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

A Schmidt number for density matrices

We introduce the notion of a Schmidt number of a bipartite density matrix, characterizing the minimum Schmidt rank of the pure states that are needed to construct the density matrix. We prove that Schmidt number is nonincreasing under local quantum operations and classical communication. We show that $k$-positive maps witness Schmidt number, in the same way that positive maps witness entanglement. We show that the family of states which is made from mixing the completely mixed state and a maximally entangled state have increasing Schmidt number depending on the amount of maximally entangled state that is mixed in. We show that Schmidt number {\it does not necessarily increase} when taking tensor copies of a density matrix $\rho$; we give an example of a density matrix for which the Schmidt numbers of $\rho$ and $\rho \otimes \rho$ are both 2.Comment: 5 pages RevTex, 1 typo in Proof Lemma 1 correcte

Non-additivity of quantum capacity for multiparty communication channels

We investigate multiparty communication scenarios where information is sent from several sender to several receivers. We establish a relation between the quantum capacity of multiparty communication channels and their distillability properties which enables us to show that the quantum capacity of such channels is not additive.Comment: 4 pages, 1 figur

Classical information deficit and monotonicity on local operations

We investigate classical information deficit: a candidate for measure of classical correlations emerging from thermodynamical approach initiated in [Phys. Rev. Lett 89, 180402]. It is defined as a difference between amount of information that can be concentrated by use of LOCC and the information contained in subsystems. We show nonintuitive fact, that one way version of this quantity can increase under local operation, hence it does not possess property required for a good measure of classical correlations. Recently it was shown by Igor Devetak, that regularised version of this quantity is monotonic under LO. In this context, our result implies that regularization plays a role of "monotoniser".Comment: 6 pages, revte

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