On the Nature of the Born Probabilities

The Born-rule, which assigns probabilities 2 2 a a p ψ ψ = 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 20 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.


Introduction
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 ψ of some appropriate Hilbert space H and physical, observable quantities as eigenvalues a ∈  in the spectrum A σ ⊆  of self-adjoint operators

( )
A L H ∈ on that Hilbert space. The eigenvalue-eigenstate postulate says that, whenever a system is found to have a value has ever been discovered experimentally. The measurement postulate is incompatible with a further postulate of quantum mechanics, namely the unitary evolution of the quantum state. Ever since the establishment of the theory there have been different interpretations and extensions of quantum physics in order to solve this incompatibility, known as the "measurement problem".
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 [2] or, passing to the epistemic/instrumentalist side, subjective degrees of belief [3] or, yet represent something else, like rational preferences in the decision-theoretic explanations of the many-worlds interpretation [4] [5] [6] [7]. Because the formalism of quantum physics shows the kind of ontological under determination it does, the Born-rule does as well. In this paper we follow the original, basic formalism and will give a possible answer to the question of the nature of the probabilities.

Some Formalism
Given the resolution of a state H ψ  in the eigenbasis , We say that the operator  in (2)

Measurement
Assume there is a density matrix  and basis (eigen)-states , a A e a σ  , corresponding to some self-adjoint operator . Assume in addition that there is an additional system  with orthonormal basis states { } n N n ≤ , which we assume originally to be in the base state 0 0 0 =  . A measurement of  by the probe  is an operation U on the joint system joint 0 0 where U is unitary II UU * = 2 . A general unitary transformation on a tensor-product, expressed in the respective bases, can be written as a matrix , , , an a n an a n nn nn U u a n a n A n n where the operators nn A ′ are given by Conversely, we can choose any set of operators n A satisfying the resolution of the identity-condition (5) to define a measurement on an initial joint state joint 0 0 = ⊗  . We now have the necessary elements in place to give the main argument.

The Born-Rule
Assume there is a quantum system  in a, not necessarily normalized, pure II.
k k a n n a a a a n N a M k m a M P P P P P P * * Therefore we can write (6) in the following form Comparing Equation (8) with Equation (2), we see that   can be viewed as a mixed state with probabilities which is the Born-rule.

Conclusions
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 1 , n p N n N = ≤ , to the mixed state in (6).
This is the important step, which helps to avoid the kind of tautological argument based on the reduced density matrix and Gleason's theorem. It bases on a kind of symmetry of the probe states , n n N ≤ , due to a lack of knowledge before observation.
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 H ψ as epistemic probabilities for different possible measurement outcomes.
If a quantum state ψ is given to us a priori, then it seems that the Bornprobabilities are objectively given with it and there might be a reluctance to embrace Laplace's principle as fundamental [9]. It seems to us that we can accept the principle as deeply rooted in our intuition and therefore to be a first principle. Authors, concerned with the many worlds interpretation in [6] [7] for instance, try to give more objective physical justifications for Laplace's principle, mainly by describing the symmetry, which it bases on, in more physical terms.
We are not sure that these ideas are really more fundamental or whether they are not the same intuition vested in different garments. We should in all this always remember that quantum states are practically given to us by making preparations, i.e. by correlating them with other systems in laboratories. It is a wonderful fact, however, that the Born-probabilities are confirmed by the corresponding frequencies, if repeated experiments are being done. This would probably not be further impressive, if quantum states would just be results of gathering measurement information. But they can also arise from an initial state by Schrödinger evolution. We at least can say that nature seems to "play the game".