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Journal of Optics B Quantum and Semiclassical Optics

Remarks on entanglement swapping

Daegene Song

Published Oct 1, 2003• 2 citations
Publisher

Abstract

In two partially entangled states, entanglement swapping by Bell measurement will yield the weaker entanglement of the two. This scheme is optimal because the average entanglement cannot increase under local operation and classical communication. However, for more than two states, this scheme does not always yield the weakest link. We consider projective measurements other than Bell-type measurement and show, numerically, that while Bell measurement may not be unique, it is indeed optimal among these projective measurements. We also discuss the non-uniqueness of Bell measurements.

DOI
10.1088/1464-4266/6/1/l02

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In two partially entangled states, entanglement swapping by Bell measurement will yield the weaker entanglement of the two. This scheme is optimal because the average entanglement cannot increase under local operation and classical communication. However, for more than two states, this scheme does not always yield the weakest link. We consider projective measurements other than Bell-type measurement and show, numerically, that while Bell measurement may not be unique, it is indeed optimal among these projective measurements. We also discuss the non-uniqueness of Bell measurements.
We consider entanglement-swapping schemes with general (rather than maximally) entangled bipartite states of arbitary dimension shared pairwise between three or more parties in a chain. The intermediate parties perform generalized Bell measurements with the result that the two end parties end up sharing an entangled state which can be converted into maximally entangled states. We obtain an expression for the average amount of maximal entanglement concentrated in such a scheme and show that in a certain reasonably broad class of cases this scheme is provably optimal and that, in these cases, the amount of entanglement concentrated between the two ends is equal to that which could be concentrated from the weakest link in the chain.
Suppose that two distant parties Alice and Bob share an entangled state ρAB\rho_{AB}ρAB​, and they want to exchange the subsystems of ρAB\rho_{AB}ρAB​ by local operations and classical communication (LOCC). In general, this LOCC task (i.e. the LOCC transformation of ρAB→VρABV\rho_{AB} \to V\rho_{AB} VρAB​→VρAB​V with VVV being a swap operator) is impossible deterministically, but becomes possible probabilistically. In this paper, we study how the optimal probability is related to the amount of entanglement in the framework of positive partial transposed (PPT) operations, and numerically show a remarkable class of states whose probability is the smallest among every state in two quantum bits.
The nonlocal correlations between quantum states in an entangled system are essential to many quantum communications applications. A basic quantum operation, which permits the distribution of entanglement between two initially uncorrelated systems, is entanglement swapping. Here we present a rigorous formulation of entanglement swapping of any two partially mixed two-qubit states without limiting ourselves to any specific type of state or noise. Further, for two important classes of the input states, Bell diagonal and pure states, we describe how the concurrence of the final state is related to the concurrence of the initial states. First, we consider Bell diagonal states, and find bounds on the concurrence of the final state in terms of the concurrences of the initial states. These bounds are important for communications applications because polarization mode dispersion in fibers produces Bell diagonal states up to a local unitary rotation. Second, we show that swapping pure states occasionally results in a state of higher concurrence than either of the initial states. In addition, we find that two pure states are most likely to be capable of swapping to a state of increased concurrence when the two initial states have similar concurrences. Our analysis offers a completely general framework for investigating the behavior of any pair of two-qubit states when used for entanglement swapping.
Entanglement swapping is a key primitive for distributing entanglement over quantum networks, but different measurement outcomes can produce end-to-end states with different entanglement, requiring branch-dependent processing or the rejection of unfavorable outcomes. We characterize all projective swapping measurements with full-Schmidt-rank vectors such that, for every pair of pure input links, all outcomes yield the same end-to-end state up to local-unitary corrections. Within this family, the measurements that maximize the average G-concurrence for every input pair are built from complex Hadamard operators, and every outcome individually attains the optimum. Classifying the underlying complex Hadamard operators that preserve optimal deterministic swapping gives one class for d=2,3d=2,3d=2,3, exactly 727272 classes for d=5d=5d=5, and uncountably many whenever d=4kd=4kd=4k. We show further that for d=2,3d=2,3d=2,3, the corrected end-to-end state in a swapping chain is independent of the swapping order, and discuss noise robustness under depolarizing noise and arbitrary convex input contamination. For pure inputs, these schemes retain every outcome while achieving optimal G-concurrence and therefore eliminate outcome-based postselection.
The basic entanglement swapping protocol allows to achieve the main aim of deterministically projecting two qubits, which have never interacted, onto a maximally entangled state. For deterministic swapping the key ingredient is the maximal entanglement initially contained in two pairs of qubits and the ability to measure onto a Bell basis. In other words, the basic and deterministic entanglement swapping scheme involves three maximal level of entanglement. In this work we address probabilistic entanglement swapping processes performed with different amounts of initial entanglement. Besides, we suggest a non Bell measuring-basis, thus we introduce a third entanglement level in the process. Additionally, we propose the \emph{unambiguous state extraction} scheme as the local mechanism for probabilistically achieving the main aim. Amalgamating these three elements allows us to design four strategies for performing probabilistic entanglement swapping. Surprisingly, we encountered an twofold entanglement matching effect related to the concurrence of the measuring-basis. Specifically, the maximal probability of accomplishing the main aim becomes a constant for concurrences higher than or equal to the matching entanglement values. Thus, we show that maximal entanglement in the measuring-basis is not required for attaining the main aim with optimal probability.
Entanglement has recently been one of the most essential elements in the development of various quantum technologies. In fact, a swapping protocol was introduced to create a long-distance entanglement from multiple shorter ones. Extending the previous work, this paper provides a more detailed numerical analysis to help create long-distance entanglement out of the two non-maximal three-level states. Specifically, it shows that while the protocol does not always yield optimal results, namely, the weaker link, there is a substantial number of states that yield an optimal result. Moreover, we discuss the numerical approach in showing the existence of states that provide a result close to the optimal outcome, which may be useful in realizing the long-distance entanglement used in quantum technology.