Separability and distillability in composite quantum systems-a primer

Abstract

Quantum mechanics is already 100 years old, but remains alive and full of challenging open problems. On one hand, the problems encountered at the frontiers of modern theoretical physics like Quantum Gravity, String Theories, etc. concern Quantum Theory, and are at the same time related to open problems of modern mathematics. But even within non-relativistic quantum mechanics itself there are fundamental unresolved problems that can be formulated in elementary terms. These problems are also related to challenging open questions of modern mathematics; linear algebra and functional analysis in particular. Two of these problems will be discussed in this article: a) the separability problem, i.e. the question when the state of a composite quantum system does not contain any quantum correlations or entanglement and b) the distillability problem, i.e. the question when the state of a composite quantum system can be transformed to an entangled pure state using local operations (local refers here to component subsystems of a given system).Although many results concerning the above mentioned problems have been obtained (in particular in the last few years in the framework of Quantum Information Theory), both problems remain until now essentially open. We will present a primer on the current state of knowledge concerning these problems, and discuss the relation of these problems to one of the most challenging questions of linear algebra: the classification and characterization of positive operator maps.

Spin Squeezing Inequalities and Entanglement ofNQubit States

“…Spin squeezing of, say a few ions to 10 7 atoms is nowadays routinely achieved in such systems. The standard tool to detect the generated forms of multipartite entanglement [2,3] provides the so-called spin squeezing parameter 2 introduced in Ref. [4].…”

Hilbert’s 17th Problem and the Quantumness of States

“…For polynomial functions, these criteria are related to Hilbert's 17th problem, which in its simplest form states that not every positive semidefinite polynomial must be a sum-of-squares (SOS) of other polynomials [6]. Our criteria have physical meaning of generalized squeezing conditions, and may be interpreted as nonclassicality witnesses (in analogy to entanglement witnesses, [7]). We show that every generic nonclassical state can be detected by a SOS polynomial.…”

“…Therefore, geometrically the problem of distinguishing between classical and nonclassical states amounts to the operational description of the simplex of measures in the space of all P distributions. We note that one encounters closely related problems in the study of quantum entanglement in multipartite systems (see [7] and references therein), with the difference being that the convex subset of classically correlated states is not a simplex.…”

Scaling of Entanglement in Finite Arrays of Exchange-Coupled Quantum Dots

“…9 Central to all these remarkable achievements is the concept of the quantum entanglement. [10][11][12][13][14] Just like energy, the entanglement has now been regarded as a controllable and a fungible physical resource. 15 Experimentally, entanglements have already been produced for up to four photons 16,17 and even between two macroscopic states such as two superconducting qubits, each of which contains as many as 10 9 electrons.…”