Quantum-Based Modelling of Database

States

Ingo Schmitt, Günther Wirsching, and Matthias Wolff

Abstract Database design of real-world scenarios requires complex data structures in order to adequately model complex real-world objects. Complex data structures can be constructed by a recursive use of elementary data types and data type constructors. The mathematics behind quantum mechanics provides us an interesting theory combining concepts from linear algebra, probability calculus, and logic. In order to make the mathematics of quantum mechanics available for database structures and states we develop a mapping of concepts from type theory of databases to the mathematics of quantum mechanics.

1 Introduction

The mathematics behind quantum mechanics [1] provides us a formalism that combines very elegantly concepts from probability calculus, linear algebra, and logic. The semantics of a quantum system is expressed by a normalized ket vector in an inner product space. Here we show how to model complex data structures of a database state as a normalized ket vector of an inner product space, see also [2]. Furthermore, we show how to read a database vector by use of the statistics of quantum measurement. Our database mapping to the mathematics of quantum mechanics proposed in the following is restricted to Þnite dimensional and real inner product spaces. For a query language based on our mapping and quantum logic as well as quantum measurement, we refer to [3].

Please note that we do not propose to perform a mapping onto a physical quantum computer. Instead, the proposed mapping is on a conceptual level rather

I. Schmitt ( ) á M. Wolff

Brandenburg University of Technology Cottbus-Senftenberg, Cottbus, Germany e-mail: schmitt@b-tu.de

G. Wirsching

Catholic University of EichstŠtt-Ingolstadt, EichstŠtt, Germany e-mail: gunther.wirsching@ku.de

D. Aerts et al. (eds.), Quantum-Like Models for Information Retrieval and Decision-Making, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-25913-6_6

than on an implementation level. The beneÞt from doing so is to bridge wellknown database modelling concepts into another formalism. This is very promising because the rich theory of linear algebra and quantum logic [4] provides us powerful concepts and gives us a deep understanding of certain database problems. For example, the relation between entanglement and functional dependencies between database values, and reasoning from databases based on quantum logic are not well understood so far. On a conceptual level we are able to develop and to prove interesting new theorems.

2 Motivating Example: Car Dealership

In the following we develop an example for demonstrating a quantum-based data modelling of a database structure and the measurement of its state. As example we use the managed data objects of a car dealership. From the view of a database designer, cars are complex-structured objects. Every car is composed of different technical components as shown in Fig. 1. Furthermore, a service booklet containing a record of car inspections exists for every car.

Some properties of components of the car dealership are listed in Table 1 deÞning the state of a car. Furthermore, some atomic conditions for measurements based on these properties are given in Table 2.