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...
In this paper, we discuss extremal extensions of entanglement witnesses based on Choi's map. The constructions are based on a generalization of the Choi map, from which we construct entanglement witnesses. These extremal extensions are powerful in terms of their capacity to detect entanglement of positive under partial transpose (PPT) entangled states and lead to unearthing of entanglement of new PPT states. We also use the Cholesky-like decomposition to construct entangled states which are revealed by these extremal entanglement witnesses.
Motivated by the notions of $k$-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al (Phys. Rev. A, 69:022308), we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4):343--351), we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure (Lett. Math. Phys. 15(3): 255--260) to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.Comment: Proved a conjecture proposed in earlier version. Added Theorem 2.3. Added Section 3 based on quantum de Finetty theorem. Background materials taken from 1506.06526 and 1504.07054. Comments welcom
In this paper we describe a new connection between UPB (unextendable product bases) and P (positive) maps which are not CP (completely positive). We show that inner automorphisms of the set of P maps which are not CP, produce extremal extensions of these maps that help in entanglement detection. By constructing such an extension of the well-known Choi map, we strengthen its power to unearth PPT (positive under partial transpose) entangled states. We further show that the class of maps generated from the Choi map via an inner automorphism naturally detects the entanglement of states in the orthogonal complement of certain UPB. This brings out a hitherto undiscovered connection between the Choi map and UPB. We also show that certain other recently considered extremal extensions are obtainable by such extensions of the Choi map.
Suppose a set of m-partite, m ≥ 3, pure orthogonal fully separable states is given. We consider the task of distinguishing these states perfectly by local operations and classical communication (LOCC) in different k-partitions, 1 < k < m. Based on this task, it is possible to classify the sets of product states into different classes. For tripartite systems, a classification of the sets with explicit examples is presented. Few important cases related to the aforesaid task are also studied when the number of parties, m ≥ 4. These cases never appear for a tripartite system. However, to distinguish any LOCC indistinguishable set, entanglement can be used as resource. An important objective of the present study is to learn about the efficient ways of resource sharing among the parties. We also find an interesting application of multipartite product states which are LOCC indistinguishable in a particular k-partition. Starting from such product states, we constitute a protocol to distribute bound entanglement between two spatially separated parties by sending a separable qubit.
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