For a continuous set of pure states which are distinguishable, as it is the case for a single classical particle, the discrete coordinates in Eq. (1.48) have to be replaced by functions on the continuous set of pure states (phase space ):
The corresponding set of event vectors reads:
; ( " We have thus seen that the general description of measurements by means of the state space
and the set of event vectors is broad enough to incorporate also the measurement statistics of classical physics. In the representation by means of operators on a Hilbert space classical physics is associated with simultaneously diagonalisable Effects and density operators. A more detailed investigation of the statistical model in classical physics comprising the ideas in this subsection can be found in the first chapter of [Hol82].
Hidden variables
If the class of measurements carried out on a classical system is (artificially) restricted and thus not all pure states can be distinguished, the unique decomposition of mixed states cannot be accomplished anymore. In this case the effective state space is no longer a simplex. In fact can assume the form of any convex compact set when the restriction is chosen appropriately, cf.
[Hol82], p.34. This fact is used by so-called hidden variable theories which represent the state space of quantum mechanical systems by a classical statistical model with a restricted class of measurements. These models however have problems to account for correlations which occur in measurements on composed systems [ADR82].
1.6 Observables and States in Higher Dimensions
We saw that for quantum mechanical two-level systems (dim + ) the state space corre-sponds to the whole three dimensional unit ball centred at , . In terms of density operators this is the convex hull of all projectors. Quantum mechanical systems with levels posses a state space which is a proper subset of the unit ball in
dimensions. In order to understand this let us first investigate the relation between the set of all states and the set of all event vectors. This will lead us to the insight that there exist in fact different classes of candidates for pure states. Basically there are two choices for the set of states – one of them leads to Quantum Mechanics.
1.6 Observables and States in Higher Dimensions 19 Relation between and
On our way to reconstruct Quantum Mechanics we would like to show that
. (1.54)
This relation is fulfilled in Q.M., where the set of all states is given by "
$
and the event vectors correspond to positive operators on with . The following arguments do not constitute a proof of Eq. (1.54), which cannot be given here. Rather they should be seen as a motivation which might lead to the formulation of propositions and proofs.
Because pure states are the extremal elements of the state space they lie on the boundary of . Since we are free to choose the values of and in , cp. proposition 1.2.2, we assume that we can manage to make the boundary of to be a subset of the boundary of , thus (cp.
Eq. (1.29))
for all pure states ,. (1.55)
Furthermore, it is convenient to choose and , such that
)
for all pure states , (1.56)
For Eq. (1.54) to hold, there must correspond to each pure state an event vector. If this is the case it is possible to construct measurements with event vectors corresponding to the mixed states as well (see below).
Preparation of pure states
Which measurements do employ event vectors , where , is a pure state? As we shall see: those measurements which serve to prepare the pure states . Such measurements belong to the class of repeatable measurements, i.e. measurements which show the same result upon repetition, cp. [BLM91], p. 44.
Definition 3 A measurement is called repeatable , if
%8" $
whenever %8" $#? * , (1.57) where is the state of the m-system before the first measurement and is the state after the first measurement, provided its result was “ ”. is the set of all possible measurement results.
This means, if two repeatable measurements are carried out one after the other on the same single quantum system, the second measurement will show (with probability ) the same result as the first.
Definition 4 A repeatable measurement will be called a preparation measurement of the pure state , if the state of the m-system after an execution of the measurement with result ” ” reads
.
20 Generalised Measurements and the Axioms of Q.M.
A preparation measurement thus transforms the state into the pure state
. In particular the initial state is thus not changed by the transformation. State transformations are discussed in the subsection 1.10.
In order to be able to prepare pure states there must exist a selection procedure as offered by preparation measurements. When measurement result occurs, one knows that the state at hand is the pure state . In addition the value “ ” can be verified by executing the measure-ment again. This verification will not destroy the prepared state
. The repeatability of the measurement indicates, that the value corresponds to arealproperty of the m-system in state
since it represents a predictable result of a measurement.9
In fact in order to produce pure states (unambiguously affiliated to the measurement result) the requirement of repeatability can be relaxed. As we will see below (in subsection 1.13), in theory there are non-repeatable measurements10which transfer the state of the m-system into a pure state in one-to-one correspondence with the measurement result. It is not obvious to the author, however, whether these measurements can be experimentally realized.
