LOCC protocols for entanglement transformations (original) (raw)

Protocols for entanglement transformations of bipartite pure states

Physical Review A, 2003

We present a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via local operations and classical communication. This framework unifies and greatly simplifies previous works. A necessary condition for "pure contraction" transformations is given. Finally, constructive protocols to achieve both probabilistic and deterministic entanglement transformations are presented.

Classification of nonasymptotic bipartite pure-state entanglement transformations

Physical Review A, 2002

Let {|ψ , |φ } be an incomparable pair of states (|ψ |φ), i.e., |ψ and |φ cannot be transformed to each other with probability one by local transformations and classical communication (LOCC). We show that incomparable states can be multiple-copy transformable, i.e., there can exist a k, such that |ψ ⊗k+1 → |φ ⊗k+1 , i.e., k + 1 copies of |ψ can be transformed to k + 1 copies of |φ with probability one by LOCC but |ψ ⊗n |φ ⊗n ∀n ≤ k. We call such states k-copy LOCC incomparable. We provide a necessary condition for a given pair of states to be k-copy LOCC incomparable for some k. We also show that there exist states that are neither k-copy LOCC incomparable for any k nor catalyzable even with multiple copies. We call such states strongly incomparable. We give a sufficient condition for strong incomparability. We demonstrate that the optimal probability of a conclusive transformation involving many copies, pmax |ψ ⊗m → |φ ⊗m can decrease exponentially with the number of source states m, even if the source state has more entropy of entanglement. We also show that the probability of a conclusive conversion might not be a monotonic function of the number of copies. Fascinating developments in quantum information theory [1] and quantum computing [2] during the past decade has led us to view entanglement as a valued physical resource. Consequently, recent studies have largely been devoted towards its quantification in appropriate limits (finite or asymptotic), optimal manipulation, and transformation properties under local operations and classical communication (LOCC) [3, 4, 5, 6, 7, 8]. Since the specific tasks that can be accomplished with entanglement as a resource is closely related to its transformation properties, it is of importance to know what transformations are allowed under LOCC. Suppose Alice and Bob share a pure state |ψ (source state), which they wish to convert to another entangled state |φ (target state) under LOCC. A necessary and sufficient condition for this transformation to be possible with certainty (denoted by |ψ → |φ) has been obtained by Nielsen [3]. If such a deterministic transformation is not possible but |ψ has at least as many Schmidt coefficients as |φ , then one

General entanglement-assisted transformation for bipartite pure quantum states

Journal of Physics A: Mathematical and Theoretical, 2007

We introduce the general catalysts for pure entanglement transformations under local operations and classical communications in such a way that we disregard the profit and loss of entanglement of the catalysts per se. As such, the possibilities of pure entanglement transformations are greatly expanded. We also design an efficient algorithm to detect whether a k × k general catalyst exists for a given entanglement transformation. This algorithm can as well be exploited to witness the existence of standard catalysts.

Universal entanglement transformations without communication

2003

We show that in the presence of finite catalysts, any pure bipartite entangled state can be converted into any other, to unlimited accuracy, without the use of any communication, quantum or classical. We call this process embezzling entanglement because it involves removing a small amount of entanglement from the catalyst in a physically unnoticeable way.

Entanglement transformation between sets of bipartite pure quantum states using local operations

Journal of Mathematical Physics, 2012

Alice and Bob are given an unknown initial state chosen from a set of pure quantum states. Their task is to transform the initial state to a corresponding final pure state using local operations only. We prove necessary and sufficient conditions on the existence of such a transformation. We also provide efficient algorithms that can quickly rule out the possibility of transforming a set of initial states to a set of final states.

Partial recovery of entanglement in bipartite-entanglement transformations

Physical Review A, 2002

Any deterministic bipartite entanglement transformation involving finite copies of pure states and carried out using local operations and classical communication (LOCC) results in a net loss of entanglement. We show that for almost all such transformations, partial recovery of lost entanglement is achievable by using 2 × 2 auxiliary entangled states, no matter how large the dimensions of the parent states are. For the rest of the special cases of deterministic LOCC transformations, we show that the dimension of the auxiliary entangled state depends on the presence of equalities in the majorization relations of the parent states. We show that genuine recovery is still possible using auxiliary states in dimensions less than that of the parent states for all patterns of majorization relations except only one special case. Entanglement, shared among spatially separated parties, is a critical resource that enables efficient implementations of several quantum information processing [2] and distributed computation [3] tasks. To better exploit the power of entanglement, considerable effort has been put into understanding its transformation properties [4]-[7] and characterizing transformations allowed under local operations and classical communication (LOCC) . A central question is: what happens to the overall entanglement during transformations? In the asymptotic limit involving infinite number of copies of pure states, entanglement can be *

Deterministic transformations of three-qubit entangled pure states

Physical Review A

The states of three-qubit systems split into two inequivalent types of genuine tripartite entanglement, namely the Greenberger-Horne-Zeilinger (GHZ) type and the W-type. A state belongs to one of these classes can be stochastically transformed only into a state within the same class by local operations and classical communications. We provide local quantum operations, consisting of the most general two-outcome measurement operators, for the deterministic transformations of three-qubit pure states in which the initial and the target states are in the same class. We explore these transformations, originally having the standard GHZ and the standard W states, under the local measurement operators carried out by a single party and p (p = 2, 3) parties (successively). We find a notable result that the standard GHZ state cannot be deterministically transformed to a GHZ-type state in which its all bipartite entanglements are nonzero, i.e., a transformation can be achieved with unit probability when the target state has at least one vanishing bipartite concurrence.

Entanglement transformations using separable operations

Physical Review A, 2007

We study conditions for the deterministic transformation |ψ −→ |φ of a bipartite entangled state by a separable operation. If the separable operation is a local operation with classical communication (LOCC), Nielsen's majorization theorem provides necessary and sufficient conditions. For the general case we derive a necessary condition in terms of products of Schmidt coefficients, which is equivalent to the Nielsen condition when either of the two factor spaces is of dimension 2, but is otherwise weaker. One implication is that no separable operation can reverse a deterministic map produced by another separable operation, if one excludes the case where the Schmidt coefficients of |ψ and are the same as those of |φ . The question of sufficient conditions in the general separable case remains open. When the Schmidt coefficients of |ψ are the same as those of |φ , we show that the Kraus operators of the separable transformation restricted to the supports of |ψ on the factor spaces are proportional to unitaries. When that proportionality holds and the factor spaces have equal dimension, we find conditions for the deterministic transformation of a collection of several full Schmidt rank pure states |ψj to pure states |φj .