Hidden-measurements interpretation

The hidden-measurements interpretation (HMI), also known as the hidden-measurements approach, is a realistic interpretation of quantum mechanics.

The basis of the hidden-measurements interpretation (HMI) is the hypothesis that a quantum measurement involves a certain amount of unavoidable fluctuations in the way the measuring system interacts with the measured entity. As a consequence, the interaction is not a priori given in a quantum measurement, but is each time selected (that is, actualized, through a weighted symmetry breaking processes) when the experiment is executed; and since different measurement-interactions can produce different outcomes, this can explain why the output of a quantum measurement can only be predicted in probabilistic terms.[1] (One should not think however of these hidden measurement-interactions to be something similar to, or to be describable in the same way as, the fundamental interactions (fundamental forces) of the standard model of particle physics, mediated by bosonic elementary entities).

## Contents

* 1 History
* 2 Description
* 3 Advantages
* 3.1 The Born rule
* 3.2 The extended Bloch representation
* 3.3 Realistic hypothesis
* 3.4 Unification of quantum and classical probability
* 3.5 Natural quantum formalism
* 4 Notes

## History

The hidden-measurements interpretation was proposed in the 1980s by the Belgian physicist Diederik Aerts,[1] and was subsequently developed over the years thanks to the work of Aerts and of a number of collaborators, such as Bruno Van Bogaert, Thomas Durt, Bob Coecke, Frank Valckenborgh, Bart D'Hooghe, Sven Aerts, Sandro Sozzo and Massimiliano Sassoli de Bianchi.[2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].

## Description

In principle the HMI can be looked upon as a hidden-variables theory. However, contrary to standard hidden-variables approaches, the variables are not associated with the state of the measured entity, but with the measurement-interactions taking place between the latter and the measuring system. In other words, in the HMI the state of the physical entity, as formulated by quantum mechanics, is considered to provide a complete description. This means that the HMI is not an attempt to come back to a classical view of our physical reality, but it contains a simple explanation of the quantum probabilities as being due to a ‘lack of knowledge about these uncontrollable fluctuations on the interaction between the measuring apparatus and the entity, occurring at a non-spatiotemporal (or pre-spatiotemporal) level of our physical reality’. Hence, they would be of an epistemic and not ontological nature.

It is important to emphasize that the HMI is not in conflict with the existing no-go theorems, and this is precisely because if considered as a hidden-variables theory, the hidden variables are not associated with the state of the entity.

## Advantages

### The Born rule

The main strength of HMI is its ability to derive, in a non-circular way, the Born rule - the prescription for determining the probability of obtaining a given outcome in a quantum measurement. Consequently, HMI offers a convincing possible solution to the longstanding measurement problem.[1][24][33][34] (See also the video presentation: Solving the measurement problem.) Its weakness is that the existence of the hidden-measurement interactions, characterizing the overall dynamics of a quantum measurement, remain for the time being a hypothesis awaiting experimental confirmation.

### The extended Bloch representation

The natural mathematical framework for the HMI, in which the Born rule can be derived, is the extended Bloch representation (EBR) of quantum mechanics.[33][35] It uses a generalized Poincaré-Bloch sphere to geometrically representing the states (for a N-dimensional quantum system the generalized Bloch sphere is (N2−1)-dimensional), in which it is also possible to represent the “potentiality regions” of quantum measurements (describing the available hidden measurement-interactions) and how a superposition state evolve (in a non-unitary way) during the latter, to transition to an outcome state.

The EBR also allows to investigate the geometry of superposition and entangled states, explaining in particular how the entanglement correlations are created (again, via the selection of a non-local/non-spatial measurement-interactions) in joint measurements on bipartite systems.[35] Note that, as it considers density operators to be also representative of genuine states, the EBR, and the associated HMI, is to be considered a completed version of standard quantum mechanics, which allows not to give up the general physical principle saying that a composite entity exists, and therefore is in a genuine state, if and only if its components also exist, and therefore are in well-defined states.[36]

### Realistic hypothesis

There are circumstances in which the hidden-measurements are not just a hypothesis, but a fact.

