SUPERPOSITIONS > Origins
Quantum states can be represented by vectors belonging to a so-called
Hilbert space. Vectors provide an ideal framework to represent statistical regularities, especially when
incompatible observables are involved. To see how this works, let

be a vector of unit length in the plane and consider its projections on two orthogonal axes

and

(figure 5). By Pythagoras' theorem:
Now suppose that we measure an observable
A that can only take two values,

and

. The probabilities of getting

and

vary depending on the way the measured system has been prepared.
Figure 5. A unit vector in the plane and its projections on the axes.
Nevertheless, since

and

represent mutually exclusive outcomes exhausting the spectrum of the possible occurrences, we know
a priori that, in any case, the corresponding probabilities meet the condition:
The predictive formalism of quantum theory is based on the parallelism between eq. (1) and (2). It is summarized by the following rules:
I.
The values

that an observable
A can take are represented by orthogonal axes

spanning a
state space.
II. Each experimental procedure which ‘prepares’ a system before a measurement of A is represented by a
state vector 
belonging to the
state space and having unit length.
III. (
Born rule) If we measure the observable A, the probability

to get the value

is given by the square modulus of the projection of

onto the

axis:
Figure 6a and b. Representation of the 'up' eigenvector of Sz . The antiparallel vector would do the job as well - we arbitrarily choose vectors belonging to the positive unit semicircle to represent quantum states.
An important corollary of rules I-III is the following: the
state vector 
which corresponds to the special case in which we can predict
with certainty (i.e. with probability 1) that the measurement of A will yield the value

is the unit vector lying on the

axis (

). Following
Dirac's notation, this vector is denoted

and is called an
eigenstate (or eigenvector) of
A.
Let's apply these simple rules to the Stern-Gerlach experiments discussed in the section on
experimental evidence. We are concerned with two
spin observables:
Sz, which can take the two ‘values’

and

; and
Sx, which can take the two ‘values’

and

. In the first step of the experiment (
figure 2) only those atoms are selected for which a measurement of
Sz yields

. For these atoms, we are certain to find

if the measurement of
Sz is repeated. Therefore, the associated
state vector is

. If, on the sample of atoms thus prepared, we now measure
Sx, we find

for half of the atoms and

for the other half (step 2 of
figure 3). Can this result be represented in our
state space? Yes, it is sufficient to take the two axes corresponding to

and

(see rule I) as forming an angle of π/4 with the axes corresponding to

and

(figure 6a). According to rule III, in order to find out the probability of finding

we have to project the
state vector 
onto the axis corresponding to

(figure 6a). In the same way we compute the probability that the measurement of
Sx will yield

. We get:
The result of the third experiment (
figure 3) can be straightforwardly formalized in the same way, observing that the first two apparatus ‘prepare’ the
state 
(figure 6b).
Let's see how
interference phenomena are taken into account by this simple model. Exploiting the elementary
rules for the composition of vectors in the plane, the vectors

and

can be rewritten as follows:
Notice that both these
states predict the same
probability distribution for a measurement of
Sx:

. Therefore, the two
states (or, to be more precise, the corresponding preparations) cannot be distinguished
a posteriori by performing such a measurement. The two preparations yield nonetheless opposite statistical results if a measurement of
Sz is performed (figure 7). In the former case one
always finds ‘

’; whereas in the latter, this
never happens. This ‘never’ is the signature of
interference (compare to the bright and dark fringes of the
double-slit experiment). Notice that the only mathematical difference between the two states (3) is the sign of the
superposition: plus in the former case, minus in the latter. This sign expresses the
phase relation existing between

and

. (Using complex notation,

or

depending on whether φ = 0 or φ = π; it is worth making a comparison with
waves.)
Figure 7. Comparison of the probability distributions predicted by two different superpositions of Sx' eigenstates.
So far, We have been concerned with a special class of observables, namely
those observables that can be represented by different orthogonal
frames within
the same state space. The
probability distributions associated with such observables are rigidly connected to one another. In the case of
Sz and
Sx, for instance, we have seen that, given an equiprobable distribution of ‘up’ and ‘down’ for
Sx, the statistics of
Sz necessarily follows one of the two distributions represented in figure 7.
However, there exist observables whose respective
probability distributions are
a priori completely independent. A typical example is provided by the
spin observables of two distinct particles. Particle 1 can be prepared in such a way that

and particle 2 in such a way that

. (This
cannot be done if
Sx and
Sz are measured on the
same particle, since
Sx and
Sz are
incompatible observables.)
In general, however, once two quantum systems have interacted, they
become correlated in such a way that measurement results can no longer
be predicted by just assigning an
individual state vector to each partner system. Instead, a unique, global
state vector has to be associated with the whole pair. This
state is constructed by combining the individual
state vectors, but belongs to a larger
state space. An
entangled state of two
spin ½ particles is the following:
The statistical predictions associated with

are very important. The
probability of finding ‘up’ or ‘down’ if
Sz (or any other
spin component) is measured on each particle separately is 1/2. Furthermore, if one of the particles is found to have
spin ‘up’ in the z direction, the result of any
spin measurement performed subsequently
on the other particle will follow the statistics predicted by

. If, conversely, the first particle is found to have
spin ‘down’ in the z direction, then the statistics of the other particle will be that corresponding to

. Due to the peculiar form of state (4), analogous correlations exist for
any spin component. Thus, if, for example, particle 1 is found to have spin ‘up’ in the x direction, particle 2 will follow the predictions of

; and so on (the situation is represented in
figure 4). Indeed,

preserves its mathematical form if expressed in terms of the
eigenvectors of
Sx:

.
The
probability distributions observed in measurements performed on a single particle are fundamentally
different depending on whether the particle has or has not an
entangled partner. In the former case, as shown by figure 8,
interference effects are washed out (this can readily be derived from the above definition of

by using formulas (3); see also
complementarity).
In the toy-model used for our presentation, the
state space was the real plane of Euclidean geometry and the outcomes were
represented by axes. However, a complete analysis of all possible
Stern-Gerlach measurements shows that this space is insufficient to
provide a complete representation. A richer geometrical structure, the
Hilbert space, defined on the field of complex number, must be introduced. Within this structure the outcomes are represented by
subspaces, and the observables by combinations of
projectors over these subspaces.
Figure 8. Probability distributions encountered when the tested particle (particle 1) has an entangled partner.