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arXiv:2007.04020 · Physical Review Research

Robust semi-device-independent certification of all pure bipartite maximally entangled states via quantum steering

Harshank Shrotriya, Kishor Bharti, L. C. Kwek

Published Jul 27, 2021• 17 citations
arXivHTMLPDF

Abstract

The idea of self-testing is to render guarantees concerning the inner workings of a device based on the measurement statistics. It is one of the most formidable quantum certification and benchmarking schemes. Recently it was shown [A. Coladangelo, K. T. Goh, and V. Scarani, Nat. Commun. 8, 15485 (2017)] that all pure bipartite entangled states can be self-tested in the device-independent scenario by employing subspace methods introduced by Yang and Navascu\'es [Phys. Rev. A 87, 050102(R) (2013)]. Here, we have adapted their method to show that any bipartite pure entangled state can be certified in the semi-device-independent scenario through quantum steering. Analogous to the tilted Clauser-Horne-Shimony-Holt inequality, we use a steering inequality called the tilted steering inequality for certifying any pure two-qubit entangled state. Furthermore, we use this inequality to certify any bipartite pure entangled state by certifying two-dimensional subspaces of the qudit state by observing the structure of the set of assemblages obtained on the trusted side after measurements are made on the untrusted side. As a feature of quantum state certification via steering, we use the notion of assemblage-based robust state certification to provide robustness bounds for the certification result in the case of pure maximally entangled states of any local dimension.

