The ansatz for the mathematical theory of quantum physics is to represent a physical quantity as a self-adjoint operator
in the space of linear operators over a state space
, which carries the structure of a complex Hilbert space of some dimension
. The values, which this quantity can assume in an experiment, are the corresponding eigenvalues
of
. We have to find a way to assign probabilities to these eigenvalues, a task which is equivalent to assigning probabilities to the orthogonal projection operators
, which project states in
to eigenstates corresponding to the
. If agents assign a probability
to a projection operator
, which is common to two families of orthogonal projectors,
and do corresponding experiments, they would like to be sure that, if they find the frequency
, the results describe the same event
. This is a non-contextuality condition (note that the set of (conditional) probabilities over realized values is contextual, as a theorem by Kochen–Specker [
8] and an example by Hardy [
9] show). A famous theorem from Gleason [
10] says that to any non-contextual measure
on the sets of projectors
over a Hilbert space
of dimension
, there exists a unique positive semi-definite, self-adjoint operator
of trace class one, called density operator, such that
for all
. This is the Born-rule. The theorem defines the appropriate measures as well as the quantum states, which are identified with the density-operators
. There is a special class
of density-operators, called pure states. Every vector,
defines a corresponding pure state
, which is the projection operator onto
The set of density operators,
, is the set of all convex combinations of pure states
. The weights corresponding to the convex combinations are the second category of probabilities mentioned above. As important as Gleason’s theorem of course is, because it technically defines the right quantum-probabilities, it does not say much about their nature/interpretation.
2.2. Probabilities of Pure States
We now consider a system
represented by a pure state
with resolution in the eigenbasis
of a self-adjoint operator
,
We can form the corresponding pure state
with matrix-entries
. The Born rule then assigns probabilities
to the projectors
. Assume that there is an additional system
with orthonormal basis states
which is initially in the base state
. A measurement of some state
by the probe
is an operation
on the joint system
where
is unitary
(this follows from the fact that a general interaction evolution
is unitary). A general unitary transformation on a tensor-product, expressed in the respective bases, can be written as a matrix
where the operators
are given by
. We denote the diagonal sub-block
simply by
. Since
is unitary, we have
Conversely, we can choose any set of operators satisfying the resolution of the identity-condition (5) to define a measurement on an initial joint state . We now have the necessary elements in place to give the main argument.
Assume there is a second system
with basis
and an observer who would like to know in what state
the system
is in, by making an appropriate measurement
on the joint system
. If that is possible in the first place, then, having no additional knowledge, the observer does not, a priori, know in what state
, the probe will be after the measurement and before observation, leading to permutation-symmetry. Let the underlying pure state
have coefficients
(since the rational numbers
are dense in
, the choice of
is general enough). The probe
can be chosen appropriately coarse-grained (this coarse-graining is first introduced in [
11] in the context of many-worlds) such that
. The observer is after the measurement and before observation in a situation where, by lack of further information, she will by Laplace’s principle of indifference a priori attribute to each outcome
equal probability
. This attribution is equivalent to maximizing the entropy function
. The observer can therefore write down in the spirit of (1) an average of outcomes
For our purpose, we now chose the operators
to be the scaled projectors
to the basis-states
. Note that we have replaced the simple-index
by the double-index
. This choice is consistent with the demands of a measurement, since the
satisfy (5)
Therefore, we can write (6) in the following form
Comparing Equation (8) with Equation (1), we see that
can be viewed as a mixed state with probabilities
which is the Born-rule.
Before we turn to consider the frequencies, let’s have a look at composite systems To show the principle it is sufficient to look at binary systems.) The state may be mixed or pure and we can apply the findings in a straightforward way. The single components are given by the partial trace . If the state has the form , then we are in the separable case and can apply the results in 2.1 and 2.2 to each individual component, which can be pure or mixed. In case is entangled, then the partial trace always produces a mixed state. When we now consider frequencies, then the individual quantum systems might be single or composite, what is important is that they are temporally separable in order to allow for statistics.
2.3. Frequencies
The theory so far does only cover single trials. Assume there is a density-operator
and a complete set of projectors
. To find probabilities for a sequence of different outcomes
of
experiments on
(this is done on
identically prepared systems) we can apply Gleason’s theorem to the tensor product [
5]
to get
with
So the outcomes of repeated measurements are identically and independently distributed (i.i.d.). The probability for outcome
to occur
times,
,
, is given by the multinomial distribution
The individual counting functions
are binomially distributed and hence
. For large
the averages,
, of the statistical counting functions approach the expectation values and therefore
The fact that
is due to the law of large numbers. The frequencies with their implied principle of indifference (14) indeed replicate the probabilities. This is achieved by a strong assumption in the theory, reflected in Equations (11), (13), and (14). It is the independence condition for the multi-trial states
. Actually, it is itself a consequence of the assumption that agents have maximal information about a system of
copies of a quantum state [
5]. Independence implies serial permutation-symmetry, i.e., the fact that it does not count in which sequence the results occur. So in the case of multiple-trials the theory uses a stronger assumption than permutation-symmetry to obtain the compatible frequency-probabilities (14). Can we weaken the assumption?
It is remarkable that, due to the (infinite) de-Finetti theorem [
12], the assumption of independence can be weakened to the one of exchangeability, to still allow reasonable statistics. Exchangeability stands for permutation-symmetry of the joint distribution of
trials
and for consistency from step
to
,
. If satisfied, it can uniquely represent an
-trial state
by an integral over product states of form (10) by means of a measure
on
(The measure
belongs to the second category of probabilities.) For states
of form (15) the statistical approach (14) works with some suitable adjustments (while the distributions are directly integral averages over the product state-distributions, there holds a law of large numbers only conditional to a suitable
-algebra [
13]). Whether we work with states of maximal information (10) or states of form (15), in any case permutation-symmetry and the principle of indifference are key features of the frequencies (14), derived in multiple-trials.