Let us stipulate some names for the objects we are concerned with. The time immediately before the measurement is called . The states of the system to be measured (m-system) are represented as ,
or . In this context the ’s are just symbols for the states. At this point let us pretend not to know anything about the mathematical representation of quantum mechanical states in calculations. The set of possible measurement results
Because the outcomes of a quantum mechanical measurement seem in general to resist an exact prediction, we have to assign probabilities to the possible outcomes. It will also be necessary to assign probabilities to composed events such as “result or result occurs”.
All composed events can be represented by subsets of . For example the event “ or ” is represented by the subset . Elementary events as “ ” are represented by subsets with one element like
. Events can also be composed by “and”, “or” and “not”. If two events are represented by the subsets then “first or second event” translates into “ ”, “first and second event” reads “ ” and “not the first event (all events but the first)” is represented by the complement of : . From now on we identify the events and the subsets of
which represent the respective events. The events form an algebra which is closed under countably many unifications ( ), countably many intersections ( ) and the formation of the complement.
It also contains
itself, which stands for the composed event: “one of the outcomes of the measurement will come out”. This is a tautology and it will always occur, therefore is called the certain event. The technical name for the algebra of all events (composed and elementary) is “sigma algebra”!#"
$
.
We want to assign to each event a probability. Regardless of the interpretation of probability there are basic properties of such probability assignments (probability distributions).
Definition 1 A function% which assigns to each element of a sigma algebra !#" $ a real num-ber, such that
1$ for countably many, pairwise disjoint
1
,.!#"
$
(1.3) is called a probability distribution or probability measure.
Properties (1.1-1.3) are referred to as Kolmogorov’s axioms.
Definition 2 The function%8"9 $ defined by
%&":
1.2 General Representation of Observables and States 5
%&": $
is the probability that event occurs, provided that also event occurs. We note, that in the Baysian view of probabilities, the defining properties of probability distributions (1.1-1.3) and of conditional probabilities (1.4) are not mere axioms but they can be derived from consistency arguments. Bayesianism interprets probabilities as subjective degrees of conviction (degrees of confidence, degrees of credence). Although the degrees of conviction are subjective they are assumed to be reasonable, i.e. all probability assignments have to be coherent. For example, if someone assigns to the event the probability , i.e. he is sure that will occur, he cannot at the same time be sure that the opposite of will happen. When saying%&"'
$/
he is forced to assign the probability
*
to the complement of . Kolmogorov’s axioms are equivalent to a coherent assignment of degrees of conviction, cp. chapter 3 of [HP89]. Bayesianism has been employed lately for a subjective interpretation of Q.M. [CFS01].
Now we have the means to discuss the statistics of an experiment. Surely the probability to obtain a certain measurement outcome (say ) depends on the initial state (denoted by ) of the system to be measured:
%8":
$/
%&": $
Let us consider the case where the system is at the beginning of the measurement either in state
or in state with probabilities
and
, respectively. This is called a mixed state, let us denote it by . The probability to find as measurement outcome is then given by The right hand side of (1.5) follows from considering the conditional probabilities that hap-pens under the condition that initial state is
or the initial state is . If we are allowed to represent the mixed state by
for shows that the probability to measure has to be a convex-linear function of the initial state. In other words: The conditional probabilities that enter the total probability to obtain a measure-ment result if the initial state is not precisely known, are correctly represented by writing the initial state as a sum of the possible initial states weighted by their probabilities and postulating that the probability of is a convex linear function of the initial state.
How does this representation effect our notion of a state? Apparently there are states which can be expressed as convex combinations of other states. Whether conversely all convex combi-nations of states again represent states of physical systems is not obvious. Because this cannot be excluded a priori, it is reasonable to suppose that the set of all states, a physical system can assume, is convex, i.e. with two states
and , it also contains the state
"
$
for , * . States which can be written as non-trivial convex combination ( ? * ) are called mixed states. States which cannot be expressed as non-trivial convex combination are called pure states. The decomposition of a mixed state into a convex combination of pure states is in general not unique if more than two pure states exist. Two convex combinations which correspond to different preparation procedures of the state before the measurement but are rep-resented by the same mixed state cannot be distinguished by any measurement on the m-system alone. This follows from the convex linearity of the probability distribution of the outcomes of
6 Generalised Measurements and the Axioms of Q.M.
any measurement as function of the initial state of the m-system:
5 1 % 1 1 5 Mixed states as can be understood as equivalence classes of all preparation procedures of the initial state which cannot be distinguished by any measurement of the m-system, cp. [Kra83], p.6. This theoretically derived structure is confirmed by experiments. For example, a photon can be prepared in a mixed state by sending it through a polariser either in x-direction or in y-direction with probability% -* , respectively. Here x and y are spatial directions which are perpendicular. Such a photon leads to the same probability distribution of results of any measurement as one that has been sent either through a -polariser or a -polariser with prob-ability% -* (state ), where and are any two perpendicular directions different from x and y. Even though these preparation procedures can be clearly distinguished by looking at the experimental set up, the resulting states of the photon are indistinguishable by means of measurements carried out on the photon alone. The state of the photon as a system which is separated from the polariser, does only describe the properties of the photon and not of the po-lariser. Therefore and which show the same properties with respect to all measurements on the photon but are connected to different settings of the polariser are identified. If a certain decomposition of a mixed state is distinguished because the preparation procedure is known, the mixed state is called a Gemenge5.
