In quantum physics a measurement is an experimental situation in which a physical entity undergoes an indeterministic and irreversible change, called the collapse of the wave function, or reduction of the state vector.
The existence of processes of change of this kind is considered to be a problem because even if we perfectly know the initial state of the entity, that is, its state before the measurement, we cannot predict with certainty what will be its final state, that is, the state at the completion of the measurement process.
The best we can do is to attach probabilities to the different possible final states, by means of a rule of correspondence, called the Born rule.
Solving the (quantum) measurement problem is about explaining what goes on, “behind the scenes,” when a system is subjected to a measurement context. What produces such an abrupt change of the entity’s state? Why such change is non-deterministic and irreversible? Why the Born rule is so successful in determining the probabilities of the different possible outcomes?
It is generally believed that convincing answers to the above questions are yet to be found. Not only: many believe that these answers cannot be found, because the so-called hidden-variables theories, which assume that quantum mechanics would be an incomplete theory, have been opposed by the well-know no-go theorems: impossibility proofs which forbid the construction of theories where the quantum indeterminism would be replaced by a deeper deterministic description.
In other terms, the no-go theorems forbid the replacement of quantum mechanics by a more fundamental theory in which the physical entities would always possess well-defined properties, that is, in which the probabilities to have or not to have a certain property would only take the values o or 1. But, is it really so?
In the following video presentation you will discover that, contrary to what is generally believed, the central problem of quantum mechanics can actually be solved by using a hidden-variables argument. But for this, we have to follow the intuition of the Belgian physicist Diederik Aerts, that already in the eighties of the last century proposed to associate the hidden-variables not to the state of the measured entity, but to its interaction with the measuring apparatus, in what is today known as the hidden-measurements interpretation of quantum mechanics.
Then, the no-go theorems no longer apply and it becomes possible not only to conceptually explain the nature of a quantum measurement, but also to derive, in a non-circular way, the Born rule.
If you are a professional physicist, and would like to deepen your understanding of the solution presented in the above video, we recommend the reading the following open source article:
Diederik Aerts & Massimiliano Sassoli de Bianchi, The Extended Bloch Representation of Quantum Mechanics and the Hidden-Measurement Solution to the Measurement Problem, Annals of Physics 351, Pages 975–1025 (2014).
If you have a good scientific background, but would like to read a less technically involved text, we recommend the following article, written as a dialogue (also available on the arXiv:1406.0620 [quant-ph]):
Diederik Aerts & Massimiliano Sassoli de Bianchi, Many-Measurements or Many-Worlds? A Dialogue, Foundations of Science. DOI: 10.1007/s10699–014–9382-y (December 2014).
If you have no specific knowledge about physics, and would like to read a more pedestrian text on the hidden-measurement interpretation, we recommend the reading the following booklet (a trailer of which can be viewed on YouTube):
A final recommendation is the video entitled: The physics of spaghetti — Heisenberg’s Uncertainty Principle and Quantum Non-spatiality (Non-locality).