_{1}

The Born-rule, which assigns probabilities to measurement outcomes, is one of the fundamental axioms of quantum physics. It dates back to the time of the establishment of the formalism of quantum physics in the first half of the 20th century. From the beginning, and particularly in connection with the development of different interpretations of the theory, there has been a desire/need to better understand the true nature of the Born-probabilities. Are they classical/epistemic of origin or are they irreducible and of on tic stature as a kind of intrinsic propensities of physical systems? We show that, by only using the mathematical formalism of the original theory, we find a possible answer.

The formalism of quantum physics has been developed during the first decades of the 20^{th} century. It describes a physical system as an element

This is the Born-rule [

There are some questions, which naturally arise with regard to the Born-rule. Firstly, why are there probabilities in the first place and secondly, what kind of probabilities are they? Both questions are intimately linked to interpretations of quantum mechanics and have in this context found various answers. Focusing on the second question we find the opinions, starting on the realist side, that the probabilities might be objective, irreducible properties of quantum systems, as in the GRW interpretations [

Given the resolution of a state

We say that the operator

^{1}. Gleason’s theorem tells us that we are looking at the right probabilities. But it is per se not helpful to better understand the nature of the Born-probabilities.

Assume there is a density matrix

where ^{2}. A general unitary transformation on a tensor-product, expressed in the respective bases, can be written as a matrix

where the operators

Conversely, we can choose any set of operators

Assume there is a quantum system

Assume there is a second system

^{3}. The probe ^{4} such that

For our purpose we now chose the operators

Therefore we can write (6) in the following form

Comparing Equation (8) with Equation (2), we see that

which is the Born-rule.

We have in the above derivation not made use of any specific interpretation of quantum mechanics, but relied on two basic assumptions only. The first one is the formalism of density operators and generalized measurement with classical or epistemic probabilities arising in mixed states (2). The second one is Laplace’s principle of indifference in order to introduce the concept of probabilities and to assign concrete probability-values

We have found that, given any not necessarily normalized pure state, it is possible to define an observer with an appropriately coarse-grained probe-system^{5} who, by lack of further knowledge, will assign exactly the Born-probabilities, as classical probabilities in the sense of (2), to finding the system in one of the basis-states, after the measurement and before observation. In other words, there is the possibility to interpret the normalized amplitudes of an arbitrary state

If a quantum state

Schlatter, A. (2017) On the Nature of the Born Probabilities. Journal of Modern Physics, 8, 756- 760. https://doi.org/10.4236/jmp.2017.85047