As all convex sets, the set of states of a system can be embedded into a vector space.
This means
1 1 1
can be interpreted as linear combination of vectors
1
with real coefficients
1
. We will thus in the following represent the states of systems as elements of a vector space over the field of real numbers . For this purpose we take the smallest vector space which contains the set of states. This vector space is given by the linear span of the states which consists of all finite linear combinations of states:
5 1 1 1 1
, 1
, (1.8)
We note that the set of states always contains a basis of the linear space in which we embed it. Therefore the convex linear mapping%&":
$
from into the real numbers can be uniquely expanded to a linear mapping from
into the reals (a linear form on
). For brevity let us here only investigate the case of finitely many linearly independent states, thus the dimension of is finite. All linear forms on are then of the form " linear forms on (dual of ) . Hence, given a measurement and a corresponding probability distribution%8":B
$
there exists for each event an event vector"'
$ ,
5“Gemenge” is the german word for an inhomogeneous mixture.
1.2 General Representation of Observables and States 7 The rules for assigning such vectors to events are determined by Kolmogorov’s axioms (1.1)-(1.3). For each measurement with event algebra !#" $ there must exist a map ; !#" $ ,
! for countably many, pairwise disjoint
1
,.!#"
$
(1.12) In a first step we employ these rules to further determine the set of states and the assigning of vectors to events, i.e. the mapping
;
!#"
$
. This will lead to a quite general struc-ture. In order to further specify and the mapping we will study in a second step the change of states due to measurements and postulate the existence of so-called “preparation measure-ments”. Preparation measurements represent a means to produce pure states . The structures we will find among others are that of density operators and positive operator valued measures (POVM). The first represent states in Quantum Mechanics, the latter generalised observables (i.e. measurable quantities). If we only invoke the subclass of generalised observables repre-sented by commuting positive operators we obtain structures reminiscent of classical physics.
It is convenient to look at the implications of rule (1.11) first. Without restriction of gen-erality the vector associated with the certain event
can be said to point in direction of some normalised vector ,
"
$/
(1.13)
where is a positive, real number. By (1.11) the first component of the state vector ., is now determined:
form an orthonormal basis of . From Eq. (1.10) one obtains the requirement
where represents the set of all event vectors (for all measurements which can be carried out on a particular m-system). In order to obtain the previous equation, both – the event vector and the state – have been expressed w.r.t. the basis "
$ of . The components of the event
is the inner product in the dimensional subspace spanned by
3 3
. Now in order to fulfil Eq. (1.15), constraints on the angle defined via the inner product
could be postulated. For the sake of simplicity we start by allowing all angles ,
*
,+.-$
. This way we shall arrive at a correct description of the simplest quantum mechanical systems. In order to model more complex systems one has to restrict the angle .
8 Generalised Measurements and the Axioms of Q.M.
Proposition 1.2.1 The set of states of the observed system is bounded, i.e. there exists a real number * , s.t. for all , .
Proof: could only be unbounded in a direction perpendicular to all event vectors (w.r.t.
the inner product on ), otherwise Eq. (1.15) would not hold. Since all probability distribu-tions would then be independent from this direction, it would correspond to a non-measurable property of the m-system and would not enter the state (which represents only measurable
prop-erties).
The component of w.r.t. the basis vector is already determined by condition (1.11):
. Let the remaining components be bounded by
where is a positive, real number. The set of states then reads
This particular set of states is called instead of in order to remind us, that we allow to point in any direction. As a consequence the angle in (1.16) is not restricted. In general the set of states can be any convex subset of with appropriate constants and .
Eq. (1.18) leads to a reformulation of requirement (1.15). Now, the inner product assumes its maximum if both vectors point in the same direction and its minimum, if they point in opposite directions: Because we allowed to point in any direction these values can be realized by preparing the corresponding physical states. This has the following consequences for the event vectors.
( These two inequalities are equivalent to the statement:
( (
(1.22) In particular this implies ( (+* . The space of event vectors thus reads
1.2 General Representation of Observables and States 9 We note again that was obtained under the assumption that the set of states is given by , i.e.
can point in any direction. In general the set of states will be a subset of , as a consequence there can be more event vectors than contained in , such that (1.15) still holds, thus
. (1.24)
Proposition 1.2.2 The set of probability distributions for measurements on the m-system al-lowed by and is not restricted by any special choice of the positive constants and in and .
In other words we are free to choose * as we like.
Proof: Given a probability distribution with%&":
$B
& ! , where
% , is
the vector assigned to the event ,!#" $ and
''% , represents the state of the m-system. Then this probability distribution can also be expressed by means of
%&"'B $ Both vectors are represented w.r.t. the basis "
$ of .
The coordinates of the new state vector and the event vector
Hence a probability distribution%&": $
, which is expressed by states , and event vectors
B, , can also be represented by states , and B, , where and are obtained from in (1.18) and in (1.23), respectively, by replacing the positive constants and by positive constants and
.
Let us sum up the investigation so far. Due to the rule for conditional probabilities, the probability distribution of a measurement of the m-system can be written as a linear function of the initial state of the m-system, if the space of states is represented by a convex subset of a linear space. Kolmogorov’s axioms lead to a state space which is contained in
and the space of event vectors which contains the set
10 Generalised Measurements and the Axioms of Q.M.
Here and are arbitrary positive, real numbers. The mapping which assigns vectors to the events must have the following properties:
;
1 $ for countably many, pairwise disjoint
1 , !#"
$
(1.33)