Experimental realisations of preparation measurements on the other hand do exist. E.g. the Stern-Gerlach experiment prepares pure spin states of the silver atoms (ignoring other degrees of freedom than spin).In order to be able to prepare each pure state and to verify its preparation by measurements without changing the state again, we postulate the existence of a large class of preparation measurements.
Postulate: To each pure state , there exists a preparation measurement with a result “ ” which transfers an arbitrary initial state , into whenever the result ” ” occurs.
Without strict proof we assume that this means: To each pure state ,# there exists an event vector
, . This is plausible, because the repeatability of the measurement implies
%&" $
(1.59)
which is fulfilled11for , since according to Eq. (1.56).
With the event vectors
and
for preparing the pure states and also any convex combination of and is an event vector. This can be easily seen by observing
9Einstein, Podolsky and Rosen have coined the notion of an “element of reality” for observables (measurable properties) which show a predictable value when measured, cp. [EPR35]. The reality criterion of EPR is quoted in subsection 2.2.2. Busch, Lahti and Mittelstaedt consider the repeatability of the measurement as the basis of the value-objectification of the observable in the final state of the m-system, i.e. the observable possesses a definite value in the final state, cp.[BLM91], p.56. How one definite value out of several possible values (measurement result ) is actually reached constitutes the key problem of the quantum theory of measurement, referred to as the measurement problem.
10They project on an over-complete set of states as for example the coherent states of a harmonic oscillator.
11Other choices of with are more cumbersome because they would lead to a prohibition of certain pure states in the neighbourhood of .
1.6 Observables and States in Higher Dimensions 21 that given two preparation measurements , containing event vectors and , respec-tively, one can always design a measurement with event vector " $ . Such a measurement can for example be accomplished by carrying out measurement or with probabilities and by employing a random generator. The new event “ ” consists of either measurement showing result “ ” or leading to “ ”.
Therefore with all pure states also all convex combinations of pure states (i.e. all mixed states) correspond to event vectors. Hence Eq. (1.54) seems to hold. This has important conse-quences for the determination of the set of possible states and the set of event vectors. We are going to study these consequences in the mathematical language of Quantum Mechanics.
Translation into the language of Quantum Mechanics
Since we eventually want to compare our results to the rules of Quantum Mechanics we are going to express them in the language of operators on a Hilbert space . As in the case of qubits where
;=
dim is equal to+ , there is also for+
an isometric isomorphism mapping state- and event vectors which are elements of onto self-adjoint trace-class operators on . It is given by
Here "% $ is the orthonormal basis of with respect to which we represented the state and
event vectors, cp. Eq. (1.14). The are generators of the group " $ of special unitary transformations on (dim
Let us check whether there are generators which obey Eq. (1.61). The trace of the product of two operators tr forms an inner product on "
$
. SU( ) is generated by
operators
, which can always be ortho-normalised with respect to the trace-inner product and the norm it induces. Thus tr
1 1 , for /
. Furthermore, it is well known that generators of SU( ) are traceless, therefore tr
1
6*
.
In the definition (1.60) of the isomorphism we assumed that dim
. Starting with the general description and an arbitrary but finite
;=
dim , and thus the group SU( ) has to be chosen such that
. The number of generators employed in the isomorphism is then limited to instead of
as in definition (1.60). The domain of has to be limited accordingly to a subset of "
$
. is clearly an isometric isomorphism since it maps the ONB
"%
$ onto another ONB, namely " $ . Hence also preserves the values of the inner product:
Now lets look at several candidates for pure states. Pure states necessarily have to satisfy two conditions:
22 Generalised Measurements and the Axioms of Q.M.
1. Like all states each pure state has to give rise to probability one for the secure event
The last line results from
2. Pure states have maximal norm, thus
tr
(1.63)
In the following we use up our freedom in choosing the constants and . As above we require
and in addition ( . The conditions on pure states then read:
tr
tr / (1.64)
These conditions expressed by means of the spectral decomposition
Among the solutions to these equations we find all orthogonal projections (projectors) with rank one, which are characterised by
and tr
. They are the objects we are after, becauserank-one projectors represent the pure states in Quantum Mechanics.In fact restricting topositive operators, i.e. non-negative eigenvalues
1
(+*
, the solution to Eq. (1.65) allows only rank-one projectors as pure states. In general there is unfortunately a multitude of candidates for pure states. Apart from the class of projectors there is the class ofindefinite operatorswith positive as well as negative eigenvalues which obey Eq. (1.65). For example, for
all
operators with eigenvalues
" ( $ " ( $ +
(1.66) fulfil conditions (1.64). The set of eigenvalues solving Eq. (1.65) can be represented as vectors
,
pointing to the intersection of a hyperplane of and the sphere
.
