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arXiv:2107.02937 · Physical Review A

Certification of incompatible measurements using quantum steering

Shubhayan Sarkar, Debashis Saha, Remigiusz Augusiak

Published Oct 21, 2022• 17 citations
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Abstract

The authors propose a simple one-sided device-independent scheme for certification of any set of genuinely incompatible projective measurements, including mutually unbiased bases, and the maximally entangled state of two qudits.
DOI
10.1103/physreva.106.l040402

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Recently, a different form of quantum steering, i.e., certification of quantum steering in a one-sided semi-device-independent way, has been formulated [C. Jebarathinam et al., Phys. Rev. A 108, 042211 (2023)]. In this work, we use this phenomenon to provide operational simultaneous correlations in mutually unbiased bases as quantified by the measures in [S. Wu et al., Sci. Rep. 4, 4036 (2014)]. First, we show that for any bipartite state, such measure of simultaneous correlations in two mutually unbiased bases can be operationally identified as exhibiting one-sided semi-device-independent steering in a two-setting scenario. Next, we demonstrate that for two-qubit Bell-diagonal states, quantifying one-sided semi-device-independent steerability provides an operational quantification of information-theoretic quantification of simultaneous correlations in mutually unbiased bases. Then, we provide a different classification of two-qubit separable states with the above-mentioned information-theoretic quantification of simultaneous correlations in mutually unbiased bases and the quantification of one-sided semi-device-independent steerability. Finally, we invoke quantum steering ellipsoid formalism to shed intuitions on the operational characterization of simultaneous correlations in complementary bases of two-qubit states via one-sided semi-device-independent steerability. This provides us with a geometric visualization of the results.
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.
Given two sets finite S0S_0S0​ and S1S_1S1​ of quantum states. We show necessary and sufficient conditions for distinguishing them by a measurement.