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arXiv:2610.42017

Logical Bell Inequalities for Marginal Wigner Negativity

Sanchit Srivastava, Selman Ipek, Cihan Okay, Shohini Ghose

quant-phSubmitted Oct 30, 2026• Announced Nov 2, 2026
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Abstract

We construct logical Bell inequalities whose violations are determined by the Wigner function of a reduced quantum state. For odd-prime qudits, each phase-space point of any nonempty proper subsystem defines an inequality, violated precisely when the corresponding Gross Wigner value is negative. This correspondence allows us to quantify the contextuality that a reduced state guarantees for the composite system. The normalized violation gives a lower bound on the contextual fraction in the complete Weyl measurement scenario, valid for every joint state with that marginal, irrespective of its correlations. We identify a family of product states that saturates this bound. For these states, one marginal Wigner value fixes the contextual fraction of the entire system.

AI Summary

The paper asks how much contextuality of a composite odd-prime-qudit system can be certified from a reduced state's Wigner negativity, independently of correlations with its complement. It constructs logical Bell inequalities for every phase-space point $u$ of any nonempty proper subsystem $A$, of dimension $D_A=d^k$. A symplectic spread supplies $D_A+1$ Weyl settings on $A$, each paired with one fixed setting on the complement. The selected-event probability sum has noncontextual lower bound $1$ and quantum value $1+D_AW_{\rho_A}(u)$. Thus violation is equivalent to negativity of that marginal Wigner value and implies $\mathrm{CF}(e_\rho)\geq[-D_AW_{\rho_A}(u)]_+$ for the complete Weyl measurement scenario. The proof combines a spread-projector identity, the phase-space characterization of complete-scenario deterministic assignments, and convexity of contextual fraction. The bound is exact for product states whose $A$-marginal has Wigner value $-\delta/D_A$ at one point and equal positive values elsewhere, with a Wigner-nonnegative complementary state: their contextual fraction is $\delta$, for $0\leq\delta\leq1$. Separable constructions with identical prescribed marginals attain contextual fractions $\delta$ and $1$, showing how correlations can change the global resource. The selected contexts alone always admit noncontextual models; certification relies on valid-outcome constraints throughout the complete Weyl scenario. Subsystem size does not monotonically strengthen the bound. Tightness for general marginals and stronger joint use of multiple negative values remain open. The construction is restricted to odd-prime local dimensions.

