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The hidden-measurements interpretation (HMI), also known as the hidden-measurements approach, is a realistic interpretation of quantum mechanics that was proposed in the eighties of the foregoing century by the Belgian physicist Diederik Aerts, 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.

The hypothesis at the basis of the interpretation is 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 each time selected (that is, actualized) when the experiment is executed; and since different measurement-interactions can produce different outcomes, this explains why the output of a quantum measurement can only be predicted in probabilistic terms.

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 terms, 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 a tentative 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 measuring apparatus and the entity’, and hence being 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 precisely because if considered as a hidden-variables theory, the hidden-variables are not associated with the state of the entity.

The strength of the HMI is its ability to derive, in a non-circular way, the Born rule, that is, the law determining the probability of obtaining a given outcome in a quantum measurement. Thus, the HMI offers a convincing solution to the longstanding measurement problem. It’s weakness is that the existence of the hidden measurement-interactions, characterizing the overall dynamics of a quantum measurement, remain for the time being an hypothesis awaiting experimental confirmation.

However, there are ambits in which hidden-measurements are not just a hypothesis, but a fact. Indeed, it is possible to conceive macroscopic quantum machines, working at room temperature, whose properties are surprisingly quantum, or quantum-like, and this precisely because their behavior is governed by a hidden-measurements mechanism. This makes it also possible to propose models of macroscopic quantum situations that violate Bell inequalities. Another domain 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 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 cannot be easily discriminated at the conscious level, not for this should be considered less real.

Another advantage of the HMI is to allow a unified view of quantum and classical probabilities, as both can be shown to result from our lack of knowledge and control about the interaction that is actualized during an experiment, be it a classical “game of chance” experiment, or a quantum measurement. 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 terms, according to the HMI, the quantum mechanical measurement problem and the classical Bertrand's paradox, would be just the two sides of a same coin.

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 them quantum or classical.

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) that such process of change cannot be predicted in advance. But at the same time, it also involves a discovery aspect, as 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 the discovery and creation aspects.

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 only be when they are detected by a measuring apparatus a spatial position for them would 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. To quote Aerts:

"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."