On a matrix inequality related to the distillability problem

We investigate the distillability problem in quantum information in \bbC^d\ox\bbC^d. A special case of the problem has been reduced to proving a matrix inequality when $d=4$. We investigate the inequality for two families of non-normal matrices. We prove the inequality for the first family with $d=4$ and two special cases of the second family with $d\ge4$. We also prove the inequality for all normal matrices with $d>4$.Comment: 19 pages, comments are welcom

Separability and distillability in composite quantum systems -a primer-

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.Comment: 11 pages latex, 1 eps figure. Final version, to appear in J. Mod. Optics, minor typos corrected, references adde

Violation of Bell's inequalities implies distillability for N qubits

We consider quantum systems composed of $N$ qubits, and the family of all Bell's correlation inequalities for two two-valued measurements per site. We show that if a $N$-qubit state $\rho$ violates any of these inequalities, then it is at least bipartite distillable. Indeed there exists a link between the amount of Bell's inequality violation and the degree of distillability. Thus, we strengthen the interpretation of Bell's inequalities as detectors of useful entanglement.Comment: 6 pages, 3 figures, REVTEX. List of authors extended. Partially rewritten, a rather qualitative explanation of the results. Conclusions unchange

Bell inequalities and distillability in N-quantum-bit systems

The relation between Bell inequalities with two two-outcome measurements per site and distillability is analyzed in systems of an arbitrary number of quantum bits. We observe that the violation of any of these inequalities by a quantum state implies that pure-state entanglement can be distilled from it. The corresponding distillation protocol may require that some of the parties join into several groups. We show that there exists a link between the amount of the Bell inequality violation and the size of the groups they have to form for distillation. Thus, a strong violation is always sufficient for full N-partite distillability. This result also allows for a security proof of multi-partite quantum key distribution (QKD) protocols.Comment: REVTEX, 12 pages, two figure

Attainable entanglement of unitary transformed thermal states in liquid-state nuclear magnetic resonance with the chemical shift

Recently, Yu, Brown, and Chuang [Phys. Rev. A {\bf 71}, 032341 (2005)] investigated the entanglement attainable from unitary transformed thermal states in liquid-state nuclear magnetic resonance (NMR). Their research gave an insight into the role of the entanglement in a liquid-state NMR quantum computer. Moreover, they attempted to reveal the role of mixed-state entanglement in quantum computing. However, they assumed that the Zeeman energy of each nuclear spin which corresponds to a qubit takes a common value for all; there is no chemical shift. In this paper, we research a model with the chemical shifts and analytically derive the physical parameter region where unitary transformed thermal states are entangled, by the positive partial transposition (PPT) criterion with respect to any bipartition. We examine the effect of the chemical shifts on the boundary between the separability and the nonseparability, and find it is negligible.Comment: 9 pages, 1 figures. There were mistakes in the previous version. The main results don't change, but our motivation has to be reconsidere

Majorization criterion for distillability of a bipartite quantum state

Bipartite quantum states are classified into three categories: separable states, bound entangled states, and free entangled states. It is of great importance to characterize these families of states for the development of quantum information science. In this paper, I show that the separable states and the bound entangled states have a common spectral property. More precisely, I prove that for undistillable -- separable and bound entangled -- states, the eigenvalue vector of the global system is majorized by that of the local system. This result constitutes a new sufficient condition for distillability of bipartite quantum states. This is achieved by proving that if a bipartite quantum state satisfies the reduction criterion for distillability, then it satisfies the majorization criterion for separability.Comment: 4 pages, no figures, REVTEX. A new lemma (Lemma 2) added. To appear in Physical Review Letter