Construction of noisy bound entangled states and the range criterion

Abstract

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...

Distinguishability classes, resource sharing, and bound entanglement distribution

“…To detect the entanglement of the state ρ ABC in that bipartition, we apply the technique described in Refs. [89,90]. We use the celebrated Choi map [91], Λ : M 3 → M 3 , and a unitary operator U for the purpose of the detection of entanglement.…”

Practical construction of positive maps which are not completely positive

“…We consider now an example of a Bound Entangled State, which are known to be entangled whilst having a positive partial transpose (see [14] or [16,Section 6.11]). We take the example from [12], with…”

Separability and entanglement in superpositions of quantum states

“…Sometimes, it is possible to find a separable state for a given entangled state, such that any nontrivial convex combination is entangled. In such a scenario, we can say that the entangled state is unconditionally robust in the direction of that separable state [29,30,34,38,42]. Here we note that for an arbitrary pair of a pure entangled state and a pure product state, any convex combination of the states with some nonzero probabilities produces entangled states only: all pure entangled states are unconditionally robust in the direction of all pure product states [34].…”

“…In discussions about robustness of entanglement, one often considers convex combinations of an entangled state with a separable state, where the latter is deemed as "noise" [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. By taking such a convex combination, one examines how robust the given entangled state is against the chosen noise: how much mixing of the noise does the entangled state tolerate, so that the newly produced state remains entangled.…”