Introduction

Abstract

The development of quantum mechanics has been one of the greatest scientific achievements of the early twentieth century. In spite of its remarkable success in explaining and predicting an amazing number of properties of our physical world, its interpretation has raised strong controversies among a wide community of scientists and philosophers. One of the hottest points of discussion is the meaning of the so-called quantum entanglement that, for systems of two or many particles, allows in particular the possibility for each particle of the system to be simultaneously located at different spatial positions. Entangled states display a special kind of correlations. Generally speaking, differently from the statistical correlations that are usually found in classical probability theory, quantum entanglement cannot be understood in terms of statistically distributed hidden variables and must involve the possibility for quantum systems of particles to be simultaneously in different single particle pure quantum states. Entangled states therefore present facets of the quantum worlds which are even more complicated than the famous example of a superposition of states in the so-called Schrödinger’s cat which is simultaneously classically dead and alive. The peculiar phenomenology of quantum mechanics goes far beyond this paradoxical case: in contrast to the usual chain rules of classical conditional probability, the probability for a physical event to occur in a quantum framework is computed by the interference of the complex-valued amplitudes corresponding to the different classical states. In dynamical processes, these classical positional states are described by paths that the system can follow during its evolution. This description of the physical world is commonly known as Feynman integral and implicitly requires that the system be simultaneously in different classical states at all intermediate times [1]. The mathematical counterpart of this picture is that quantum states of a composite system are described by a tensor product structure where each product entry represents a component of the system. In this picture, entanglement is encoded in quantum superpositions, that is linear combinations of completely decomposed tensors. In this sense, if the tensor product involves different states of a given component which are localized in far and causally separated spatial regions, a single component of the system may be simultaneously located in different places.