No-cost Bell Nonlocality Certification from Quantum Tomography and Its Applications in Quantum Magic Witnessing

TL;DR

The paper proposes a constructive framework to derive XYZ Bell inequalities using only Pauli tomography, enabling no-cost certification of Bell nonlocality directly from standard tomographic data. By formulating a linear-programming approach, it generates inequalities tailored to specific $N$-qubit states and Pauli measurement subsets, and analyzes their robustness via critical visibilities. It connects these inequalities to quantum magic witnessing through stabilizer maxima and demonstrates both fixed-measurement and arbitrary-settings magic witnessing cases, supported by experimental data reanalysis of four-qubit systems. The work unifies state tomography with fundamental nonlocality tests, provides practical tools for reinterpreting archival data, and highlights the broader utility of tomography-driven Bell tests for quantum information tasks. Overall, it shows that no additional experimental cost is required to certify nonlocality and, in many cases, to witness quantum magic, making the approach attractive for fundamental studies and real-world quantum technologies.

Abstract

Tomographic measurements are the standard tool for characterizing quantum states, yet they are usually regarded only as means for state reconstruction or fidelity measurement. Here, we show that the same Pauli-basis measurements (X, Y, Z) can be directly employed for the certification of nonlocality at no additional experimental cost. Our framework allows any tomographic data - including archival datasets -- to be reinterpreted in terms of fundamental nonlocality tests. We introduce a generic, constructive method to generate tailored Bell inequalities and showcase their applicability to certify the non-locality of states in realistic experimental scenarios. Recognizing the stabilizer nature of the considered operators, we analyze our inequalities in the context of witnessing quantum magic - a crucial resource for quantum computing. Our approach requires Pauli measurements only and tests the quantum magic solely through the resources present in the state. Our results establish a universal standard that unifies state tomography with nonlocality certification and its application to quantum magic witnessing, thereby streamlining both fundamental studies and practical applications.

Figures (3)

• Figure 1: Can one use the tomography-type data to certify quantum resources at no additional cost? During many quantum experiments, an experimenter has to certify the state that is being prepared in a lab. One way of doing that is by performing a set of measurements in a standard basis. For $N$-qubit systems, this is usually done by the measurements of the Pauli observables. Then the state's fidelity or full tomography can be established. Here, we explore our capabilities of exploiting this data to directly violate Bell inequalities, not only infer Bell-type correlations, and to witness quantum magic at no additional cost with a single function combining the obtained data.
• Figure 2: Optimal critical visibility of the $|\psi(\alpha)\rangle$ states as a function of $\alpha$ (solid line and red points) for the XYZ Bell inequalities presented in this work together with the experimental violation factors (blue points). The coloured regions marked with Roman numbers correspond to different optimal inequalities which vary depending on the state's parameter. For region I, which encompasses the product of two-qubit Bell states and Dicke state $|\mathrm{D}^2_4\rangle$, the optimal inequality is presented in Table \ref{['tab:4_qubits']} under # 5. By further increasing $\alpha$, we switch to region II containing the four-qubit state $|\psi_4\rangle$ with the corresponding optimal inequality (\ref{['ineqpsi4']}). The last region labelled by III is governed by the Mermin inequality (see # 1 in Table \ref{['tab:4_qubits']}). For analytic expressions, see the main text. The right axis on the above plot denotes the violation factors $Q/L$. Experimental data for the chosen states gathered in PhysRevLett.117.210504PhysRevLett.107.080504Schmid_2010 are represented by the blue points, with the diameter representing the estimated error, and lead to a clear violation.
• Figure 3: Generic critical visibilities in XYZ and optimal Bell inequalities. Panel $a)$ shows the XYZ inequality's critical visibilities, and the optimal critical visibilities for $10^5$ pure Haar random states for $m=3$. Histograms of the marginal distributions are shown in blue on the corresponding sides of the plot. Results for the states previously studied in section \ref{['sec:four_qubits']} are denoted by red points. Panel $b)$ shows an estimated probability density function of a relative difference between the critical visibilities from the upper panel. It shows that for a generic state, the $v^{\mathrm{XYZ}}_{\mathrm{crit}}$ for Pauli measurements is on average $\approx 20\%$ worse than the $v^{opt}_{\mathrm{crit}}$. The loss of visibility with such a constrained class of measurement settings is surprisingly low.