Using our example we demonstrate data type construction. Assume that the following elementary data types lt-dt ,2 yoc-dt ,3 kilometre-dt , anddate-dt are given. The data types for a car and for an entry of the service booklet are constructed as follows:

car-dt := tuple (lt : lt-dt , yoc : yoc-dt )

entry-dt := tuple (kilometre : kilometre-dt , date : date-dt ) .

Data type construction is recursive. Starting points are elementary data types:

Ð An elementary data type is a data type.

Ð Let dt1 to dtN be data types and name1 to nameN labels. Then tuple (name1 : dt1 , . . . , nameN : dtN ) is a data type.

Ð If dt is a data type, then set ( dt ) is a data type.

In our example we deÞne the data type service-booklet-dt and the data type car-dealership-dt by applying the set data type constructor on top of the tuple data type constructor:

service-booklet-dt := set ( entry-dt )

car-dealership-dt := set ( car-dt ) .

Analogously we can deÞne all data types required for our example.

5 Quantum-Based Data Type Constructors

After introducing data type constructors we will map them onto concepts of quantum mechanics. The idea is to deÞne a bijection that relates values of constructed data types to vectors of an inner product space.

5.1 Tuple Data Type Constructor

The tuple data type constructor groups a Þxed number of components. Every component consists of a label and a data type. Let us be given the labels namej and

the values Vjj Dom( dtj ) of the components j as well as the corresponding

i

ket vectors |Vjji . Following the mathematics of quantum mechanics, several inner

2lt-dt stands for license tag .

3yoc-dt stands for year of construction .

124 I. Schmitt et al.

product spaces are combined to a larger one by applying the tensor product. For a tuple data type construction

T-dt := tuple name1 : dt1 , . . . , nameN : dtN ,

the ket vectors and their inner product spaces are combined by use of the tensor product:

QDom( T-dt ) = QDom( dt1 ) . . . QDom( dtN )

Dom( T-dt ) → QDom( T-dt )

V11i , . . . , VNNi → |V11i . . . |VNNi .

The ket vector |V11i . . . |VNNi can be shortly notated as |V11i . . . VNNi . If dl is the number of basis ket vectors of component l, that is, the number of dimensions, then

the tensor product needs d1 · . . . · dN many basis ket vectors. That is, with respect to the tensor product the number of dimensions is multiplied and not added as in the case of the Cartesian product.

Due to

V1V2V3|i1i2i3 = V1|i1 V2|i2 V3|i3

we obtain

V1V2V3|P 123|V1V2V3 = V1|P 1|V1 V2|P 2|V2 V3|P 3|V3 .

Fig. 4 Sets as one of 23 corners over the

{a, b, c}-cube (for illustration purpose, the distances between origin and the corners are not all one)

Thus, measuring a tuple value equals the product of the component measurements. Sometimes we want to measure only one component. In that case we use the identity 1 = |i ON B |i i| as projector for the no-care-components where ON B

stands for orthonormal basis:

V1V2V3| P 1 1 1 |V1V2V3 = V1|P 1|V1 .

5.2 Set Data Type Constructor

The set data type constructor is used on top of an orthonormal data type dt with QDom( dt ) = {|V1 , . . . , |Vk } being an orthonormal basis:

S-dt := set( dt ).

Every element of Dom( S-dt ) is a subset of Dom( dt ). The idea of our quantum modelling approach is the usage of the superposition principle, see Fig. 4. To construct the vector space QDom ( S-dt ), we have to collect all superpositions of ket vectors from QDom ( dt ). Mathematically, this leads to the vector space of all linear combinations of ket vectors from QDom ( dt ), which is called the span of QDom ( dt ). The ket vectors of the set elements are superimposed:

The mapping does not change the number of basis ket vectors, that is, the number of dimensions.4 Please note that the set data type constructor yields a non-orthogonal data type.

Let us be given a superposition ket vector |S . We want to test by quantum measurement if the value Vj mapped to |Vj QDom( dt ) is a member of the set S. For measurement we use the projector P = |Vj Vj |:

4The special case of an empty set is discussed later on.

Table 5 illustrates an example for the set measurement. As a result we obtain 2/3 = 1/3 + 1/3.

The empty set needs special consideration. The problem is that there is a superposition of nothing, that is, the empty set cannot be represented by any vector of the given data type. Therefore, we insert an additional basis vector |N U LL into QDom ( dt ). The ket vector |N U LL is only used if an empty set needs to be encoded.

Assume a non-orthogonal data type no-dt with

QDom( no-dt ) = {|V1 , . . . , |Vk }

is given. However, the set data type constructor is not deÞned for non-orthogonal data types. One solution is to orthogonalize the non-orthogonal data type. This can be easily realized by applying a tuple construction together with an auxiliary orthogonal data type aux-dt of dimension k. So, you obtain the orthogonalized data type o-dt by constructing:

o-dt := tuple (id : aux-dt , value : no-dt )

and assigning bijectively a unique number from dom( aux-dt ) to every value from dom( no-dt ):

QDom ( o-dt ) {|1 , . . . , |k} QDom ( no-dt )

QDom( no-dt ) → QDom( o-dt )

|Vi → |i |Vi .