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arXiv:2503.13282 · npj Quantum Information

Quantum bounds and device-independent security with rank-one qubit measurements

Lorenzo Coccia, Matteo Padovan, Andrea Pompermaier, Mattia Sabatini, Marco Avesani, Davide G. Marangon, Paolo Villoresi, Giuseppe Vallone

Published Jan 21, 2026• 2 citations
arXivHTMLPDF

Abstract

Device-independent (DI) quantum protocols use Bell inequality violations to ensure security or certify quantum properties without assumptions on the devices’ internal workings. In this work, we study the role of rank-one qubit positive operator-valued measures (POVMs) in DI scenarios. This class includes all qubit extremal POVMs, i.e., those measurements that cannot be realized as mixtures of others, as well as part of non-extremal POVMs, recently shown to be useful in sequential quantum protocols. We demonstrate that any rank-one POVM can generate correlations in bipartite scenarios that saturate a Tsirelson inequality when two parties share an arbitrary entangled two-qubit state and perform specific self-tested measurements. For extremal POVMs, such saturation enables explicit computation of guessing probability and worst-case conditional von Neumann entropy. From the Tsirelson inequality, we establish a randomness certification method that facilitates numerical simulations and we validate it through a proof-of-concept experiment with three-outcome POVMs and tilted entangled states.

DOI
10.1038/s41534-025-01175-x

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Device-independent (DI) quantum protocols use Bell inequality violations to ensure security or certify quantum properties without assumptions on the devices’ internal workings. In this work, we study the role of rank-one qubit positive operator-valued measures (POVMs) in DI scenarios. This class includes all qubit extremal POVMs, i.e., those measurements that cannot be realized as mixtures of others, as well as part of non-extremal POVMs, recently shown to be useful in sequential quantum protocols. We demonstrate that any rank-one POVM can generate correlations in bipartite scenarios that saturate a Tsirelson inequality when two parties share an arbitrary entangled two-qubit state and perform specific self-tested measurements. For extremal POVMs, such saturation enables explicit computation of guessing probability and worst-case conditional von Neumann entropy. From the Tsirelson inequality, we establish a randomness certification method that facilitates numerical simulations and we validate it through a proof-of-concept experiment with three-outcome POVMs and tilted entangled states.
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In the device-independent scenario, positive operator-valued measurements (POVMs) can certify more randomness than projective measurements. This paper self-tests a three-outcome extremal qubit POVM in the X-Z plane of the Bloch sphere by achieving the maximal quantum violation of a newly constructed Bell expression C'3, adapted from the chained inequality C3. Using this POVM, approximately 1.58 bits of local randomness can be certified, which is the maximum amount of local randomness achievable by an extremal qubit POVM in this plane. Further modifications of C'3 produce C''3, enabling the self-testing of another three-outcome extremal qubit POVM. Together, these POVMs certify about 2.27 bits of global randomness. Both local and global randomness surpass the limitations certified from projective measurements. Additionally, the Navascués-Pironio-Acín hierarchy is employed to compare the lower bounds on global randomness certified by C3 and several other inequalities. As the extent of violation increases, C3 demonstrates superior performance in randomness certification.
Among certification techniques, those based on the violation of Bell inequalities are appealing because they do not require assumptions on the underlying Hilbert space dimension and on the accuracy of calibration methods. Such device-independent techniques have been proposed to certify the quality of entangled states, unitary operations, projective measurements following von Neumann's model and rank-one positive-operator-valued measures (POVM). Here, we show that they can be extended to the characterization of quantum instruments with post-measurement states that are not fully determined by the Kraus operators but also depend on input states. We provide concrete certification recipes that are robust to noise.
Device-independent quantum information is attracting significant attention, particularly for its applications in information security. This interest arises because the security of device-independent protocols relies solely on the observed outcomes of spatially separated measurements and the validity of quantum physics. Sequential scenarios, i.e., where measurements occur in a precise temporal order, have been proved to enhance performance of device-independent protocols in some specific cases by enabling the reuse of the same quantum state. In this work, we propose a systematic approach to designing sequential quantum protocols for device-independent security. Our method begins with a bipartite self-testing qubit protocol and transforms it into a sequential protocol by replacing one measurement with its non-projective counterpart and adding an additional user thereafter. We analytically prove that, with this systematic construction, the resulting ideal correlations are secure in the sense that they cannot be reproduced as a statistical mixture of other correlations, thereby enabling, for example, the device-independent certification of all the randomness present in the observed correlations. The general recipe we provide can be exploited for further development of new device-independent quantum schemes for security.
Multiparty quantum cryptography based on distributed entanglement will find its natural application in the upcoming quantum networks.The security of many multipartite device-independent (DI) protocols, such as DI conference-key agreement, relies on bounding the von Neumann entropy of the parties' outcomes conditioned on the eavesdropper's information, given the violation of a multipartite Bell inequality.We consider three parties testing the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality and certify the privacy of their outcomes by bounding the conditional entropy of a single party's outcome and the joint conditional entropy of two parties' outcomes.From the former bound, we show that genuine multipartite entanglement is necessary to certify the privacy of a party's outcome, while the latter significantly improves previous results.We obtain the entropy bounds thanks to two general results of independent interest.The first one drastically simplifies the quantum setup of an N -partite Bell scenario.The second one provides an upper bound on the violation of the MABK inequality by an arbitrary N -qubit state, as a function of the state's parameters.
Certification of quantum systems and their properties has become a field of intensive study. Here, taking advantage of the one-sided device-independent (1SDI) scenario (also known as the quantum steering scenario), we propose a self-testing scheme for all bipartite entangled states using a single family of steering inequalities with the minimal number of two measurements per party. Building on this scheme we then show how to certify all rank-one extremal measurements, including nonprojective d2{d}^{2}d2-outcome measurements, which in turn can be used for certification of the maximal amount of randomness, that is, 2log2d2{\mathrm{log}}_{2}d2log2​d bits. Finally, in the particular case of d=3d=3d=3, we propose an extended Bell scenario that transforms a 1SDI scheme to ``almost device independent.''