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arXiv:1707.06534 · New Journal of Physics

Self-testing multipartite entangled states through projections onto two systems

Ivan Šupić, Andrea Coladangelo, Remigiusz Augusiak, Antonio Acín

Published Aug 7, 2018• 92 citations
arXivHTMLPDF

Abstract

Finding ways to test the behaviour of quantum devices is a timely enterprise, especially in light of the rapid development of quantum technologies. Device-independent self-testing is one desirable approach, as it makes minimal assumptions on the devices being tested. In this work, we address the question of which states can be self-tested. This has been answered recently in the bipartite case (Coladangelo et al 2017 Nat. Commun. 8 15485), while it is largely unexplored in the multipartite case, with only a few scattered results, using a variety of different methods: maximal violation of a Bell inequality, numerical SWAP method, stabiliser self-testing etc. In this work, we investigate a simple, and potentially unifying, approach: combining projections onto two-qubit spaces (projecting parties or degrees of freedom) and then using maximal violation of the tilted CHSH inequalities. This allows one to obtain self-testing of Dicke states and partially entangled GHZ states with two measurements per party, and also to recover self-testing of graph states (previously known only through stabiliser methods). Finally, we give the first self-test of a class of multipartite qudit states: we generalise the self-testing of partially entangled GHZ states by adapting techniques from (Coladangelo et al 2017 Nat. Commun. 8 15485), and show that all multipartite states which admit a Schmidt decomposition can be self-tested with few measurements.

DOI
10.1088/1367-2630/aad89b

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Finding ways to test the behaviour of quantum devices is a timely enterprise, especially in light of the rapid development of quantum technologies. Device-independent self-testing is one desirable approach, as it makes minimal assumptions on the devices being tested. In this work, we address the question of which states can be self-tested. This has been answered recently in the bipartite case (Coladangelo et al 2017 Nat. Commun. 8 15485), while it is largely unexplored in the multipartite case, with only a few scattered results, using a variety of different methods: maximal violation of a Bell inequality, numerical SWAP method, stabiliser self-testing etc. In this work, we investigate a simple, and potentially unifying, approach: combining projections onto two-qubit spaces (projecting parties or degrees of freedom) and then using maximal violation of the tilted CHSH inequalities. This allows one to obtain self-testing of Dicke states and partially entangled GHZ states with two measurements per party, and also to recover self-testing of graph states (previously known only through stabiliser methods). Finally, we give the first self-test of a class of multipartite qudit states: we generalise the self-testing of partially entangled GHZ states by adapting techniques from (Coladangelo et al 2017 Nat. Commun. 8 15485), and show that all multipartite states which admit a Schmidt decomposition can be self-tested with few measurements.
Self-testing is a device-independent examination of quantum devices based on correlations of observed statistics. Motivated by elegant progresses on self-testing strategies for measurements [Phys. Rev. Lett. 121, 250507 (2018)] and for states [New J. Phys. 20, 083041 (2018)], we develop a general self-testing procedure for multipartite generalized GHZ-state measurements. The key step is self-testing all measurement eigenstates for a general N-qubit multipartite GHZ-state measurement. Following our procedure, one only needs to perform local measurements on N-2 chosen parties of the N-partite eigenstate and maintain to achieve the maximal violation of tilted Clauser-Horne-Shimony-Holt (CHSH) Bell inequality for remaining two parties. Moreover, this approach is physically operational from an experimental point of view. It turns out that the existing result for three-qubit GHZ-state measurement is recovered as a special case. Meanwhile, we develop the self-testing method to be robust against certain white noise.
Self-testing refers to a device-independent way to uniquely identify an unknown quantum device based only on the observed statistics. Earlier results on self-testing of multipartite state were restricted either to Dicke states or Graph states. In this paper, we propose self-testing schemes for a large family of symmetric three-qubit states, namely the superposition of W state and GHZ state. We first propose and analytically prove a self-testing criterion for the special symmetric state with equal coefficients of the canonical bases, by designing subsystem self-testing of partially and maximally entangled state simultaneously. Then we demonstrate for the general case, the states can be self-tested numerically by the swap method combining semidefinite programming (SDP) in high precision.
Self-testing is a method with which a classical user can certify the state and measurements of quantum systems in a device-independent way. In particular, self-testing of entangled states is of great importance in quantum information processing. An understandable example is that the maximal violation of the Clauser-Horne-Shimony-Holt inequality necessarily implies that the bipartite system shares a singlet. One essential question in self-testing is that, when one observes a nonmaximum violation, how far is the tested state from the target state (which maximally violates a certain Bell inequality)? The answer to this question describes the robustness of the used self-testing criterion, which is highly important in a practical sense. Recently, J. Kaniewski derived two analytic self-testing bounds for bipartite and tripartite systems. In this Letter, we experimentally investigate these two bounds with high-quality two-qubit and three-qubit entanglement sources. The results show that these bounds are valid for various entangled states that we prepared. Thereby, a proof-of-concept demonstration of robust self-testing is achieved, which improves on the previous results significantly.
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.
Device-independent certification, also known as self-testing, aims at guaranteeing the proper functioning of untrusted and uncharacterized devices. For example, the quality of an unknown source expected to produce two-qubit maximally entangled states can be evaluated in a bipartite scenario, each party using two binary measurements. The most robust approach consists in deducing the fidelity of produced states with respect to a two-qubit maximally entangled state from the violation of the CHSH inequality. In this paper, we show how the self-testing of two-qubit maximally entangled states is improved by a refined analysis of measurement statistics. The use of suitably chosen Bell tests, depending on the observed correlations, allows one to conclude higher fidelities than ones previously known. In particular, nontrivial self-testing statements can be obtained from correlations that cannot be exploited by a CHSH-based self-testing strategy. Our results not only provide insight into the set of quantum correlations suited for self-testing, but also facilitate the experimental implementations of device-independent certifications.
Abstract Quantum entanglement is the key resource for quantum information processing. Device-independent certification of entangled states is a long standing open question, which arouses the concept of self-testing. The central aim of self-testing is to certify the state and measurements of quantum systems without any knowledge of their inner workings, even when the used devices cannot be trusted. Specifically, utilizing Bell’s theorem, one can infer the appearance of certain entangled state when the maximum violation is observed, e.g., to self-test singlet state using CHSH inequality. In this work, by constructing a versatile entanglement source, we experimentally demonstrate a generalized self-testing proposal for various bipartite entangled states up to four dimensions. We show that the high-quality generated states can approach the maximum violations of the utilized Bell inequalities, and thus, their Schmidt coefficients can be precisely inferred by self-testing them into respective target states with near-unity fidelities. Our results indicate the superior completeness and robustness of this method and promote self-testing as a practical tool for developing quantum techniques.