It is possible to conceive macroscopic quantum machines, working at room temperature, whose properties are surprisingly quantum, or quantum-like, and this is precisely because their behavior is governed by a hidden-measurements mechanism.[1][3][4][5][27][28] This makes it also possible to propose models of macroscopic quantum situations that violate Bell inequalities.[2][29]

Another situation where the hidden-measurements mechanism is more than just a hypothesis is quantum cognition, an emerging field which applies the mathematical formalism of quantum theory to model cognitive phenomena. This is because it is very natural in this ambit to consider that the hidden-measurement interactions result from our subconscious “non-logical” intrapsychic processes, which although they cannot be easily discriminated at the conscious level, should not be considered any less real.[7][25][26][31][32]

### Unification of quantum and classical probability

HMI allows a unified view of quantum and classical probabilities: Both can be shown to result from our lack of knowledge and control about the particular interaction that actually happens during an experiment. A classical “game of chance” experiment is interpreted in the same way that a quantum measurement is.[28][34]

This common origin of quantum and classical probabilities allows one to use the hidden-measurements approach to also propose a solution to a fundamental problem of classical probability theory: Bertrand's paradox. In other words, according to the HMI, the quantum mechanical measurement problem and the classical Bertrand's paradox, would be just two sides of a same coin.[30]

### Natural quantum formalism

The HMI also offers the possibility of providing a natural generalization of the quantum formalism, allowing for the investigation of quantum-like entities whose space state is not necessarily the Hilbert space. This can be done by simply varying the amount of fluctuations between the measurement apparatus and the entity considered, obtaining in this way, intermediary structures that are neither quantum nor classical, but truly in between. In this way, one also obtains a theory for the study of the mesoscopic region of our reality, the structure of which would be impossible to obtain in the ambit of orthodox theories, be they quantum or classical.[5][6]

Considering that a quantum measurement is a process which, starting from an initial pre-measurement state, produces a final post-measurement state, and that according to the HMI a quantum state is to be considered a complete description of the reality of the entity under consideration, it follows that a hidden-measurement interaction corresponds to an invasive process, able to create new elements of reality (new properties).

More precisely, a quantum measurement would involve a creation aspect because

1. it gives rise to a change in the state of the entity and
2. such a process of change cannot be predicted in advance.

At the same time, it also involves a discovery aspect, as it is clear that the statistics of outcomes can provide information about the pre-measurement state. In that sense, a quantum measurement is a process which, according to the HMI, would entail a sort of balance between discovery and creation.[17][31][32]

If a quantum measurement involves a creation aspect, resulting from the interaction of the measuring system with the measuring apparatus, then we are forced to accept that microscopic quantum entities, like electrons, protons, etc., are not permanently present in space, and that only at the moment they are detected by a measuring apparatus, would a position for them be created. In other terms, the HMI indicates that when a quantum entity, like an electron, in a non-spatial (superposition) state is detected, it is literally “dragged” or “sucked up” into space by the detection system. And this means that our physical reality would not be contained in space, but the other way around.[17]

To quote Aerts:[21]

> Reality is not contained within space. Space is a momentaneous crystallization of a theatre for reality where the motions and interactions of the macroscopic material and energetic entities take place. But other entities – like quantum entities for example – “take place” outside space, or – and this would be another way of saying the same thing – within a space that is not the three dimensional Euclidean space.