Since to each pure state there is a preparation measurement with an “event operator”
there are mutually exclusive classes of candidates for pure states. For example in the case of the operators to a negative probability when the corresponding preparation measurement is carried out on a system in state
1.7 Density Operators and Effects 23 The set of self-adjoint operators satisfying conditions (1.64) is divided into several classes of candidates for pure states due to the existence of preparation measurements for each pure state.
Such a class is “closed” in the sense that any two of its elements
1
One of these classes of candidates for pure states is the set of projectors on with
and rank one, i.e. operators which orthogonally project onto a subspace of dimension . Each rank- projector can be represented by the unit vector (“ket”)
, on which it projects in the form
. It is easy to see that the inner product of two projectors satisfies Eq. (1.68), since
tr
1
1
(1.69) The class of pure states containing all projectors on is exclusive in the sense that it does not admit pure states which are not projectors, i.e. it consists only of the projectors. This follows from the fact that the eigenvalues
1
of any such state must be positive in order to satisfy tr
(+*
for all
), . Then the only solutions to Eq. (1.65) are projectors.
To investigate the other classes would go beyond the scope of this treatise. For the same reason we will not address the interesting questions of whether the projectors form the largest class of pure states and whether they induce the largest class of probability distributions.
1.7 Density Operators and Effects
In fact we have found the state space of Quantum Mechanics. Having determined the pure states as being represented by the projectors on the Hilbert space , the set of all states can be represented and identified with the set of all convex combinations of projectors:
"
$
(1.70)
which is the set of all positive operators on with unit trace. To see this, please note that each positive operator possesses a spectral decomposition
1 1 . Because of tr
1 1
, is a convex combination of projectors. On the other hand each convex combination of projectors is a positive operator (
(+*
) and possesses unit trace.
The state space" $ is in accordance with the content of the first axiom of Q.M. Since the general representation of the set of states as subset of is quite cumbersome for higher-level systems12, i.e. + , we will not try to translate our result back and write down the pre-image
of " $ under the isomorphism . From now on we will refer to" $ as the state space
and represent states in terms of density operators, i.e. elements of"
$
. Accordingly we will omit the “hat” on .
12An example, namely the generalisation of the Bloch sphere for three-level systems can be found in [AMM97].
24 Generalised Measurements and the Axioms of Q.M.
Once the state space is fixed, the set of all event operators
This condition just means that the eigenvalues
1
of event operators should satisfy (
1
. Event operators obeying Eq. (1.72) are calledEffects. The set of all event operators adjusted to thus reads:
, "
$<;
( (+*
(1.73)
The inverse problem,given the set of Effects , what is the set of all states (i.e. mathematical objects which give rise to “probability distributions” on ) 13, is the content of a generalised form of Gleason’s theorem , which was originally formulated for the lattice of all projectors on
instead of the set of Effects .
1.8 Generalised Measurements
The statistics of any quantum mechanical measurement can be evaluated if the set of Effects associated with the possible results of the measurement and the initial state of the m-system are known. The Effects , have to be associated with observable events , !#" $ in such a way that they induce probability measure for any given state. I.e. the rules of the association have to be in agreement with Kolmogorov’s axioms, cp. (1.1)-(1.3). They are the same as the rules for assigning event vectors to events and can be transcribed from equations (1.10)-(1.12)
;
1$ for countably many, pairwise disjoint
1
,.!#"
$
(1.76) Such a mapping is called a (normalised) positive operator-valued measure (POVM) since it forms a (normalised) -additive measure, the values of which are positive operators. The adjec-tive “normalised” is often omitted. Please note that Eq. (1.75) together with Eq. (1.76) imply
5 2 " $
(1.77) This (in general non-orthogonal) decomposition of unity by the Effects associated with the elementary events ,
gives rise to an alternative, simplified definition of a POVM, according
13A rigorous formulation of the generalised Gleason’s theorem from which one can read off what it is meant by
“probability distributions” on can be found in [BLM91].
1.9 Dynamics in Quantum Mechanics 25