Quantum Ideas for Classical Systems

John R. Klauder

Noson S. Yanofsky, Mirco A. Mannucci

Everyone has lost their marbles! Anonymous Before we formally present quantum mechanics in all its wonders, we shall spend time providing some basic intuitions behind its core methods and ideas. Realizing that computer scientists feel comfortable with graphs and matrices, we shall cast quantum mechanical ideas in graph-theoretic and matrix-theoretic terms. Everyone who has taken a class in discrete structures knows how to represent a (weighted) graph as an adjacency matrix. We shall take this basic idea and generalize it in several straightforward ways. While doing this, we shall present a few concepts that are at the very core of quantum mechanics. In Section 3.1, the graphs are without weights. This will model classical deterministic systems. In Section 3.2, the graphs are weighted with real numbers. This will model classical probabilistic systems. In Section 3.3, the graphs are weighted with complex numbers and will model quantum systems. We conclude Section 3.3 with a computer science/graph-theoretic version of the double-slit experiment. This is perhaps the most important experiment in quantum mechanics. Section 3.4 discusses ways of combining systems to yield larger systems. Throughout this chapter, we first present an idea in terms of a toy model, then generalize it to an abstract point, and finally discuss its connection with quantum mechanics, before moving on to the next idea.

Akira Saitoh, Robabeh Rahimi, Mikio Nakahara

It is a popular idea that a multi-partite quantum system represented by a density matrix having no product eigenbasis possesses non-classical correlation. Supporting this idea, we define an entropic measure of non-classical correlation on the basis of the minimum uncertainty about a joint-system after we collect reports of particular local measurements. Numerical computation of this measure for several examples is performed.

Pavel Bóna

Vieri Benci

Sebastián Fortín, Manuel Gadella, Federico Holik +1 more

The standard descriptions of the quantum-to-classical transition of quantum systems focuses on the evolution of quantum states. The main idea is to appeal to the decoherence process in order to explain the disappearance of the interference terms of the quantum states. These approaches give an adequate description of several aspects of the classical limit process. However, they do not explain how the logical structure of quantum properties becomes classical. In this chapter, we present an alternative approach to the quantum-to-classical transition, based on the evolution of the observables instead of the states. We show that if the observables of the system evolve over time in such a way that they become commutative, then the logical structure of quantum properties becomes a classical logic. Moreover, we illustrate the general idea of this approach considering four different models.