Bound Entanglement Provides Convertibility of Pure Entangled States

Abstract

I show that two distant parties can transform pure entangled states to arbitrary pure states by stochastic local operations and classical communication (SLOCC) at the single copy level, if they share bound entangled states. This is the effect of bound entanglement since this entanglement processing is impossible by SLOCC alone. Similar effect of bound entanglement occurs in three qubits where two incomparable entangled states of GHZ and W can be inter-converted. In general multipartite settings composed by N distant parties, all N -partite pure entangled states are interconvertible by SLOCC with the assistance of bound entangled states with positive partial transpose.PACS numbers: 03.67. Mn, 03.65.Ud In quantum entanglement processing, the transformation of entangled states by local operations and classical communication (LOCC) is a basic task, where many intriguing aspects concerning convertibility and irreversibility appear. All bipartite pure entangled states are inter-convertible in the asymptotic transformation [1] where infinitely many identical copies of states are processed. Therefore, all bipartite pure entangled states can be used to perform the same task of entanglement processing in the asymptotic regime. However, there exists a restriction in the transformation of a single copy of bipartite pure states ( Fig. 1): two distant parties cannot increase the number of superposed terms (Schmidt rank, the rank of the reduced density matrix) by LOCC even in a stochastic manner (such stochastic LOCC is denoted by SLOCC) [2,3,4]. As a result, bipartite entanglement is classified by the Schmidt rank from the viewpoint of the convertibility at the single copy level [5]. Such restriction becomes more strict in multipartite settings. In three qubits, there are two different types of tripartite entanglement: GHZ and W type [5] [the GHZ state is (|111 +|222 )/ √ 2 and W state is (|112 +|121 +|211 )/ √ 3]. These cannot be transformed to each other by SLOCC, and such entangled states are said to be incomparable (Fig. 2). In general multipartite settings composed by N distant parties, there are many (possibly infinitely many) incomparable types of N -partite entanglement.On the other hand, a remarkable aspect of the irreversibility is the existence of bound entangled (BE) states [6]. Distant parties need to consume pure entangled states to prepare BE states, but they cannot distill pure entangled states from it any more. Much attention has been paid to this weak type of entanglement to clarify its properties and usefulness for quantum information processing. BE states by itself are useless for both quantum teleportation [7] and superdense coding [8]. However, it has been shown that BE states can activate the bound entanglement of the other state [9] and are useful for secure key distribution [10] In this paper, I show that BE states strongly influence the entanglement processing of pure states at the single copy level. Two distant parties can get capability to increase the Schmidt rank of bipartite pure entangled s...

Correlated entanglement distillation and the structure of the set of undistillable states

“…This is accomplished by proving a new entanglement activation result, which generalizes previous findings 14 and points toward an activation process involving two undistillable states which would, in particular, imply the existence of NPPT states. Entanglement activation [11][12][13][14][15][16][17][18][19][20] is the process in which an entangled state, which by itself would be useless for a given task, e.g., teleportation, can be activated and used as an a resource when processed together with a second state. In the first example of such a phenomenon, 15 it was shown that a certain PPT bound entangled state could be employed in order to increase the fidelity of teleportation of a second state.…”

Logarithmic Negativity: A Full Entanglement Monotone That is not Convex

“…The phenomenon of bound entanglement [6], i.e., nonseparable states that possess a positive partial transpose and are useless to distill pure maximally entangled states by LOCC [7,8], suggest that there are other natural restricted classes of operations. One might add bound entangled states as a free resource to LOCC operations to achieve tasks that are impossible under LOCC alone [9][10][11]. Encompassing both classes is the mathematically more natural and convenient class of positive partial transpose preserving operations (PPT-operations) [12] which have the property that they map the set of positive partial transpose states into itself just as LOCC operations map the set of separable states into itself.…”

Strong monotonicity in mixed-state entanglement manipulation

“…Here, a map Λ is called PPT-preserving operations [17] when both Λ and Γ•Λ•Γ is a completely positive (CP) map with Γ being a map of the partial transpose [18]. On the other hand, the manipulation of mixed-state entanglement still suffers some restriction even under PPT-preserving operations in single-copy settings [15,16]. This implies that there certainly exists strong monotonicity independent on the Schmidt number in mixed-state entanglement manipulation.…”

“…Concerning the conversion between bipartite pure states, the Schmidt number is a unique strong monotone (the conversion is impossible when the Schmidt rank of the target state is larger than that of the initial state, but otherwise the conversion is possible with nonzero probability [13,14]). Positive partial transpose preserving (PPT-preserving) operations can overcome the monotonicity of the Schmidt number, and therefore all pure entangled states become convertible under PPT-preserving operations [15,16]. Here, a map Λ is called PPT-preserving operations [17] when both Λ and Γ•Λ•Γ is a completely positive (CP) map with Γ being a map of the partial transpose [18].…”