On the geometric distance between quantum states with positive partial transposition and private states

We prove an analytic positive lower bound for the geometric distance between entangled positive partial transpose (PPT) states of a broad class and any private state that delivers one secure key bit. Our proof holds for any Hilbert space of finite dimension. Although our result is proven for a specific class of PPT states, we show that our bound nonetheless holds for all known entangled PPT states with non-zero distillable key rates whether or not they are in our special class.Comment: 16 page

Dynamics of quantum entanglement

A model of discrete dynamics of entanglement of bipartite quantum state is considered. It involves a global unitary dynamics of the system and periodic actions of local bistochastic or decaying channel. For initially pure states the decay of entanglement is accompanied with an increase of von Neumann entropy of the system. We observe and discuss revivals of entanglement due to unitary interaction of both subsystems. For some mixed states having different marginal entropies of both subsystems (one of them larger than the global entropy and the other one one smaller) we find an asymmetry in speed of entanglement decay. The entanglement of these states decreases faster, if the depolarizing channel acts on the "classical" subsystem, characterized by smaller marginal entropy.Comment: 10 pages, Revtex, 10 figures, refined versio

Unconditional privacy over channels which cannot convey quantum information

By sending systems in specially prepared quantum states, two parties can communicate without an eavesdropper being able to listen. The technique, called quantum cryptography, enables one to verify that the state of the quantum system has not been tampered with, and thus one can obtain privacy regardless of the power of the eavesdropper. All previous protocols relied on the ability to faithfully send quantum states. In fact, until recently, they could all be reduced to a single protocol where security is ensured though sharing maximally entangled states. Here we show this need not be the case -- one can obtain verifiable privacy even through some channels which cannot be used to reliably send quantum states.Comment: Related to quant-ph/0608195 and for a more general audienc

Measuring Multipartite Concurrence with a Single Factorizable Observable

We show that, for any composite system with an arbitrary number of finite-dimensional subsystems, it is possible to directly measure the multipartite concurrence of pure states by detecting only one single factorizable observable, provided that two copies of the composite state are available. This result can be immediately put into practice in trapped-ion and entangled-photon experiments.Comment: 4 pages; no figures; published versio

Positive maps, majorization, entropic inequalities, and detection of entanglement

In this paper, we discuss some general connections between the notions of positive map, weak majorization and entropic inequalities in the context of detection of entanglement among bipartite quantum systems. First, basing on the fact that any positive map $\Lambda:M_{d}(\mathbb{C})\to M_{d}(\mathbb{C})$ can be written as the difference between two completely positive maps $\Lambda=\Lambda_{1}-\Lambda_{2}$, we propose a possible way to generalize the Nielsen--Kempe majorization criterion. Then we present two methods of derivation of some general classes of entropic inequalities useful for the detection of entanglement. While the first one follows from the aforementioned generalized majorization relation and the concept of the Schur--concave decreasing functions, the second is based on some functional inequalities. What is important is that, contrary to the Nielsen--Kempe majorization criterion and entropic inequalities, our criteria allow for the detection of entangled states with positive partial transposition when using indecomposable positive maps. We also point out that if a state with at least one maximally mixed subsystem is detected by some necessary criterion based on the positive map $\Lambda$, then there exist entropic inequalities derived from $\Lambda$ (by both procedures) that also detect this state. In this sense, they are equivalent to the necessary criterion [I\ot\Lambda](\varrho_{AB})\geq 0. Moreover, our inequalities provide a way of constructing multi--copy entanglement witnesses and therefore are promising from the experimental point of view. Finally, we discuss some of the derived inequalities in the context of recently introduced protocol of state merging and possibility of approximating the mean value of a linear entanglement witness.Comment: the published version, 25 pages in NJP format, 6 figure

Limits for entanglement measures

We show that {\it any} entanglement measure $E$ suitable for the regime of high number of entangled pairs satisfies $E_D\leq E\leq E_F$ where $E_D$ and $E_F$ are entanglement of distillation and formation respectively. We also exhibit a general theorem on bounds for distillable entanglement. The results are obtained by use of a very transparent reasoning based on the fundamental principle of entanglement theory saying that entanglement cannot increase under local operations and classical communication.Comment: 4 pages, Revtex, typos correcte