DOI
10.1103/physrevresearch.3.033093

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The idea of self-testing is to render guarantees concerning the inner workings of a device based on the measurement statistics. It is one of the most formidable quantum certification and benchmarking schemes. Recently it was shown [A. Coladangelo, K. T. Goh, and V. Scarani, Nat. Commun. 8, 15485 (2017)] that all pure bipartite entangled states can be self-tested in the device-independent scenario by employing subspace methods introduced by Yang and Navascu'es [Phys. Rev. A 87, 050102(R) (2013)]. Here, we have adapted their method to show that any bipartite pure entangled state can be certified in the semi-device-independent scenario through quantum steering. Analogous to the tilted Clauser-Horne-Shimony-Holt inequality, we use a steering inequality called the tilted steering inequality for certifying any pure two-qubit entangled state. Furthermore, we use this inequality to certify any bipartite pure entangled state by certifying two-dimensional subspaces of the qudit state by observing the structure of the set of assemblages obtained on the trusted side after measurements are made on the untrusted side. As a feature of quantum state certification via steering, we use the notion of assemblage-based robust state certification to provide robustness bounds for the certification result in the case of pure maximally entangled states of any local dimension.
The idea of self-testing is to render guarantees concerning the inner workings of a device based on the measurement statistics. It is one of the most formidable quantum certification and benchmarking schemes. Here, we have shown that any bipartite pure entangled state can be self-tested through Quantum Steering. Analogous to the tilted CHSH inequality, we have established a new steering inequality called Tilted Steering Inequality for self-testing any pure two-qubit entangled state. We have further used this inequality to self-test any bipartite pure entangled state by certifying two-dimensional sub-spaces of the qudit state by observing the structure of the assemblage obtained on Bob's side after Alice's measurement.
We consider the problem of one-sided device-independent self-testing of any pure entangled two-qubit state based on steering inequalities which certify the presence of quantum steering. In particular, we note that in the 2-2-2 steering scenario (involving two parties, two measurement settings per party, and two outcomes per measurement setting), the maximal violation of a fine-grained steering inequality can be used to witness certain extremal steerable correlations, which certify all pure two-qubit entangled states. We demonstrate that the violation of the analogous Clauser-Horne-Shimony-Holt inequality of steering or the nonvanishing value of a quantity constructed using a correlation function called mutual predictability, together with the maximal violation of the fine-grained steering inequality, can be used to self-test any pure entangled two-qubit state in a one-sided device-independent way.
Given a Bell inequality, if its maximal quantum violation can be achieved only by a single set of measurements for each party or a single quantum state, up to local unitaries, one refers to such a phenomenon as self-testing. For instance, the maximal quantum violation of the Clauser-Horne-Shimony-Holt inequality certifies that the underlying state contains the two-qubit maximally entangled state and the measurements of one party contains a pair of anti-commuting qubit observables. As a consequence, the other party automatically verifies the set of states remotely steered, namely the "assemblage", is in the eigenstates of a pair of anti-commuting observables. It is natural to ask if the quantum violation of the Bell inequality is not maximally achieved, or if one does not care about self-testing the state or measurements, are we capable of estimating how close the underlying assemblage is to the reference one? In this work, we provide a systematic device-independent estimation by proposing a framework called "robust self-testing of steerable quantum assemblages". In particular, we consider assemblages violating several paradigmatic Bell inequalities and obtain the robust self-testing statement for each scenario. Our result is device-independent (DI), i.e., no assumption is made on the shared state and the measurement devices involved. Our work thus not only paves a way for exploring the connection between the boundary of quantum set of correlations and steerable assemblages, but also provides a useful tool in the areas of DI quantum certification. As two explicit applications, we show 1) that it can be used for an alternative proof of the protocol of DI certification of all entangled two-qubit states proposed by Bowles et al., and 2) that it can be used to verify all non-entanglement-breaking qubit channels with fewer assumptions compared with the work of Rosset et al.
Entangled two-qubit states are the core building blocks for constructing quantum communication networks. Their accurate verification is crucial to the functioning of the networks, especially for untrusted networks. In this work we study the self-testing of two-qubit entangled states via steering inequalities, with robustness analysis against noise. More precisely, steering inequalities are constructed from the tilted Clauser-Horne-Shimony-Holt inequality and its general form, to verify the general two-qubit entangled states. The study provides a good robustness bound, using both local extraction map and numerical semidefinite-programming methods. In particular, optimal local extraction maps are constructed in the analytical method, which yields the theoretical optimal robustness bound. To further improve the robustness of one-sided self-testing, we propose a family of three measurement settings steering inequalities. The result shows that three-setting steering inequality demonstrates an advantage over two-setting steering inequality on robust self-testing with noise. Moreover, to construct a practical verification protocol, we clarify the sample efficiency of our protocols in the one-sided device-independent scenario.
Abstract Robust self-testing provides a framework for certifying quantum resources under experimental imperfections. Improving robustness bounds for quantum resources such as quantum states, steering assemblages, and measurements is a constant effort that ensures relevance in the experimental realm. Despite progress in analytic self-testing methods for quantum states and measurements, extending these techniques to device-independent certification of steering assemblages has remained an open challenge, with previous work relying primarily on numerical approaches. We address this gap by developing operator inequalities for robust self-testing of quantum steering assemblages. Specifically, we consider the assemblage that achieves maximal violation of the Clauser–Horne–Shimony–Holt inequality and obtain explicit lower bounds on its certification fidelity. Our analytic approach yields results that significantly improve upon previous numerical bounds, representing the first analytic treatment of device-independent assemblage self-testing. This work demonstrates a new application of operator inequalities beyond quantum state certification and contributes to the foundational understanding of device-independent certification in steering scenarios, potentially guiding future theoretical and experimental developments.
Abstract Device-independent (DI) certification schemes are based on minimal assumptions about the quantum system under study, which makes the most desirable among certification schemes. However, they are often the most challenging to implement. In order to reduce the implementation cost one can consider semi-DI (SDI) schemes such as those based on quantum steering. Here we provide a construction of a family of steering inequalities which are tailored to large classes of d-outcomes projective measurements being a certain linear combination of the Heisenberg–Weyl operators on the untrusted side and a fixed set of known measurements on the trusted side. We then prove that the maximal quantum violation of those inequalities can be used for certification of those measurements and the maximally entangled state of two qudits. Importantly, in our self-testing proof, we do not assume the shared state to be pure, nor do we assume the measurements to be projective. Before concluding, we also show how robust to noise our self-testing statement is. We believe that our construction broadens the scope of SDI certification, paving the way for more general but still less costly quantum certification protocols.