Abstract
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.
Highlights
Nonlocality of quantum theory enables one, by performing incompatible measurements on entangled states, to create correlations not admitting a local-hidden-variable model [1]
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
Accepted in Quantum 2021-09-17, click title to verify instance, observing the maximal quantum violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality [9] uniquely certifies that the system under consideration contains the maximally entangled two-qubit state and that a pair of anticommuting qubit observables is embedded in the measurements performed [10, 11]
Summary
Let us start by briefly reviewing a Bell-type experiment. Consider a bipartite physical resource shared between two observers, called Alice and Bob. If P is generated by a local-hidden-variable model, the probabilities conditional on the hidden variables λ factorize as P (a, b|x, y, λ) = P (a|x, λ)P (b|y, λ) Such correlations are referred to as local correlations and their set, denoted by L, forms a polytope in R|A||B||X ||Y|, i.e., it is a bounded, convex set with finite number of extremal points. Q is a proper superset of L [1, 3] This means for any given quantum correlation P which is not local, there exists a hyperplane a,b,x,y βa,b,x,yP (a, b|x, y) = αL separating P and L, with βa,b,x,y being some real numbers. All the Bell inequalities considered in this work are those with αQ being strictly greater than αL
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