1. What is 'nonlocality without entanglement' (NLWE) in quantum measurements?
'Nonlocality without entanglement' (NLWE) refers to a phenomenon in quantum measurements where separable quantum measurements (where all eigenstates are non-entangled) are strictly stronger than LOCC measurements. This effect was first introduced by Bennett et al. [5] and shows that certain sets of product states can never be perfectly distinguished by any LOCC measurement. NLWE has been generalized in various directions, with connections to the notion of unextendible product basis and bound entanglement [6, 13]. Recent research has focused on certifying the effect of NLWE in a black-box setting, particularly in multipartite systems. The quantum measurement featuring NLWE can be certified using concepts and tools from self-testing [20, 21], allowing for device-independent certification of quantum resources. This result demonstrates that a strong form of nonlocal quantum correlations in networks, known as genuine network quantum nonlocality [22], can be obtained without the need for entangled measurements traditionally used in network nonlocality.
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2. What is the main goal of the research in the entanglement swapping experiment section?
The main goal of the research in the entanglement swapping experiment section is to certify, in a device-independent manner, that Bob performs a quantum measurement featuring NLWE. This means that, assuming only the independence of the two sources, the presence of such a measurement can be demonstrated from observed data alone, up to irrelevant local transformations. The researchers use the tools and concepts of self-testing to establish a form of equivalence between two experiments, where the physical experiment is compared to the reference experiment. If the physical experiment simulates the reference experiment exactly, it is considered self-tested. The reference experiment involves preparing a pair of maximally entangled qutrits and performing specific measurements on the states and measurements to be certified. The researchers aim to show that the physical experiment must be equivalent to the reference experiment, up to irrelevant local transformations.
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3. What is the main result of the reference experiment in terms of self-testing?
The main result of the reference experiment is that it is a self-test. Considering a physical experiment with unknown states |ps AB1 and |ps B2C and measurements {M a|x }, {M b1,b2|y } and {M c|z }, if the statistics of the physical experiment match exactly the statistics of the reference experiment given in Eq. (3), then all states and measurements of the physical experiment are equivalent (up to irrelevant local transformation) to the reference states and measurements. This implies that the measurement y = for Bob must feature NLWE. More formally, there exists a local isometry Ph mapping EQUATION ) where |x AB1B2C = |x 1 AB1 |x 2 B2C is a valid quantum state. The states |ps AB1 and |ps B2C to the reference states (i.e., pairs of maximally entangled qutrits), and Alice's and Charlie's measurements to the corresponding reference measurements M j : Ph M a|x |ps A1B1 M c|z |ps B2C |0000 A B 1 B 2 C = = |x AB1B2C M a|x |ph + A B 1 M c|z |ph + B 2 C = EQUATION for every a, b1, x, y then there exists isometry Ph such that EQUATION where |x ABC is a valid quantum state. The proof can be found in Appendix A and is directly inspired by the self-testing of maximally entangled pair of qutrits presented in [30].
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4. How can the construction be generalized for higher dimensions in bipartite NLWE measurements?
The construction can be generalized to higher dimensions in bipartite NLWE measurements by using the bilocality network. The local dimensions of two maximally entangled states distributed in the network must be adapted to the dimensions of the states (d A and d B respectively). These states, along with the local measurements of Alice and Charlie, can be self-tested using available methods. Alice and Charlie can remotely prepare for Bob by using their certified local measurements acting on half of the shared maximally entangled pairs. This allows them to match any of the eigenstates of the measurement M b|. Moving to the multipartite case involves a star network with N branches, where the central node performs the NLWE measurement. Each branch must self-test a maximally entangled state of two qudits and local measurements of the lateral nodes. The central NLWE is also self-tested as described earlier.
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