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arXiv:1512.02074 · Physical Review A

Device-independent parallel self-testing of two singlets

Xingyao Wu, Jean-Daniel Bancal, Matthew McKague, Valerio Scarani

Published Jun 23, 2016• 83 citations
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Abstract

Device-independent self-testing offers the possibility of certifying the quantum state and measurements, up to local isometries, using only the statistics observed by querying uncharacterized local devices. In this paper we study parallel self-testing of two maximally entangled pairs of qubits; in particular, the local tensor product structure is not assumed but derived. We prove two criteria that achieve the desired result: a double use of the Clauser-Horne-Shimony-Holt inequality and the $3\ifmmode\times\else\texttimes\fi{}3$ magic square game. This demonstrate that the magic square game can only be perfectly won by measuring a two-singlet state. The tolerance to noise is well within reach of state-of-the-art experiments.

DOI
10.1103/physreva.93.062121

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Device-independent self-testing offers the possibility of certifying the quantum state and measurements, up to local isometries, using only the statistics observed by querying uncharacterized local devices. In this paper we study parallel self-testing of two maximally entangled pairs of qubits; in particular, the local tensor product structure is not assumed but derived. We prove two criteria that achieve the desired result: a double use of the Clauser-Horne-Shimony-Holt inequality and the 3\ifmmode×\else\texttimes\fi33\ifmmode\times\else\texttimes\fi{}33\ifmmode×\else\texttimes\fi3 magic square game. This demonstrate that the magic square game can only be perfectly won by measuring a two-singlet state. The tolerance to noise is well within reach of state-of-the-art experiments.
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.
Self-testing is a method to verify that one has a particular quantum state from purely classical statistics. For practical applications, such as device-independent delegated verifiable quantum computation, it is crucial that one self-tests multiple Bell states in parallel while keeping the quantum capabilities required of one side to a minimum. In this work, we use the $3\ifmmode\times\else\texttimes\fi{}n$ magic rectangle games (generalizations of the magic square game) to obtain a self-test for nnn Bell states where the one side needs only to measure single-qubit Pauli observables. The protocol requires small input sizes [constant for Alice and O(log2n)O({log}_{2}n)O(log2​n) bits for Bob] and is robust with robustness $O({n}^{5/2}\sqrt{\ensuremath{\varepsilon}})$, where $\ensuremath{\varepsilon}$ is the closeness of the ideal (perfect) correlations to those observed. To achieve the desired self-test, we introduce a one-side-local quantum strategy for the magic square game that wins with certainty, we generalize this strategy to the family of $3\ifmmode\times\else\texttimes\fi{}n$ magic rectangle games, and we supplement these nonlocal games with extra check rounds (of single and pairs of observables).
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.
Self-testing is a device-independent technique based on non-local correlations whose aim is to certify the effective uniqueness of the quantum state and measurements needed to produce these correlations. It is known that the maximal violation of some Bell inequalities suffices for this purpose. However, most of the existing self-testing protocols for two devices exploit the well-known Clauser–Horne–Shimony–Holt Bell inequality or modifications of it, and always with two measurements per party. Here, we generalize the previous results by demonstrating that one can construct self-testing protocols based on the chained Bell inequalities, defined for two devices implementing an arbitrary number of two-output measurements. On the one hand, this proves that the quantum state and measurements leading to the maximal violation of the chained Bell inequality are unique. On the other hand, in the limit of a large number of measurements, our approach allows one to self-test the entire plane of measurements spanned by the Pauli matrices X and Z . Our results also imply that the chained Bell inequalities can be used to certify two bits of perfect randomness.
Device-independent self-testing allows a verifier to certify that potentially malicious parties hold on to a specific quantum state, based only on the observed correlations. Parallel self-testing has recently been explored, aiming to self-test many copies (i.e. a tensor product) of the target state concurrently. In this work, we show that n EPR pairs can be self-tested in parallel through n copies of the well known CHSH game. We generalise this result further to a parallel self-test of n tilted EPR pairs with arbitrary angles, and finally we show how our results and calculations can also be applied to obtain a parallel self-test of 2n EPR pairs via n copies of the Mermin-Peres magic square game.
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.