CBH theorem

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The CBH theorem is an information-theoretical reconstruction of quantum mechanics , which was developed in 2003 by Rob Clifton, Jeffrey Bub and Hans Halvorson.

According to the CBH theorem , quantum mechanics can be derived from three information-theoretical principles and a few mathematical auxiliary postulates. Based on the CBH theorem, Bub suggests that quantum mechanics should not be interpreted as a theory about the mechanical properties of non-classical waves or particles, but as a theory about the possibilities and limits of the transmission of information .

Bub's interpretation received some attention in the professional world, but is mostly rejected.

background

→ Main article: Interpretations of quantum mechanics

In terms of its empirical success, quantum mechanics is one of the best-established physical theories. However, the question of how to interpret quantum mechanics is controversial. Although the orthodox interpretation is widespread to this day, various reservations have been made against this interpretation, including in particular various points of criticism relating to the measurement problem . In addition to the orthodox interpretation, there are a number of alternative interpretations, which, however, also have conceptual and / or philosophical problems and are not generally accepted.

A fundamental problem in the interpretation of physical theories is the problem of underdetermination . Various philosophers see a further problem with the common interpretations, such as B. von Neumann's orthodox interpretation, or the De Broglie Bohm theory, in the use of a fundamentally unsuitable methodology: With these interpretations, the mathematical formalism is assumed and then an attempt is made to use the formal terms of the theory, such as B. the quantum mechanical state to give a semantic meaning. Theories interpreted in this way are also referred to as constructive theories . In other theories, such as B. the theory of relativity , however - according to Bub - another approach, the formulation as principle theory , has proven to be more successful: Einstein's starting point for the derivation and interpretation of the special theory of relativity were not the Lorentz transformations , but physical principles , the relativity principle and the Assumption of the constancy of the speed of light in all inertial systems, from which he then derived the basic equations of the theory in a second step. The assumption of a light ether , previously considered necessary, thus turned out to be superfluous.

A reconstruction of quantum mechanics, i. H. their derivation from suitably chosen physical principles is necessary. Only from the perspective of these principles can a meaningful interpretation of quantum mechanics be formulated. There are a number of different proposals for axiomatic reconstructions of quantum mechanics, with most of these approaches being based on information theoretic principles since the early 1990s . In addition to Rovelli's “Relational Interpretation” and Zeilinger's “Basic Principle of Quantum Physics”, the CBH theorem is one of the best-known information-based reconstructions.

Axioms of the CBH theorem

The CBH theorem states that quantum mechanics can be derived from the following three information-theoretical principles:

  • CBH1: There is no faster than light transmission of information between two systems possible ("no signaling"). In particular, this implies that if Alice and Bob each perform local measurements, the measurement made by Alice has no influence on the statistics of Bob's measurement results, and vice versa. The axiom CBH1 corresponds to the no-signaling theorem of quantum mechanics, but the CBH theorem is not derived from quantum mechanics, but is assumed as a fundamental principle.
  • CBH2: It is not possible to make a copy of the information contained in an unknown state ("no broadcasting"). This postulate is similar to the no-cloning theorem , but has a wider range of validity, since (in contrast to the no-cloning theorem) it is not restricted to pure states , but is assumed for any quantum mechanical states. In classical information theory, copying of information is possible, the axiom CBH2 establishes a fundamental difference between classical information theory and quantum mechanics.
  • CBH3: A secure bit commit protocol is not possible (“no bit”). With this quantum cryptographic protocol , information is transmitted in two steps: In a first step, Alice sends an encrypted bit to Bob, and in a second step, she sends further information for decryption. The bit commit protocol is said to be secure if, firstly, it is ensured that Bob cannot decrypt the value until he receives further information about decryption from Alice, and secondly, Bob can be sure that the decrypted value actually corresponds to the (unencrypted ) Corresponds to the value on which Alice was based when sending the bit.

The term “ information ” is used here in its technical sense as the quantity that is quantified by Shannon or von Neumann entropy .

