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Physical Review A

Nonlocality swapping

Li‐Yi Hsu

Published May 29, 2002• 2 citations
Publisher

Abstract

We propose a measurement scheme of generalized Bell states in which, although entanglement is destroyed, the nonlocality feature can be preserved and swapped. Here we consider two quantum systems: one consisting of two-party three-level maximal entangled pairs; the other being three-party two-level maximal entangled pairs. It is found that these local immeasurable states can be prepared without any classical communication and local operation. Moreover, nonmaximal entangled states cannot preserve nonlocality.

DOI
10.1103/physreva.65.062302

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We propose a measurement scheme of generalized Bell states in which, although entanglement is destroyed, the nonlocality feature can be preserved and swapped. Here we consider two quantum systems: one consisting of two-party three-level maximal entangled pairs; the other being three-party two-level maximal entangled pairs. It is found that these local immeasurable states can be prepared without any classical communication and local operation. Moreover, nonmaximal entangled states cannot preserve nonlocality.
Nonlocal correlations arising from measurements on tripartite entangled states can be classified into two groups, one genuinely three-way nonlocal and other local with respect to some bipartition. Still, whether a genuinely tripartite entangled quantum state can exhibit genuine three-way nonlocality remains a challenging problem as far as measurement context is concerned. Here we introduce an approach in this regard. We consider three tripartite quantum states, none of which is genuinely three-way nonlocal in a specific Bell scenario (three parties, two measurements per party, two outcomes per measurement), but they can exhibit genuine three-way nonlocality when the initial states are subjected to stochastic local operations and classical communication. So, genuine three-way nonlocality is a resource which can be revealed by using a sequence of measurements.
We generalize the entanglement swapping scheme originally proposed for two pairs of qubits to an arbitrary number q of systems composed from an arbitrary number m j of qudits. Each of the systems is supposed to be prepared in a maximally entangled state of m j qudits, while different systems are not correlated at all. We show that when a set of ∑ q j = 1 a j particles (from each of the q systems a j particles are measured) is subjected to a generalized Bell-type measurement, the resulting set of ∑ q j = 1 ( m j - a j ) particles will collapse into a maximally entangled state.
In an entanglement swapping scenario, if two sources distributing entangled states between three parties are independent, local correlations lead to a different kind of inequality than the standard Bell inequalities, known as network local models. A highly demanding task is to find out a way to involve many players nontrivially in a quantum network since measurements, in general, disturb the system. To this end, we consider here an alternative way of sharing network nonlocality when two observers initially share a maximally entangled, a nonmaximally entangled and a noisy entangled state. We report that by employing unsharp measurements performed by one of the observers, at most six pairs can sequentially demonstrate the violation of bilocal inequality while a maximum of two pairs of observers can exhibit bi-nonlocality when both the observers perform unsharp measurements. We also find the critical noise involved in unsharp measurements in each round to illustrate the bi-nonlocality for a fixed shared entangled state as a resource. We also connect the entanglement content of the shared state, quantified via von-Neumann entropy for pure states and entanglement of formation for Werner states with the maximum number of rounds showing violation of bilocal inequality. By considering elegant joint measurement at the node such that the entanglement content in the elements of the joint measurement by the third party gets reduced, we observe that the maximum number reduces to two sequential sharing of bi-nonlocality even for the maximally entangled state when the settings at each side are taken to be three and fixed.
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.
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.