of quantum mechanics, a local hidden-variable theory is a hidden-variable theory that satisfies the principle of locality. These models attempt to account for the probabilistic features of quantum mechanics via the mechanism of underlying, but inaccessible variables, with the additional requirement that distant events be statistically independent.

<span class="mw-page-title-main">Asher Peres</span> Israeli physicist

Asher Peres was an Israeli physicist. He is well known for his work relating quantum mechanics and information theory. He helped to develop the Peres–Horodecki criterion for quantum entanglement, as well as the concept of quantum teleportation, and collaborated with others on quantum information and special relativity. He also introduced the Peres metric and researched the Hamilton–Jacobi–Einstein equation in general relativity. With Mario Feingold, he published work in quantum chaos that is known to mathematicians as the Feingold–Peres conjecture and to physicists as the Feingold–Peres theory.

In quantum mechanics, the Kochen–Specker (KS) theorem, also known as the Bell–KS theorem, is a "no-go" theorem proved by John S. Bell in 1966 and by Simon B. Kochen and Ernst Specker in 1967. It places certain constraints on the permissible types of hidden-variable theories, which try to explain the predictions of quantum mechanics in a context-independent way. The version of the theorem proved by Kochen and Specker also gave an explicit example for this constraint in terms of a finite number of state vectors.

The Born rule is a postulate of quantum mechanics that gives the probability that a measurement of a quantum system will yield a given result. In one commonly used application, it states that the probability density for finding a particle at a given position is proportional to the square of the amplitude of the system's wavefunction at that position. It was formulated and published by German physicist Max Born in July, 1926.

In mathematical physics, Gleason's theorem shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non-contextuality. Andrew M. Gleason first proved the theorem in 1957, answering a question posed by George W. Mackey, an accomplishment that was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics. Multiple variations have been proven in the years since. Gleason's theorem is of particular importance for the field of quantum logic and its attempt to find a minimal set of mathematical axioms for quantum theory.

<span class="mw-page-title-main">N. David Mermin</span> American physicist

Nathaniel David Mermin is a solid-state physicist at Cornell University best known for the eponymous Hohenberg–Mermin–Wagner theorem, his application of the term "boojum" to superfluidity, his textbook with Neil Ashcroft on solid-state physics, and for contributions to the foundations of quantum mechanics and quantum information science.

Some interpretations of quantum mechanics posit a central role for an observer of a quantum phenomenon. The quantum mechanical observer is tied to the issue of observer effect, where a measurement necessarily requires interacting with the physical object being measured, affecting its properties through the interaction. The term "observable" has gained a technical meaning, denoting a Hermitian operator that represents a measurement.

<span class="mw-page-title-main">Quantum Bayesianism</span> Interpretation of quantum mechanics

In physics and the philosophy of physics, quantum Bayesianism is a collection of related approaches to the interpretation of quantum mechanics, the most prominent of which is QBism. QBism is an interpretation that takes an agent's actions and experiences as the central concerns of the theory. QBism deals with common questions in the interpretation of quantum theory about the nature of wavefunction superposition, quantum measurement, and entanglement. According to QBism, many, but not all, aspects of the quantum formalism are subjective in nature. For example, in this interpretation, a quantum state is not an element of reality—instead, it represents the degrees of belief an agent has about the possible outcomes of measurements. For this reason, some philosophers of science have deemed QBism a form of anti-realism. The originators of the interpretation disagree with this characterization, proposing instead that the theory more properly aligns with a kind of realism they call "participatory realism", wherein reality consists of more than can be captured by any putative third-person account of it.

The Pusey–Barrett–Rudolph (PBR) theorem is a no-go theorem in quantum foundations due to Matthew Pusey, Jonathan Barrett, and Terry Rudolph in 2012. It has particular significance for how one may interpret the nature of the quantum state.

Quantum contextuality is a feature of the phenomenology of quantum mechanics whereby measurements of quantum observables cannot simply be thought of as revealing pre-existing values. Any attempt to do so in a realistic hidden-variable theory leads to values that are dependent upon the choice of the other (compatible) observables which are simultaneously measured. More formally, the measurement result of a quantum observable is dependent upon which other commuting observables are within the same measurement set.

<i>Quantum Computation and Quantum Information</i> Textbook by scientists Michael Nielsen and Isaac Chuang

Quantum Computation and Quantum Information is a textbook about quantum information science written by Michael Nielsen and Isaac Chuang, regarded as a standard text on the subject. It is informally known as "Mike and Ike", after the candies of that name. The book assumes minimal prior experience with quantum mechanics and with computer science, aiming instead to be a self-contained introduction to the relevant features of both. The focus of the text is on theory, rather than the experimental implementations of quantum computers, which are discussed more briefly.

Quantum foundations is a discipline of science that seeks to understand the most counter-intuitive aspects of quantum theory, reformulate it and even propose new generalizations thereof. Contrary to other physical theories, such as general relativity, the defining axioms of quantum theory are quite ad hoc, with no obvious physical intuition. While they lead to the right experimental predictions, they do not come with a mental picture of the world where they fit.

