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HandWiki. Hidden-Measurements Interpretation. Encyclopedia. Available online: https://encyclopedia.pub/entry/31014 (accessed on 25 June 2024).

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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. (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).

standard model
quantum measurement
fundamental interactions

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]}.

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.

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 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 [math]\displaystyle{ N }[/math]-dimensional quantum system the generalized Bloch sphere is [math]\displaystyle{ (N^2-1) }[/math]-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]}

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]}

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]}

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

- it gives rise to a change in the state of the entity and
- 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.

- Aerts, D. (1986). A possible explanation for the probabilities of quantum mechanics, Journal of Mathematical Physics, 27, pp. 202-210.
- Aerts, D. (1991). A mechanistic classical laboratory situation violating the Bell inequalities with [math]\displaystyle{ 2\sqrt }[/math], exactly 'in the same way' as its violations by the EPR experiments. Helvetica Physica Acta, 64, pp. 1-23.
- 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.
- 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.
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- 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
- 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
- Coecke, B. and Valckenborgh F. (1998) Hidden Measurements, Automorphisms, and Decompositions in Context-Dependent Components. International Journal of Theoretical Physics 37; pp.311-321.
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