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Cognitive technologies

Kronecker Products

Christian Bauckhage, Rafet Sifa

Published Nov 17, 2025• 0 citations
Publisher

Abstract

Abstract Tensor products or, more specifically, Kronecker products play a fundamental role in quantum computing and can also help with modeling certain combinatorial problems. In this chapter, we therefore discuss Kronecker products and their properties. To segue into this discussion, we first introduce the important notion of pseudo Boolean functions and look at their relations to multilinear algebra. Given this use case, we can easily understand the role of different kinds of tensor products and investigate algebraic properties especially of Kronecker products.

DOI
10.1007/978-3-031-99402-9_3

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Abstract Tensor products or, more specifically, Kronecker products play a fundamental role in quantum computing and can also help with modeling certain combinatorial problems. In this chapter, we therefore discuss Kronecker products and their properties. To segue into this discussion, we first introduce the important notion of pseudo Boolean functions and look at their relations to multilinear algebra. Given this use case, we can easily understand the role of different kinds of tensor products and investigate algebraic properties especially of Kronecker products.
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Abstract This paper not only explores in detail the basic properties of the Kronecker product, such as the application of the union law, the distributive law, the transpose and the conjugate transpose, as well as the trace and the rank of the Kronecker product, but also rigorously proves these properties by means of mathematical induction or direct proofs. In addition, this paper extends these properties to Kronecker n-matrices and explores the applicability of inverse matrices and determinants to Kronecker products. More importantly, we introduce a conjecture related to the long-standing problem of distillability, which is partially solved when the number of nonzero elements is restricted. By showing the framework of the proof of the conjecture and a special case, we provide new perspectives for further research in this area.
Abstract Previously, we viewed Hopfield nets from the point of view of statistical mechanics. In this chapter, we widen our perspective one again and view them from the point of view of quantum mechanics. This requires us to introduce completely different representations of the states and energy functions of Hopfield nets and relate them to the kind of tensor product spaces we studied earlier. At this point, this different perspective on Hopfield nets is of purely theoretical interest because it generally defies practical implementations on digital computers. However, it establishes a connection between Hopfield nets and important quantum mechanical concepts. In particular, we will get to know the notion of the Hamiltonian of a physical system, its eigenstates, and eigenvalues. Since the material in this chapter is more abstract and less intuitive than in previous chapters, we develop our ideas by looking at an exemplary simple Hopfield net and support our technical claims through practical computations and code examples.
In Section 2.4 we learned that a composite quantum system, composed of two subsystems A and B , is associated with a tensor product space HA ? HB . Composite quantum systems have already played an important role in many instances in this text, especially in open systems and measurement models. In this chapter we dive deeper into tensor product spaces, discovering some strange implications of the quantum theory of composite quantum systems. As a result of its tensor product structure, quantum theory has embedded into it the phenomenon of entanglement , which is often seen as the foremost quantum feature. As early as in 1935 it puzzled Erwin Schrödinger, who introduced the German term Verschränkung [127] and Albert Einstein, who, together with Boris Podolski and Nathan Rosen, used the specific example of an entangled state to argue that quantum theory is incomplete [58]. In the 1960s John Bell demonstrated that the consequences of entanglement are incompatible with the Einstein– Podolsky–Rosen model of local realism [11], [12]. In recent years entanglement has been recognized as the resource for quantum information processing. Nowadays entanglement theory is a highly developed subject, and we can present only the mathematical basics. For the reader with a deeper interest in entanglement we recommend the recent reviews by Horodecki et al . [80] and Plenio et al . [118], which also cover the quantum information aspects of entanglement. To simplify our discussion we will assume that the Hilbert spaces are finite dimensional.