The second step of the reconstruction consists in transferring the three principles into a mathematical representation. The formal starting point of the CBH theorem is a general abstract characterization of physical theories in the context of C * algebra , which includes, among other things, as special cases the mathematical structures of classical physics as well as all variations of quantum physics (including quantum field theory ). The three CBH principles filter out those structures from this superset of mathematical structures of the C * algebra which violate the CBH principles. The remaining C * algebras have the well-known properties of quantum theory : The superposition principle and the non-commutativity of the observables of individual systems, two properties characteristic of quantum mechanics, result from the CBH theorem as a result of the “no broadcasting” principle (CBH2), and the quantum mechanical entanglement can be traced back to the “no bit” principle (CBH3). According to CBH, every physical theory that can be formulated in the context of C * algebra and that fulfills the 3 CBH principles is a quantum theory.

Other interpreters doubt that the CBH theorem fulfills all the necessary requirements that are placed on an axiomatic method for deriving quantum theories. The physicists Lee Smolin and Robert Spekkens showed that theories can also be derived from the CBH axioms that cannot be regarded as meaningful quantum theories. Furthermore, these analyzes suggest that the properties of the reconstructed quantum theories are not only due to the three CBH principles, but are also (contrary to the objectives of the CBH program) largely determined by the choice of C * algebra as the mathematical framework . Another weak point is the fact that the dynamics of quantum theory (i.e. the time evolution of the quantum mechanical state) cannot be derived from the CBH theorem.

Bubs interpretation

From the statement of the CBH theorem that the information-based CBH principles represent a sufficient basis for the derivation of quantum theories, J. Bub derives the thesis that quantum theories are not theories about mechanical properties of physical objects (e.g. particles or waves ) should be interpreted, but as theories about the possibilities and limits of the transmission of information. Information should therefore be regarded as a fundamental physical quantity.

Mechanical theories of quantum phenomena (by which Bub understands hidden variables theories, such as the De Broglie-Bohm theory ), however, are not acceptable. This is especially true for theories that provide dynamic descriptions of the measurement process.

Bub's main arguments and an overview of how they were judged by the professional community are presented below.

Criteria for theory selection

Bub takes the position that quantum theories derived from the CBH theorem are to be preferred over alternative theories, since they are principle theories.

Part of a quantum theory is exactly what is needed to derive it, i.e. H. to reconstruct its mathematical formalism, and the theory is valid to the extent that its empirical principles are valid.

Additional structures, as postulated in hidden variable theories, have the same status as the light ether: Their introduction is not based on empirical principles, but on insufficiently justified philosophical a priori assumptions, which are not suitable as a basis for the formulation of physical theories be.

This line of argument is criticized by the philosophers HR Brown, C. Timpson, Amit Hagar, M. Hemmo and others: On the one hand, the CBH principles CBH2 and CBH3 are nowhere near as well empirically verified as is the case with the principles of relativity is. Furthermore, these interpreters deny that principle theories are fundamentally preferable to constructive theories. The reconstruction of theories on the basis of empirical principles is only of heuristic benefit in deriving the theory, but these practical advantages are bought at the price of limitations in the explanatory power of the theory. You refer to the example of thermodynamics . Classical thermodynamics can be derived completely from its main principles, so it is a principle theory. However, the deeper justification for classical thermodynamics is provided by statistical physics , a constructive theory. Analogous to this, a formulation as a constructive theory should also be aimed for for quantum theory.

Measurement problem

The non-commutativity of quantum mechanical C * algebra means that the state of a quantum system cannot be represented in a phase space , i.e. H. no uniquely defined catalog of physical properties can be assigned to the system. This uncertainty is well known from quantum mechanics, but it involves the question, known as a measurement problem , of how the theory can be reconciled with the occurrence of clearly defined pointer positions of measurement devices. Bub draws the conclusion that there is an inevitable limit in the quantum-theoretical description of measuring devices, that measuring devices must therefore be viewed in principle as unanalysable black boxes .