References

  1. ↑ Gilligan-Lee, Ciarán M (2021-10-01). "Computation in a general physical setting". Journal of Physics A: Mathematical and Theoretical . 54 (39): 394001. arXiv: 2108.11454 . Bibcode:2021JPhA...54M4001G. doi:10.1088/1751-8121/ac2007. ISSN   1751-8113. S2CID   237304127.
  2. ↑ Brukner, Časlav; Zeilinger, Anton (2003). "Information and Fundamental Elements of the Structure of Quantum Theory". In Castell, Lutz; Ischebeck, Otfried (eds.). Time, Quantum and Information. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 323–354. arXiv: quant-ph/0212084 . Bibcode:2003tqi..conf..323B. doi:10.1007/978-3-662-10557-3_21. ISBN   978-3-642-07892-7. S2CID   16831915 . Retrieved 2022-02-19.{{cite book}}: |journal= ignored (help)
  3. ↑ Czartowski, Jakub; Życzkowski, Karol (2021-04-26). "Bipartite quantum measurements with optimal single-sided distinguishability". Quantum . 5: 442. arXiv: 2010.14868 . Bibcode:2021Quant...5..442C. doi:10.22331/q-2021-04-26-442. ISSN   2521-327X. S2CID   225094378.
  4. 1 2 3 4 5 Mermin, N. David (1997-03-01). Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics . 28 (1): 131–135. Bibcode:1997SHPMP..28..131M. doi:10.1016/S1355-2198(97)85538-4. ISSN   1355-2198.{{cite journal}}: CS1 maint: untitled periodical (link)
  5. ↑ Flammia, Steven T.; Silberfarb, Andrew; Caves, Carlton M. (2005-12-01). "Minimal Informationally Complete Measurements for Pure States". Foundations of Physics . 35 (12): 1985–2006. arXiv: quant-ph/0404137 . Bibcode:2005FoPh...35.1985F. doi:10.1007/s10701-005-8658-z. ISSN   1572-9516. S2CID   119382552.
  6. ↑ Brandt, Howard E. (May 1999). "Positive operator valued measure in quantum information processing". American Journal of Physics . 67 (5): 434–439. Bibcode:1999AmJPh..67..434B. doi: 10.1119/1.19280 . ISSN   0002-9505.
  7. 1 2 Terzian, Joseph E.; Bennett, Charles H.; Mann, Ady; Wootters, William K. (August 2005). "Obituary: Asher Peres". Physics Today . 58 (8): 65–66. Bibcode:2005PhT....58h..65A. doi: 10.1063/1.2062925 .
  8. ↑ Clifton, Rob (1995-01-01). Foundations of Physics . 25 (1): 205–209. doi:10.1007/BF02054666. ISSN   1572-9516. S2CID   189837484.{{cite journal}}: CS1 maint: untitled periodical (link)
  9. 1 2 Ballentine, Leslie E. (March 1995). American Journal of Physics . 63 (3): 285–286. doi:10.1119/1.17946. ISSN   0002-9505.{{cite journal}}: CS1 maint: untitled periodical (link)
  10. 1 2 Baez, John C. (1994-05-10). "week33". This Week's Finds in Mathematical Physics . Retrieved 2020-04-10.
  11. ↑ Nielsen, Michael A. (2005-01-05). "Asher Peres". michaelnielsen.org. Retrieved 2018-02-21.
  12. ↑ Nielsen, Michael A.; Chuang, Isaac L. (2010-12-09). Quantum Computation and Quantum Information: 10th Anniversary Edition . Cambridge University Press. pp.  118. ISBN   9781107002173. OCLC   844974180.
  13. ↑ Rau, Jochen (2021). Quantum Theory: An information processing approach (1st ed.). Oxford: Oxford University Press. p. 152. ISBN   978-0-19-265027-6. OCLC   1256446911.
  14. 1 2 Mayer, Meinhard E. (2008-01-11). Physics Today . 47 (12): 65–66. doi:10.1063/1.2808757. ISSN   0031-9228.{{cite journal}}: CS1 maint: untitled periodical (link)
  15. ↑ Sudbery, Tony (April 1994). "Ordinary questions, extraordinary answers". Physics World . 7 (4): 65–68. doi:10.1088/2058-7058/7/4/42. ISSN   0953-8585.
  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. ↑ Conway, John; Kochen, Simon (2006-11-22). "The Free Will Theorem". Foundations of Physics . 36 (10): 1441–1473. arXiv: quant-ph/0604079 . Bibcode:2006FoPh...36.1441C. doi:10.1007/s10701-006-9068-6. ISSN   0015-9018. S2CID   12999337.
  18. ↑ Fuchs, Christopher A.; Stacey, Blake C. (2019). "QBism: Quantum Theory as a Hero's Handbook". In Rasel, E. M.; Schleich, W. P.; Wölk, S. (eds.). Proceedings of the International School of Physics "Enrico Fermi". Vol. 197. IOS Press. pp. 133–202. arXiv: 1612.07308 . doi:10.3254/978-1-61499-937-9-133. ISBN   9781614999379. OCLC   1086375617.
  19. ↑ Watrous, John (2018). The Theory of Quantum Information. Cambridge University Press. ISBN   9781316853122. OCLC   1034577167.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.