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arXiv:2112.10868 · Physical Review A

Self-testing of multipartite Greenberger-Horne-Zeilinger states of arbitrary local dimension with arbitrary number of measurements per party

Shubhayan Sarkar, Remigiusz Augusiak

Published Mar 7, 2022• 19 citations
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Abstract

Device-independent certification schemes have gained a lot of interest lately, not only for their applications in quantum information tasks, but also their implications towards foundations of quantum theory. The strongest form of device-independent certification, known as self-testing, often requires for a Bell inequality to be maximally violated by specific quantum states and measurements. In this work, using the techniques developed recently [S. Sarkar et al., npj Quantum Inf. 7, 151 (2021)], we provide a self-testing scheme for the multipartite Greenberger-Horne-Zeilinger states of arbitrary local dimension that does not rely on self-testing results for qubit states and that exploits the minimal number of two measurements per party. This makes our results interesting as far as practical implementation of device-independent certification methods is concerned. Our self-testing statement relies on maximal violation of a Bell inequality proposed recently [R. Augusiak et al., New J. Phys. 21, 113001 (2019)].

DOI
10.1103/physreva.105.032416

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Device-independent certification schemes have gained a lot of interest lately, not only for their applications in quantum information tasks, but also their implications towards foundations of quantum theory. The strongest form of device-independent certification, known as self-testing, often requires for a Bell inequality to be maximally violated by specific quantum states and measurements. In this work, using the techniques developed recently [S. Sarkar et al., npj Quantum Inf. 7, 151 (2021)], we provide a self-testing scheme for the multipartite Greenberger-Horne-Zeilinger states of arbitrary local dimension that does not rely on self-testing results for qubit states and that exploits the minimal number of two measurements per party. This makes our results interesting as far as practical implementation of device-independent certification methods is concerned. Our self-testing statement relies on maximal violation of a Bell inequality proposed recently [R. Augusiak et al., New J. Phys. 21, 113001 (2019)].
Abstract In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in Salavrakos et al (2017 Phys. Rev. Lett. 119 040402) a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger–Horne–Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled Greenberger–Horne–Zeilinger-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.
Abstract In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in Salavrakos et al (2017 Phys. Rev. Lett. 119 040402) a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger–Horne–Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled Greenberger–Horne–Zeilinger-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.
Device-independent certification of quantum states enables the characterization of states within a device under minimal physical assumptions. A major problem in this regard is to certify quantum states using minimal resources. Aiming to address this problem, we consider a multipartite quantum steering scenario involving an arbitrary number of parties, of which only one is trusted, meaning that the measurements performed by this party are known. Consequently, the self-testing scheme is almost device-independent. Importantly, all the parties can only perform two measurements each, which is the minimal number of measurements required to observe any form of quantum nonlocality. Then, we propose steering inequalities that are maximally violated by three major classes of genuinely multipartite entangled (GME) states: graph states of arbitrary local dimension, Schmidt states of arbitrary local dimension, and NNN-qubit generalized W states. Using the proposed inequalities, we then provide an almost device-independent certification of the above GME states. Restricting to qubits, we also lift our almost device-independent scheme to device-independent self-testing.
Self-testing is an examination for characterizing unknown quantum devices based on correlations of observed statistics in a black-box scenario. With development of large-scale quantum networks, the requirement for self-testing multipartite entangled measurements has become very demanding. We develop here a general procedure for self-testing arbitrary generalized Greenberger-Horne-Zeilinger-state (GHZ-state) measurements which applies to any number of parties. Moreover, it turns out that the existing result for the three-qubit GHZ-state measurement is recovered as a special case. Our results will motivate operational certification of quantum devices related to device-independent quantum information tasks for various scenarios in complicated quantum networks.
Graph states-one of the most representative families of multipartite entangled states-are important resources for multiparty quantum communication, quantum error correction, and quantum computation. Device-independent certification of highly entangled graph states plays a prominent role in quantum information processing tasks. Here we have experimentally demonstrated device-independent certification for multipartite graph states by adopting the robust self-testing scheme based on scalable Bell inequalities. Specifically, the prepared multi-qubit Greenberger-Horne-Zeilinger (GHZ) states and linear cluster states achieve a high degree of Bell violation, which are beyond the nontrivial bounds of the robust self-testing scheme. Furthermore, our work paves the way to the device-independent certification of complex multipartite quantum states.
Practical robust self-testing of multipartite entanglement has so far been restricted to small-scale systems due to error bounds that degrade severely with system size. In this work, we establish multipartite self-testing with robustness independent of the size of the quantum network. We derive a fully analytic, device-independent self-testing bound for nnn-qubit Greenberger-Horne-Zeilinger (GHZ) states. The bound scales linearly with the observed violation error and lies universally within a constant factor of two from a theoretical upper bound. Furthermore, the operator-inequality framework reduces the verification of the conjectured optimal bound to a highly efficient numerical check, which we perform up to n=100n=100n=100. Consequently, GHZ entanglement can be certified under a fixed noise level in arbitrarily large systems, enabling scalable device-independent verification.