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arXiv:2607.21288

Device-Independent Self-Testing of the Three-Qubit CCZ Hypergraph State

Yunguang Han, Xin Kuang, Xingyuan Bu, Yukun Wang

quant-phSubmitted Jul 23, 2026• Announced Jul 24, 2026• 0 citations
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Abstract

The three-qubit CCZ state is the smallest rank-three hypergraph state and an elementary entangled magic resource. Its cubic phase is governed by generalized stabilizers that are not Pauli strings, so standard graph-state self-testing arguments do not apply directly. We show that twenty correlators, all obtainable from five of the eight global input triples in the tripartite two-input, two-output scenario, determine this state and the action of the Pauli $X/Z$ measurements up to local isometries. The proof fixes eight equally weighted computational branches and propagates conditional $X$-flip relations across the branch cube, recovering the minus sign of the $111$ amplitude. These five-context correlations are nonlocal, but the canonical Pauli measurements cannot attain the largest quantum value of any Bell inequality that they violate: whenever they maximize a Bell expression, its local bound has the same value. Introducing an independent third measurement makes self-testing from maximal Bell violation possible. We construct an explicit Bell inequality whose maximal quantum violation self-tests the CCZ state and all three local measurements. An exact sum-of-squares decomposition proves the quantum bound, and its equality conditions yield an analytic SWAP extraction. Together, these results give two explicit device-independent self-tests of the CCZ state and demonstrate that determining a state and its measurements from several correlator equalities is distinct from identifying them through the maximal violation of a single Bell inequality.

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26 pages, 2 figures

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The three-qubit CCZ state is the smallest rank-three hypergraph state and an elementary entangled magic resource. Its cubic phase is governed by generalized stabilizers that are not Pauli strings, so standard graph-state self-testing arguments do not apply directly. We show that twenty correlators, all obtainable from five of the eight global input triples in the tripartite two-input, two-output scenario, determine this state and the action of the Pauli X/ZX/ZX/Z measurements up to local isometries. The proof fixes eight equally weighted computational branches and propagates conditional XXX-flip relations across the branch cube, recovering the minus sign of the 111111111 amplitude. These five-context correlations are nonlocal, but the canonical Pauli measurements cannot attain the largest quantum value of any Bell inequality that they violate: whenever they maximize a Bell expression, its local bound has the same value. Introducing an independent third measurement makes self-testing from maximal Bell violation possible. We construct an explicit Bell inequality whose maximal quantum violation self-tests the CCZ state and all three local measurements. An exact sum-of-squares decomposition proves the quantum bound, and its equality conditions yield an analytic SWAP extraction. Together, these results give two explicit device-independent self-tests of the CCZ state and demonstrate that determining a state and its measurements from several correlator equalities is distinct from identifying them through the maximal violation of a single Bell inequality.
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Quantum self-testing is the task of certifying quantum states and measurements using the output statistics solely, with minimal assumptions about the underlying quantum system. It is based on the observation that some extremal points in the set of quantum correlations can only be achieved, up to isometries, with specific states and measurements. Here, we present a new approach for quantum self-testing in Bell non-locality scenarios, motivated by the following observation: the quantum maximum of a given Bell inequality is, in general, difficult to characterize. However, it is strictly contained in an easy-to-characterize set: the \emph{theta body} of a vertex-weighted induced subgraph (G,w)(G,w)(G,w) of the graph in which vertices represent the events and edges join mutually exclusive events. This implies that, for the cases where the quantum maximum and the maximum within the theta body (known as the Lovász theta number) of (G,w)(G,w)(G,w) coincide, self-testing can be demonstrated by just proving self-testability with the theta body of GGG. This graph-theoretic framework allows us to (i) recover the self-testability of several quantum correlations that are known to permit self-testing (like those violating the Clauser-Horne-Shimony-Holt (CHSH) and three-party Mermin Bell inequalities for projective measurements of arbitrary rank, and chained Bell inequalities for rank-one projective measurements), (ii) prove the self-testability of quantum correlations that were not known using existing self-testing techniques (e.g., those violating the Abner Shimony Bell inequality for rank-one projective measurements). Additionally, the analysis of the chained Bell inequalities gives us a closed-form expression of the Lovász theta number for a family of well-studied graphs known as the Möbius ladders, which might be of independent interest in the community of discrete mathematics.
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.
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.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2)(2,2,2)(2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3)(2,2,3)(2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.
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.