Hidden variable theories (such as the De Broglie Bohm theory) also do not offer a valid basis for an analysis of the measurement process - according to Bub: For every empirically adequate hidden variable theory, one can be measured in terms of its own Predictions formulate equivalent CBH theory, the description of the measurement process in the context of the hidden variable theory is therefore without empirical basis and therefore to be rejected. The search for a solution to the measurement problem is therefore just as pointless as the search for the light ether, it is ultimately a pseudo-problem.

Bub's position on the measurement problem is criticized by the philosopher Amit Hagar: There is no reason to view measurement instruments as a black box. Measuring instruments are ordinary physical objects, and the description of measuring processes is just as much the subject of physical research as the investigation of other physical processes. There are alternative theories, such as B. the Ghirardi – Rimini – Weber theory (GRW theory; see: Interpretations of Quantum Mechanics , Section Dynamic Collapse Theories), which allow a description of the measurement process and deviate in its empirical predictions from CBH theories (although an experimental one is Evidence of these deviations is not possible with the technical means available today). The assessment of such theories is not a question of principle, but a task to be clarified experimentally.

Information as a physical quantity

Bub presents CBH quantum theories as physical theories, from which he deduces that information should be regarded as a fundamental physical quantity. Quantum theories are therefore theories about the representation and manipulation of information. Not the description of the properties of material objects, but the elaboration of quantum theories from an information-theoretical perspective is the adequate objective of physics.

There is a broad spectrum of different views on the question of the relationship between information theory and physics, or between the basic concepts of information and matter . Some physicists, such as B. J. Preskill or R. Landauer , argue that the transmission and processing of information always presupposes a physical substrate and is thus determined by physical laws, which is why the properties of information can be completely traced back to the properties of physical objects. Bub's interpretation, on the other hand, corresponds to the opposite position that physics can be reduced to information. The best-known representative of this school of thought, the physicist JA Wheeler , formulated his “it from bit” thesis in 1990, according to which all physical entities, such as B. elementary particles, force fields, even space-time, have an information-theoretical origin.

Other artists, such as BG Jaeger, C. Timpson or A. Duwell consider both points of view to be untenable extreme positions: Information cannot be reduced to physical laws, since even technical definitions of “information” are based on concepts that presuppose conscious (human) action. For example, according to Shannon, the information content of messages is based on a coding that must be agreed between a sender and a recipient. The assumption that information can be reducible to physics is therefore based on a questionable physicalistic worldview. The opposite thesis that physics can be traced back to information is also rejected by many interpreters. In particular, information should not be regarded as a physical substance (i.e. as an object or as physical matter), and information is therefore not suitable as a basic term for describing the properties of matter. This means that Bub's basic assumption that physics can be traced back to information-theoretical concepts is inconclusive and should therefore be rejected.

literature

  • G. Jaeger, Entanglement, Information, and the Interpretation of Quantum Mechanics , Springer (2009).
  • CG Timpson, Philosophical Aspects of Quantum Information Theory , in D. Rickles (ed.), The Ashgate Companion to the New Philosophy of Physics (Ashgate 2008). arxiv : quant-ph / 0611187 .
  • Horst Völz : That is information. Shaker Verlag, Aachen 2017, ISBN 978-3-8440-5587-0 .
  • Horst Völz : Description of the world. Space, time, temperature and information - aspects, positions, debates. Shaker Verlag, Aachen 2018, ISBN 978-3-8440-6323-3 .

