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Graph-Theoretic Approach for Self-Testing in Bell Scenarios

Kishor Bharti, Maharshi Ray, Zhen‐Peng Xu, Masahito Hayashi, L. C. Kwek, Adán Cabello

Published Sep 26, 2022• 14 citations
Publisher

Abstract

Self-testing is a technology to certify states and measurements using only the statistics of the experiment.Self-testing is possible if some extremal points in the set B Q of quantum correlations for a Bell experiment are achieved, up to isometries, with specific states and measurements.However, B Q is difficult to characterize, so it is also difficult to prove whether or not a given matrix of quantum correlations allows for self-testing.Here, we show how some tools from graph theory can help to address this problem.We observe that B Q is strictly contained in an easy-to-characterize set associated with a graph, (G).Therefore, whenever the optimum over B Q and the optimum over (G) coincide, self-testing can be demonstrated by simply proving self-testability with (G).Interestingly, these maxima coincide for the quantum correlations that maximally violate many families of Bell-like inequalities.Therefore, we can apply this approach to prove the self-testability of many quantum correlations, including some that are not previously known to allow for self-testing.In addition, this approach connects self-testing to some open problems in discrete mathematics.We use this connection to prove a conjecture [M.Araújo et al., Phys.Rev. A, 88, 022118 (2013)] about the closed-form expression of the Lovász theta number for a family of graphs called the Möbius ladders.Although there are a few remaining issues (e.g., in some cases, the proof requires the assumption that measurements are of rank 1), this approach provides an alternative method to self-testing and draws interesting connections between quantum mechanics and discrete mathematics.

DOI
10.1103/prxquantum.3.030344

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Self-testing is a technology to certify states and measurements using only the statistics of the experiment.Self-testing is possible if some extremal points in the set B Q of quantum correlations for a Bell experiment are achieved, up to isometries, with specific states and measurements.However, B Q is difficult to characterize, so it is also difficult to prove whether or not a given matrix of quantum correlations allows for self-testing.Here, we show how some tools from graph theory can help to address this problem.We observe that B Q is strictly contained in an easy-to-characterize set associated with a graph, (G).Therefore, whenever the optimum over B Q and the optimum over (G) coincide, self-testing can be demonstrated by simply proving self-testability with (G).Interestingly, these maxima coincide for the quantum correlations that maximally violate many families of Bell-like inequalities.Therefore, we can apply this approach to prove the self-testability of many quantum correlations, including some that are not previously known to allow for self-testing.In addition, this approach connects self-testing to some open problems in discrete mathematics.We use this connection to prove a conjecture [M.Araújo et al., Phys.Rev. A, 88, 022118 (2013)] about the closed-form expression of the Lovász theta number for a family of graphs called the Möbius ladders.Although there are a few remaining issues (e.g., in some cases, the proof requires the assumption that measurements are of rank 1), this approach provides an alternative method to self-testing and draws interesting connections between quantum mechanics and discrete mathematics.
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 device-independent technique that allows for verification of unknown quantum systems only when observed correlations achieve some quantum supremum. In this paper, we present self-testing schemes for a class of multipartite Bell inequalities with a specific structure, which we refer to as the unit. Our research is based on the graph-theoretic approach, where correlations are characterized via Lov\'asz theta body in the framework of exclusivity graphs. The derived statements are subsequently applied to prove the self-testability of NNN-partite Mermin-Ardehali-Belinski\ifmmode \check{i}\else \v{i}\fi{}-Klyshko (MABK) family of Bell inequalities. In particular, we take special cases to demonstrate that the maximal violation of the multipartite MABK inequality is a self-test for the corresponding multipartite Greenberger-Horne-Zeilinger (GHZ) state. Moreover, we propose a device-independent verification algorithm for distributed blind quantum computation in the noisy intermediate-scale quantum era combined with multipartite GHZ state self-testing, where multiple quantum servers are involved and work in collaboration to reduce the burden on a single server.
Quantum nonlocality has recently been intensively studied in connection to device-independent quantum information processing, where the extremal points of the set of quantum correlations play a crucial role through self-testing. In most protocols, the proofs for self-testing rely on the maximal violation of the Bell inequalities, but there is another known proof based on the geometry of state vectors for self-testing a maximally entangled state. We extend the geometrical proof to the case of partially entangled states. We show that, when a set of correlators in the simplest Bell scenario satisfies a condition, the geometry of the state vectors of the realization is uniquely determined. However, the realization becomes self-testable only when another unitary observable exists on the geometry. Applying this result, we propose self-testing protocols by intentionally adding one more measurement. This geometrical scheme for self-testing is superior in that, by using this as a building block and repeatedly adding measurements, a realization with an arbitrary number of measurements can be self-tested. Besides the application, we also attempt to describe nonlocal correlations by guessing probabilities of distant measurement outcomes. In this description, the quantum set is also convex, and a large class of extremal points is identified by the uniqueness of the geometry.
In device-independent quantum information, correlations between local measurement outcomes observed by spatially separated parties in a Bell test play a fundamental role. Even though it is long-known that the set of correlations allowed in quantum theory lies strictly between the Bell-local set and the no-signaling set, many questions concerning the geometry of the quantum set remain unanswered. Here, we revisit the problem of when the boundary of the quantum set coincides with the no-signaling set in the simplest Bell scenario. In particular, for each Class of these common boundaries containing k zero probabilities, we provide a (5−k) -parameter family of quantum strategies realizing these (extremal) correlations. We further prove that self-testing is possible in all nontrivial Classes beyond the known examples of Hardy-type correlations, and provide numerical evidence supporting the robustness of these self-testing results. Candidates of one-parameter families of self-testing correlations from some of these Classes are identified. As a byproduct of our investigation, if the qubit strategies leading to an extremal nonlocal correlation are local-unitarily equivalent, a self-testing statement provably follows. Interestingly, all these self-testing correlations found on the no-signaling boundary are provably non-exposed. An analogous characterization for the set M of quantum correlations arising from finite-dimensional maximally entangled states is also provided. En route to establishing this last result, we show that all correlations of M in the simplest Bell scenario are attainable as convex combinations of those achievable using a Bell pair and projective measurements. In turn, we obtain the maximal Clauser-Horne-Shimony-Holt Bell inequality violation by any maximally entangled two-qudit state and a no-go theorem regarding the self-testing of such states.
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.
Abstract Quantum nonlocality has recently been intensively studied in connection to device-independent quantum information processing, where the extremal points of the set of quantum correlations play a crucial role through self-testing. In most protocols, the proofs for self-testing rely on the maximal violation of the Bell inequalities, but there is another known proof based on the geometry of state vectors to self-test a maximally entangled state. We present a geometrical proof in the case of partially entangled states. We show that, when a set of correlators in the simplest Bell scenario satisfies a condition, the geometry of the state vectors is uniquely determined. The realization becomes self-testable when another unitary observable exists on the geometry. Applying this proven fact, we propose self-testing protocols by intentionally adding one more measurement. This geometrical scheme for self-testing is superior in that, by using this as a building block and repeatedly adding measurements, a realization with an arbitrary number of measurements can be self-tested. Besides the application, we also attempt to describe nonlocal correlations by guessing probabilities of distant measurement outcomes. In this description, the quantum set is also convex, and a large class of extremal points is identified by the uniqueness of the geometry.