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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

The utility of entanglement swapping in quantum communications

Brian T. Kirby, Siddhartha Santra, Vladimir S. Malinovsky, Michael Brodsky

Published Sep 13, 2016• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.2237760

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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.
We consider entanglement swapping, a key component of quantum network operations and entanglement distribution. Pure entangled states, which are the desired input to the swapping protocol, are typically mixed by environmental interactions, causing a reduction in their degree of entanglement. Thus an understanding of entanglement swapping with partially mixed states is of importance. Here we present a general analytical solution for entanglement swapping of arbitrary two-qubit states. Our result provides a comprehensive method for analyzing entanglement swapping in quantum networks. First, we show that the concurrence of a partially mixed state is conserved when this state is swapped with a Bell state. Then, we find upper and lower bounds on the concurrence of the state resulting from entanglement swapping for various classes of input states. Finally, we determine a general relationship between the ranks of the initial states and the rank of the final state after swapping.
Entanglement swapping is gaining widespread attention due to its application in entanglement distribution among different parts of quantum appliances. We investigate the entanglement swapping for pure and noisy systems, and argue different entanglement quantifiers for quantum states. We explore the relationship between the entanglement of initial states and the average entanglement of final states in terms of concurrence and negativity. We find that if initial quantum states are maximally entangled and we make measurements in the Bell basis, then average concurrence and average negativity of final states give similar results. In this case, we simply obtain the average concurrence (average negativity) of the final states by taking the product of concurrences (negativities) of the initial states. However, the measurement in non-maximally entangled basis during entanglement swapping degrades the average swapped entanglement. Further, the product of the entanglement of the initial mixed states provides an upper bound to the average swapped entanglement of final states obtained after entanglement swapping. The negativity work well for weak entangled noisy states but concurrence gives better results for relatively strong entanglement regimes. We also discuss how successfully the output state can be used as a channel for the teleportation of an unknown qubit.
We study the entanglement purification via entanglement swapping in terms of optimal output state concurrence. A strict formula that governs the process is first given analytically for the case where both input entangled pairs are in arbitrary pure states. Then, it is generalized, with strong numerical evidence, to the case with one of the entangled pairs being in an arbitrary mixed state. The formula subsequently gives a formal upper bound of entanglement purification via entanglement swapping, as well as how the maximum entanglement purification can be attained by engineering the pure state entangled pair. Moreover, the formalism of the upper bound in turn suggests that entanglement swapping is most suited for reducing the influence of pure dissipative noise, which can be applied to extend the lifespan of stationary qubits in quantum communications.
We study the role of average concurrence in entanglement swapping in quantum networks. We begin with qubit pure states, and there is a very simple rule governing the propagation of average concurrence in multiple swaps. We find a similarly simple rule for average concurrence when creating a Greenberger-Horne-Zeilinger state from three entangled pairs. We look at examples of mixed qubit states and find that the relation for pure states gives an upper bound on what is possible with mixed states. We then move on to qudits, where we make use of the I-concurrence. Here the situation is not as simple as for qubits, but in some cases relatively straightforward results can be obtained.
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.