## Notes

1. ↑ 1.0 1.1 1.2 1.3 Aerts, D. (1986). A possible explanation for the probabilities of quantum mechanics, Journal of Mathematical Physics, 27, pp. 202-210.
2. ↑ 2.0 2.1 Aerts, D. (1991). A mechanistic classical laboratory situation violating the Bell inequalities with 22, exactly 'in the same way' as its violations by the EPR experiments. Helvetica Physica Acta, 64, pp. 1-23.
3. ↑ 3.0 3.1 Aerts, D., Durt, T. and Van Bogaert, B. (1993). A physical example of quantum fuzzy sets and the classical limit. Tatra Mountains Mathematical Publications, 1, pp. 5-15.
4. ↑ 4.0 4.1 Aerts, D., Durt, T. and Van Bogaert, B. (1993). Quantum probability, the classical limit and nonlocality. In K. V. Laurikainen and C. Montonen (Eds.), Symposium on the Foundations of Modern Physics 1992: The Copenhagen Interpretation and Wolfgang Pauli (pp. 35-56). Singapore: World Scientific.
5. ↑ 5.0 5.1 5.2 Aerts, D. and Durt, T. (1994). Quantum, classical and intermediate: a measurement model. In K. V. Laurikainen, C. Montonen and K. Sunnaborg (Eds.), Symposium on the Foundations of Modern Physics. Gives Sur Yvettes, France: Editions Frontieres.
6. ↑ 6.0 6.1 Aerts, D. and Durt, T. (1994). Quantum, classical and intermediate, an illustrative example. Foundations of Physics, 24, pp. 1353-1369.
7. ↑ 7.0 7.1 Aerts, D. and Aerts, S. (1995). Applications of quantum statistics in psychological studies of decision processes. Foundations of Science, 1, pp. 85-97.
8. ↑ Coecke, B. (1995). A hidden measurement representation for quantum entities described by finite-dimensional complex Hilbert spaces. Foundations of Physics 25 (8), 1185-1208,
9. ↑ Coecke, B. (1995). Hidden measurement model for pure and mixed states of quantum physics in Euclidean space. International Journal of Theoretical Physics 34 (8), 1313-1320.
10. ↑ Coecke, B. (1995). Generalization of the proof on the existence of hidden measurements to experiments with an infinite set of outcomes. Foundations of Physics Letters 8 (5), 437-447.
11. ↑ Aerts, D., Aerts, S., Coecke, B., D'Hooghe, B., Durt, T. and Valckenborgh, F. (1997). A model with varying fluctuations in the measurement context. In M. Ferrero and A. van der Merwe (Eds.), New Developments on Fundamental Problems in Quantum Physics (pp. 7-9). Dordrecht: Springer.
12. ↑ Aerts, D., Coecke, B., D'Hooghe, B. and Valckenborgh, F. (1997). A mechanistic macroscopical physical entity with a three dimensional Hilbert space quantum description. Helvetica Physica Acta, 70, pp. 793-802.
13. ↑ Aerts, D., Coecke, B., Durt, T. and Valckenborgh, F. (1997). Quantum, classical and intermediate I & II. Tatra Mountains Mathematical Publications, 10, p. 225; p. 241.
14. ↑ Coecke, B. (1997) Classical representations for quantum-like systems through an axiomatics for context dependence. Helvetica Physica Acta 70; pp.442-461. arXiv:quant-ph/0008061
15. ↑ Coecke, B. (1997) A classification of classical representations for quantum-like systems. Helvetica Physica Acta 70; pp.462-477. arXiv:quant-ph/0008062
16. ↑ Aerts, D. (1998). The hidden measurement formalism: what can be explained and where paradoxes remain. International Journal of Theoretical Physics, 37, pp. 291-304.
17. ↑ 17.0 17.1 17.2 Aerts, D. (1998). The entity and modern physics: the creation-discovery view of reality. In E. Castellani (Ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics (pp. 223-257). Princeton: Princeton University Press.
18. ↑ Coecke, B. (1998) A Representation for Compound Quantum Systems as Individual Entities: Hard Acts of Creation and Hidden Correlations. Foundations of Physics 28; pp.1109-1135. arXiv:quant-ph/0105093