Individual evidence

  1. ↑ a b C.G. Timpson, Philosophical Aspects of Quantum Information Theory , in D. Rickles (ed.), The Ashgate Companion to the New Philosophy of Physics (Ashgate 2008), p. 245 ff. Arxiv : quant-ph / 0611187 .
  2. ↑ Alexei Grinbaum: Reconstruction of Quantum Theory (PDF; 170 kB) , Brit. J. Phil. Sci. 8 (2007), pp. 387-408.
  3. ↑ C. Rovelli: Relational Quantum Mechanics , International Journal of Theoretical Physics 35 (1996), pp. 1637-1678. arxiv : quant-ph / 9609002 .
  4. ↑ A. Zeilinger: A foundational principle for quantum mechanics (PDF; 595 kB) , Found. Phys. 29 (1999), pp. 631-643.
  5. ^ R. Clifton, J. Bub and H. Halvorson, Characterizing Quantum Theory in Terms of Information-Theoretic Constraints , Foundations of Physics 33 (2003), p. 1561.
  6. ↑ JASmolin, Can Quantum Cryptography Imply Quantum Mechanics? , Quantum Information and Computation 5 (2005), p. 161. arxiv : quant-ph / 0310067 .
  7. ↑ R. Spekkens, in defense of the epistemic view of quantum states: A toy theory , Phys. Rev. A 75 (2007), p. 032110. arxiv : quant-ph / 0401052 .
  8. ↑ a b J. Bub, Why the quantum? , Stud. Hist. Phil. Mod. Phys. 35 (2004), p. 241 arxiv : quant-ph / 0402149 .
    J. Bub, Quantum Mechanics is about Quantum Information ,, Found. Phys. 35 (2005), p. 541. arxiv : quant-ph / 0408020
  9. ↑ If the information-theoretic constraints are to legitimate a conception of quantum mechanics, their epistemic status should be secure, but that is not the case. They are not empirically discovered constraints. The evidence for the constraints is indirect and challengeable. The information-theoretic constraints are predictions of the standard theory. Only insofar as the standard theory is a reliable device for making predictions are the constraints justified. We have no direct empirical evidence that they are true. in A. Duwell, Re-conceiving quantum theories in terms of information-theoretic constraints , Studies in History and Philosophy of Modern Physics 38 (2007) p. 181.
  10. ↑ HRBrown, C. Timpson, Why special relativity should not be a template for a fundamental reformulation of quantum mechanics , in W. Demopoulos and I. Pitowsky (editors), Physical Theory and Its Interpretation: Essays in Honor of Jeffrey Bub, Vol. 72 of The Western Ontario Series in Philosophy of Science, Springer (2006). (PDF; 136 kB)
  11. ↑ As to thesis II, we shall also argue that interpreting quantum mechanics as a principle theory is not the right epistemological stance given the theoretical basis of quantum mechanics, and that information theoretic interpretations of quantum mechanics as a principle theory don't warrant abandoning alternative constructive dynamical theories, in particular theories which differ empirically from no collapse quantum mechanics. in A. Hagar, M. Hemmo, Explaining the Unobserved — Why Quantum Mechanics Ain't Only About Information , Found. Phys. 36 (2006), p. 1295.
  12. ^ A. Hagar, Experimental metaphysics2: The double standard in the quantum information approach to the foundations of quantum theory , Studies in History and Philosophy of Modern Physics 38 (2007) 906-919.
  13. ^ J. Preskill, Physics 229: Advanced mathematical methods of physics — Quantum computation and information, California Institute of Technology (1998). (Online) (PDF; 205 kB)
  14. ↑ Landauer, R., The physical nature of information , Phys. Lett. A 217 (1996), p. 188.
  15. ↑ "Every 'it', every particle, every field of force, even the spacetime continuum itself, derives its way of action and its very existence entirely, even if in some contexts indirectly, from the detector-elicited answers to yes-or- no questions, binary choices, bits. Otherwise stated, all things physical, all its ... must in the end submit to an information-theoretical description. " From JA Wheeler, " Sakharov revisited; It from bit ” , in LV Keldysh and V. Yu. Fainberg (Eds.), Proceedings of the first international Sakharov conference on physics, Vol. 2 (Nova Science Publishers; New York, 1991), p. 751.
  16. ↑ G. Jaeger, Entanglement, Information, and the Interpretation of Quantum Mechanics , Springer (2009), chap. 4.7. google books
  17. ^ A. Duwell, Quantum information does exist , Studies in History and Philosophy of Modern Physics 39 (2008), p. 195.
  18. ↑ G. Jaeger, Entanglement, Information, and the Interpretation of Quantum Mechanics , Springer (2009), p. 188 and p. 234 ff.