We derive a general framework that connects every positive map with a corresponding witness for partial separability in multipartite quantum systems. We show that many previous approaches were intimately connected to the witnesses derived from partial transposition and that such criteria can easily be outperformed in higher dimensions by non-decomposable maps. As an exemplary case we present a witness that is capable of detecting genuine multipartite entanglement in bound entangled states.
While entanglement is believed to be an important ingredient in understanding quantum many-body physics, the complexity of its characterization scales very unfavorably with the size of the system. Finding super-sets of the set of separable states that admit a simpler description has proven to be a fruitful approach in the bipartite setting. In this paper we discuss a systematic way of characterizing multiparticle entanglement via various relaxations. We furthermore describe an operational witness construction arising from such relaxations that is capable of detecting every entangled state. Finally, we also derive an analytic upper-bound on the volume of biseparable states and show that the volume of the states with a positive partial transpose for any split rapidly outgrows this volume. This proves that simple semi-definite relaxations in the multiparticle case cannot be an equally good approximation for any scenario.
All the $n(2n+3)$ mean and covariance parameters of an $n$-mode Gaussian states are expressed in terms of the expectation values of the same number of conjugates of the total number observable. This permits a complete tomography of the state. The same is applied to outputs of a Gaussian channel corresponding to selected coherent states to perform the complete tomography of the channel. This leads to some interesting problems concerning the distribution of the number operator and also tomographic complexity
In this work we consider bipartite noisy bound entangled states with positive partial transpose, that is, such a state can be written as a convex combination of an edge state and a separable state. In particular, we present schemes to construct distinct classes of noisy bound entangled states which satisfy the range criterion. As a consequence of the present study we also identify noisy bound entangled states which do not satisfy the range criterion. All of the present states are constituted by exploring different types of product bases.One of the key developments within the theory of quantum entanglement [1,2], is the invention of bound entangled states [3]. These states are mixed entangled states from which entanglement in pure form cannot be extracted by local operations and classical communication [4]. This holds true even if large number of identical copies of the state are shared among spatially separated parties. Since the discovery of bound entangled states, there is no simple technique to identify such states. Therefore, it is highly nontrivial to present new classes of bound entangled states. For a given bipartite quantum state if the state produces negative eigenvalue(s) under partial transpose then it guarantees inseparability of that state [5]. The problem arises when the given state remains positive under partial transpose (PPT). In such a situation it is not always easy to conclude whether the state is separable or inseparable (entangled). Generally, for an arbitrary bipartite PPT state if the dimension of the corresponding Hilbert space is greater than 6 then it is difficult to say whether the state is separable or inseparable [6]. In fact, the problem of determining any density matrix -separable or entangled is a NP-hard problem [7]. However, if a PPT state is entangled then the state must be bound entangled [4]. On the other hand existence of bound entangled states with negative partial transpose is conjectured and remains open till date [8][9][10].Application of the range criterion is quite effective approach to prove the inseparability a given PPT state [3]. For a given bipartite density matrix ρ, if the state is separable then there exists a set of product states {|θ i 1 ⊗ |θ i 2 } that spans the range of ρ while the set of product states {|θ i 1 ⊗ |θ * i 2 } spans the range of ρ t . Here, the superscript t denotes the partial transpose operation (considering second subsystem) and * denotes the complex conjugation in a basis with respect to which the partial transpose is taken. Any state which violates the range criterion is an entangled state. However, there exist several classes of PPT entangled states which satisfy the range criterion [11,12]. Evidently, detection of such states are one of the troublesome tasks in the entanglement theory. Therefore, to understand these states in a better way, it is important to constitute such states. Note that a full-rank state trivially satisfy the range criterion. So, it is significant to understand the forms of distinct classes of low-rank bound entang...
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