Jan Samsonowicz

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.

Entangled symmetric states of N qubits with all positive partial transpositions

From both theoretical and experimental points of view symmetric states constitute an important class of multipartite states. Still, entanglement properties of these states, in particular those with positive partial transposition (PPT), lack a systematic study. Aiming at filling in this gap, we have recently affirmatively answered the open question of existence of four-qubit entangled symmetric states with PPT and thoroughly characterized entanglement properties of such states [J. Tura et al., Phys. Rev. A 85, 060302(R) (2012)]. With the present contribution we continue on characterizing PPT entangled symmetric states. On the one hand, we present all the results of our previous work in a detailed way. On the other hand, we generalize them to systems consisting of an arbitrary number of qubits. In particular, we provide criteria for separability of such states formulated in terms of their ranks. Interestingly, for most of the cases, the symmetric states are either separable or typically separable. Then, edge states in these systems are studied, showing in particular that to characterize generic PPT entangled states with four and five qubits, it is enough to study only those that assume few (respectively, two and three) specific configurations of ranks. Finally, we numerically search for extremal PPT entangled states in such systems consisting of up to 23 qubits. One can clearly notice regularity behind the ranks of such extremal states, and, in particular, for systems composed of odd numbers of qubits we find a single configuration of ranks for which there are extremal states.

Four-qubit entangled symmetric states with positive partial transpositions

We solve the open question of the existence of four-qubit entangled symmetric states with positive partial transpositions (PPT states). We reach this goal with two different approaches. First, we propose a half-analytical-half-numerical method that allows to construct multipartite PPT entangled symmetric states (PPTESS) from the qubit-qudit PPT entangled states. Second, we adapt the algorithm allowing to search for extremal elements in the convex set of bipartite PPT states [J. M. Leinaas, J. Myrheim, and E. Ovrum, Phys. Rev. A 76, 034304 (2007)] to the multipartite scenario. With its aid we search for extremal four-qubit PPTESS and show that generically they have ranks (5,7,8). Finally, we provide an exhaustive characterization of these states with respect to their separability properties.

Separability, entanglement, and full families of commuting normal matrices

We reduce the question of whether a given quantum mixed state is separable or entangled to the problem of existence of a certain full family of commuting normal matrices whose matrix elements are partially determined by components of the pure states constituting a decomposition of the considered mixture. The method reproduces many known entanglement and/or separability criteria, and provides yet another geometrical characterization of mixed separable states.

On the field nature of measuring process

Abstract Basing upon the philosophical assumption that it is not possible to develop a theory of measurement independent from the physical nature of the measured object, we formulate the topological theory of measurement. The model of measurement is a fibre bundle space with the base space D representing topological properties of measured quantity (measurement quantity base space) and fibres F corresponding to the algebraic properties of space values of measured quantity (properties of scale). We introduce a join operation ∗ of two base spaces defining base space of measurement of two quantities ( D=D 1 ∗D 2 ) . This topological model of measurement is consistent with the field theory.

K-THEORY FOR IM-BUNDLES

Introduction By an im-bundle we mean amy quasi-bundle (i.e. any singular vector bundle) which cam be represented as sin image of an endomorphism of a locally trivial bundle. Some properties of the category ImVB of im-bundles are presented in [4] and [5]. Any im-bundle £ over a space X determines a decomposition of this base space into the sets over which the dimensions of fibres are constant. By a filtered space denoted X we mean any space X together with a fixed decomposition of such kind. Observe that in a natural way we can define the semiring ImVB(X) (with respect to Whitney sum and tensor product) of isomorphism classes of im-bundles which give the same decomposition of X. Then the K-functor may be applied. Let us denote K(ImVB(X)) by K (X). The main i 111 theorem of this paper says that