We introduce a classification of mixed three-qubit states, in which we define the classes of separable, biseparable, W-and GHZ-states. These classes are successively embedded into each other. We show that contrary to pure W-type states, the mixed W -class is not of measure zero. We construct witness operators that detect the class of a mixed state. We discuss the conjecture that all entangled states with positive partial transpose (PPTES) belong to the W -class. Finally, we present a new family of PPTES "edge" states with maximal ranks. 03.65.Bz,03.65.Ca, 03.67.Hk The rapidly increasing interest in quantum information processing has motivated the detailed study of entanglement. Whereas entanglement of pure bipartite systems is well understood, the classification of mixed states according to the degree and character of their entanglement is still a matter of intensive research (see [1]). It was soon realised, that the entanglement of pure tripartite quantum states is not a trivial extension of the entanglement of bipartite systems [2,3]. Recently, the first results concerning the entanglement of pure tripartite systems have been achieved [4][5][6]. There, the main goal has been to generalize the concept of the Schmidt decomposition to three-party systems [4,5], and to distinguish classes of locally inequivalent states [6]. The knowledge of mixed tripartite entanglement is much less advanced (see, however, [7][8][9]).In this Letter we introduce a classification of the whole space of mixed three-qubit states into different entanglement classes. We provide a method to determine to which class a given state belongs (tripartite witnesses). We also discuss the characterization of entangled states that are positive under partial transposition (PPTES). Finally, we introduce a new family of PPTES for mixed tripartite qubits.Our proposal to classify mixed tripartite-qubit states is done by specifying compact convex subsets of the space of all states, which are embedded into each other. This idea vaguely resembles the classification of bipartite systems by their Schmidt number [9][10][11]. However, as shown later our classification does not follow the Schmidt number [9]. Also in this respect, entanglement of tripartite systems differs genuinely from the one of bipartite quantum systems.Before presenting our results concerning mixed states, we briefly review some of the recent results on pure threequbit states. Any three-qubit vector (pure state) can be written aswhere

We discuss quantum correlations in systems of indistinguishable particles in relation to entanglement in composite quantum systems consisting of well separated subsystems. Our studies are motivated by recent experiments and theoretical investigations on quantum dots and neutral atoms in microtraps as tools for quantum information processing. We present analogies between distinguishable particles, bosons, and fermions in low-dimensional Hilbert spaces. We introduce the notion of Slater rank for pure states of pairs of fermions and bosons in analogy to the Schmidt rank for pairs of distinguishable particles. This concept is generalized to mixed states and provides a correlation measure for indistinguishable particles. Then we generalize these notions to pure fermionic and bosonic states in higher-dimensional Hilbert spaces and also to the multi-particle case. We review the results on quantum correlations in mixed fermionic states and discuss the concept of fermionic Slater witnesses. Then the theory of quantum correlations in mixed bosonic states and of bosonic Slater witnesses is formulated. In both cases we provide methods of constructing optimal Slater witnesses that detect the degree of quantum correlations in mixed fermionic and bosonic states. C

A generalization of the quantum cryptographic protocol by Bennett and Brassard is discussed, using three conjugate bases, i.e. six states. By calculating the optimal mutual information between sender and eavesdropper it is shown that this scheme is safer against eavesdropping on single qubits than the one based on two conjugate bases. We also address the question for a connection between the maximal classical correlation in a generalized Bell inequality and the intersection of mutual informations between sender/receiver and sender/eavesdropper. 03.65.Bz, 03.67.-a, 03.67.Dd

We present the experimental detection of genuine multipartite entanglement using entanglement witness operators. To this aim we introduce a canonical way of constructing and decomposing witness operators so that they can be directly implemented with present technology. We apply this method to three-and four-qubit entangled states of polarized photons, giving experimental evidence that the considered states contain true multipartite entanglement. [3,4] as it gives a simple sufficient and necessary condition for entanglement. Yet, the situation is much more complicated for higher dimensional and multipartite systems, where simple necessary and sufficient conditions are not known [5].In the analysis of multipartite systems, it is important to distinguish between genuine multipartite entanglement and biseparable (triseparable, etc.) entanglement. Genuine multipartite entangled pure states cannot be created without participation of all parties. Conversely, for pure biseparable states of n parties a group of m < n parties can be found which are entangled among each other, but not with any member of the other group of n − m parties [6]. Distinction and detection of genuine multipartite entanglement poses an important challenge in quantum information science. Bell inequalities are not suited to this aim in general. Multiseparable and biseparable states violate known Bell inequalities less than npartite Greenberger-Horne-Zeilinger (GHZ) states. However, for n > 3 there exist even pure n-partite entangled states with a lower violation than biseparable states [7]. Only recently, significant progress in classifying multipartite entanglement has been achieved using entanglement witnesses [4,8]. These observables can always be used to detect various forms of multipartite entanglement, when some a priori knowledge about the states under investigation is provided [9]; they are in this sense more powerful than Bell inequalities.A witness of genuine n-partite entanglement is an observable which has a positive expectation value on states with n − 1 partite entanglement and a negative expectation value on some n-partite entangled states. The latter states and their entanglement, respectively, are said to be detected by W. Witnesses provide sufficient criteria for entanglement and for distinguishing the various classes of genuine multipartite entangled states.The goal of this Letter is twofold. First, we introduce a general scheme for the construction of multipartite witness operators and their decomposition into locally measurable observables. In this way, we demonstrate how witness operators can be implemented experimentally in a straightforward way by using local projective measurements, even for multipartite systems [10]. Then, we apply this scheme to certain states and perform the experimental detection of their multipartite entanglement, which could not be revealed by known Bell inequalities. In particular, we use this method for the characterization of the three-qubit W state [11], and the four-qubit state |Ψ (4) [12]. A wit...

We study the distillability of a certain class of bipartite density operators which can be obtained via depolarization starting from an arbitrary one. Our results suggest that non-positivity of the partial transpose of a density operator is not a sufficient condition for distillability, when the dimension of both subsystems is higher than two.Comment: 8 pages, 1 figur