19. ↑ Coecke, B. (1998) A Representation for a Spin-S Entity as a Compound System in R^3 Consisting of 2S Individual Spin-1/2 Entities. Foundations of Physics 28; pp.1347-1365. arXiv:quant-ph/0105094
20. ↑ Coecke, B. and Valckenborgh F. (1998) Hidden Measurements, Automorphisms, and Decompositions in Context-Dependent Components. International Journal of Theoretical Physics 37; pp.311-321.
21. ↑ 21.0 21.1 Aerts Diederik (1999). The stuff the world is made of: physics and reality. In D. Aerts, J. Broekaert and E. Mathijs (Eds.), Einstein meets Magritte: An Interdisciplinary Reflection (pp. 129-183). Dordrecht: Kluwer Academic.
22. ↑ Aerts, D., Aerts, S., Durt, T. and Leveque, O. (1999). Classical and quantum probability in the epsilon model. International Journal of Theoretical Physics, 38, pp. 407-429.
23. ↑ Aerts, S., (2002). Hidden measurements from contextual axiomatics, in Probing the Structure of Quantum Mechanics. Eds. D. Aerts., M. Czachor and T. Durt, World Scientific Publishers, pp. 149-164.
24. ↑ 24.0 24.1 Aerts, S. (2005). The Born rule from a consistency requirement on hidden measurements in complex Hilbert space. International Journal of Theoretical Physics, 44, pp. 999-1009.
25. ↑ 25.0 25.1 Aerts, D. and Sozzo, S. (2012). Quantum Model Theory (QMod): Modeling contextual emergent entangled interfering entities. Quantum Interaction. Lecture Notes in Computer Science, 7620, pp 126-137, 2012.
26. ↑ 26.0 26.1 Aerts, D. and Sozzo, S. (2012). Entanglement of Conceptual Entities in Quantum Model Theory (QMod). Quantum Interaction. Lecture Notes in Computer Science, 7620, pp 114-125, 2012.
27. ↑ 27.0 27.1 Sassoli de Bianchi, M. (2013). Using simple elastic bands to explain quantum mechanics: a conceptual review of two of Aerts' machine-models, Central European Journal of Physics, 11, 147-161.
28. ↑ 28.0 28.1 28.2 Sassoli de Bianchi, M. (2013). Quantum dice. Annals of Physics, 336, 56-75.
29. ↑ 29.0 29.1 Sassoli de Bianchi, M. (2014). A remark on the role of indeterminism and non-locality in the violation of Bell's inequality. Annals of Physics 342, 133-142.
30. ↑ 30.0 30.1 Aerts, D. and Sassoli de Bianchi, M. (2014). Solving the hard problem of Bertrand's paradox. Journal of Mathematical Physics, 55, 083503.
31. ↑ 31.0 31.1 31.2 Aerts, D. and Sassoli de Bianchi, M. (2014). The unreasonable success of quantum probability I: Quantum measurements as uniform fluctuations. To appear in: Journal of Mathematical Psychology. arXiv:1401.2647.
32. ↑ 32.0 32.1 32.2 Aerts, D. and Sassoli de Bianchi, M. (2014). The unreasonable success of quantum probability II: Quantum measurements as universal measurements. arXiv:1401.2650.
33. ↑ 33.0 33.1 33.2 Aerts, D. and Sassoli de Bianchi, M. (2014). The extended Bloch representation of quantum mechanics and the hidden-measurement solution to the measurement problem. Annals of Physics 351, Pages 975–1025 (Open Access).
34. ↑ 34.0 34.1 34.2 Aerts, D. & Sassoli de Bianchi M. (2015). Many-Measurements or Many-Worlds? A Dialogue. Found. Sci. 20: 399, doi:10.1007/s10699-014-9382-y. Also arXiv:1406.0620 [1]
35. ↑ 35.0 35.1 35.2 Aerts, D. & Sassoli de Bianchi. M. (2016). The Extended Bloch Representation of Quantum Mechanics. Explaining Superposition, Interference and Entanglement. J. Math. Phys. 57, 122110 (2016), doi: [2]
36. ↑ 36.0 36.1 Aerts, D., Sassoli de Bianchi. M. and Sozzo, S. (2017). The extended Bloch Representation of Entanglement and Measurement in Quantum Mechanics. Int. J. Theor. Phys. 56, 3727-3739, doi: [3]

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