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We construct logical Bell inequalities whose violations are determined by the Wigner function of a reduced quantum state. For odd-prime qudits, each phase-space point of any nonempty proper subsystem defines an inequality, violated precisely when the corresponding Gross Wigner value is negative. This correspondence allows us to quantify the contextuality that a reduced state guarantees for the composite system. The normalized violation gives a lower bound on the contextual fraction in the complete Weyl measurement scenario, valid for every joint state with that marginal, irrespective of its correlations. We identify a family of product states that saturates this bound. For these states, one marginal Wigner value fixes the contextual fraction of the entire system.
The paper asks how much contextuality of a composite odd-prime-qudit system can be certified from a reduced state's Wigner negativity, independently of correlations with its complement. It constructs logical Bell inequalities for every phase-space point uuu of any nonempty proper subsystem AAA, of dimension DA=dkD_A=d^kDA​=dk.
A symplectic spread supplies DA+1D_A+1DA​+1 Weyl settings on AAA, each paired with one fixed setting on the complement. The selected-event probability sum has noncontextual lower bound 111 and quantum value 1+DAWρA(u)1+D_AW_{\rho_A}(u)1+DA​WρA​​(u). Thus violation is equivalent to negativity of that marginal Wigner value and implies CF(eρ)≥[−DAWρA(u)]+\mathrm{CF}(e_\rho)\geq[-D_AW_{\rho_A}(u)]_+CF(eρ​)≥[−DA​WρA​​(u)]+​ for the complete Weyl measurement scenario. The proof combines a spread-projector identity, the phase-space characterization of complete-scenario deterministic assignments, and convexity of contextual fraction.
The bound is exact for product states whose AAA-marginal has Wigner value −δ/DA-\delta/D_A−δ/DA​ at one point and equal positive values elsewhere, with a Wigner-nonnegative complementary state: their contextual fraction is δ\deltaδ, for 0≤δ≤10\leq\delta\leq10≤δ≤1. Separable constructions with identical prescribed marginals attain contextual fractions δ\deltaδ and 111, showing how correlations can change the global resource.
The selected contexts alone always admit noncontextual models; certification relies on valid-outcome constraints throughout the complete Weyl scenario. Subsystem size does not monotonically strengthen the bound. Tightness for general marginals and stronger joint use of multiple negative values remain open. The construction is restricted to odd-prime local dimensions.
Abstract The relation between discrete Wigner function and quantum contextuality based on graph theory has been studied, following the work in [Nature 510,351(2014)]. To do this, non‐stabilizer projectors have been introduced to a series of non‐contextuality graphs based on stabilizer projectors for a single qudit with odd prime dimension. It has been found that, for a phase space point defined by Wootters, there exists a given set of states for an odd‐prime qudit where the negative discrete Wigner function on the phase space point means its quantum contextuality under measurements on the graphs designed by a specific method. To implement this method, a subset of non‐stabilizer projectors has been found. In the union of the set of states for all phase space points, there exists a negativity‐to‐violation map between Wigner function and quantum contextuality inequality. The robustness of the equivalence under depolarizing noise has been analyzed and discussed. For demonstration purposes, the graphs with different independence numbers and the corresponding set of states have been established on a single qutrit. Different to the cited work, this method involves only a single qudit, then is experimentally feasible for a qutrit.
Nonlocality is an essential concept that distinguishes quantum from classical models and has been extensively studied in systems of qubits. For higher-dimensional systems, certain results for their two-level counterpart, like Bell violations with stabilizer states and Clifford operators, do not generalize. On the other hand, similar to continuous variable systems, Wigner negativity is necessary for nonlocality in qudit systems. We propose a new generalization of the CHSH inequality for qudits by inquiring correlations related to the Wigner negativity of stabilizer states under the adjoint action of a generalization of the qubit π / 8 -gate. A specified stabilizer state maximally violates the inequality among all qudit states based on its Wigner negativity. The Bell operator not only serves as a measure for the singlet fraction but also quantifies the volume of Wigner negativity. Additionally, we show how a bipartite entangled qudit state can serve as a witness for contextuality when it exhibits Wigner negativity. Furthermore, we identify rational-phase diagonal unitaries as the key resource that exactly reproduce the CGLMP and SATWAP violation with the maximally entangled state through simple phase-difference alignment.
Negativity of the Wigner function is arguably one of the most striking non-classical features of quantum states. Beyond its fundamental relevance, it is also a necessary resource for quantum speedup with continuous variables. As quantum technologies emerge, the need to identify and characterize the resources which provide an advantage over existing classical technologies becomes more pressing. Here we derive witnesses for Wigner negativity of single mode and multimode quantum states, based on fidelities with Fock states, which can be reliably measured using standard detection setups. They possess a threshold expectation value indicating whether the measured state has a negative Wigner function. Moreover, the amount of violation provides an operational quantification of Wigner negativity. We phrase the problem of finding the threshold values for our witnesses as an infinite-dimensional linear optimisation. By relaxing and restricting the corresponding linear programs, we derive two hierarchies of semidefinite programs, which provide numerical sequences of increasingly tighter upper and lower bounds for the threshold values. We further show that both sequences converge to the threshold value. Moreover, our witnesses form a complete family – each Wigner negative state is detected by at least one witness – thus providing a reliable method for experimentally witnessing Wigner negativity of quantum states from few measurements. From a foundational perspective, our findings provide insights on the set of positive Wigner functions which still lacks a proper characterisation.
Understanding what distinguishes quantum mechanics from classical mechanics is crucial for quantum information processing applications. In this work, we consider two notions of non-classicality for quantum systems, negativity of the Wigner function and contextuality for Pauli measurements. We prove that these two notions are equivalent for multi-qudit systems with odd local dimension. For a single qudit, the equivalence breaks down. We show that there exist single qudit states that admit a non-contextual hidden variable model description and whose Wigner functions are negative.
In this work, we show that Bell's inequality violation of arise from the fact that the condition imposed upon the development of inequality is not respected when it is applied in the idealized experiment. Such a condition is that the quantities taken by probability must be non negative, and such a condition is represented by ∣P(w)∣=P(w)|\cal{P(w)}|=\cal{P(w)}∣P(w)∣=P(w). We will also show that, when trying to define the values of the joint probabilities of $ (Z_1, Z_2, Z_3) $, through the values obtained from the $ (Z_j, Z_k) $ pairs, we find that these values are negative, so not Kolmogorov's axiom is respected: P(w)≥0\cal{P(w)}\geq0P(w)≥0 in cases where Bell's inequality is violated, and we also show that only such violation is possible if Wigner's inequality, in a certain arrangement, is violated, and that both violations are related to the violation of one of Kolmogorov's axioms. At the end of the paper, we suggest a new interpretation of the probabilities involved, in order to avoid the situation of negative probabilities and the violation of Bell's inequality and, consequently, Wigner's inequality.
Characterizing the origin of quantum computational advantage amounts to identifying the key properties, or quantum resources, that distinguish quantum computers from their classical counterparts. Fault-tolerant computation, however, requires error-correcting codes to encode logical information into a larger, physical quantum system. This makes the task of resource identification more difficult, as what constitutes a resource from the logical and physical points of view can differ significantly. Here, we introduce a framework to correctly identify quantum resources in general encoded computations. For a given quantum code, our construction provides a Wigner function that accounts for how the symmetry of the code space is contained within the symmetry of the physical space, resulting in an object capable of describing the logical content of any physical state, both within and outside the code space. We illustrate our general construction with the Gottesman–Kitaev–Preskill encoding of qudits with odd dimension. The resulting Wigner function, which we call the Zak–Gross Wigner function, is shown to correctly identify quantum resources through its phase-space negativity. For instance, it is positive for encoded stabilizer states and negative for the bosonic vacuum. We further show that its negativity provides a measure of magic for the logical content of a state and that its marginals are modular measurement distributions associated with conjugate Zak patches.