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Exact Attainability of the Svetlichny Maximum: Boundary Sufficiency, the κ-Criterion, and the Full-Family Proof for the Singly-Excited W -Family

Yong Jingxiong

Published Oct 28, 2026• 0 citations
Publisher

Abstract

We study when the maximal Svetlichny expectation S(ρ) of a three-qubit state ρattains the upper bound 4ν. Starting from the original objective, we eliminate Alice’stwo measurement directions exactly and reduce the optimization to an operator-normproblem for a complex matrix over legal measurement frames. This yields the universalbound S(ρ) ≤ 4ν, where ν is a state-dependent spectral quantity of the three-bodycorrelation tensor (the largest operator norm of its complex slice combinations), andthe identity maxF σmax(Z) = 2ν over all legal frames. The two ingredients of thebound—a complex-product encoding of the Svetlichny terms and a spectral norm ofthe correlation tensor—are well represented in the literature, and the bound is thereforeused here as the anchor of the attainability analysis rather than claimed as an isolatednovelty. The boundary condition νR = ν holds if and only if some boundary frameattains the global maximum of that spectral norm; under this condition an explicitmeasurement construction gives S(ρ) = 4ν, without assuming that the top singularvalue is simple. On a fixed frame, κ(s, d) = 0 decides whether the inner bound isattained; when ν > 0 and at least one S(ρ)-optimal frame has a positive simple topsingular value, the global criterion reads S(ρ) = 4ν if and only if there exists a framethat is both spectrally maximal and has κ = 0. The GHZ state shows that νR = ν isnot necessary. The same equality framework is then applied to the singly-excited Wstates: from the amplitudes a, b, c ≥ 0 we compute the three correlation slices termby term, reduce the maximization over complex directions to a two-variable functionF(p, r) depending only on p, r, and obtain the threshold polynomial E(A, B), withE(A, B) ≥ 0 if and only if ν = 1; written in the three two-body concurrences thesame polynomial is 1 − (C_212 + C_213 + C_223) − 2C_12 C_13 C_23, so the criterion is manifestlysymmetric under permutations of the three parties. In the violating region E < 0, ν isgiven as a radical closed form in the amplitudes, and the same maximizing point yieldssix real measurement directions that attain S(ρ) = 4ν. All closed forms and equalitydirections are confined to the singly-excited W family defined in this paper.

DOI
10.5281/zenodo.23690078

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We study when the maximal Svetlichny expectation S(ρ) of a three-qubit state ρattains the upper bound 4ν. Starting from the original objective, we eliminate Alice’stwo measurement directions exactly and reduce the optimization to an operator-normproblem for a complex matrix over legal measurement frames. This yields the universalbound S(ρ) ≤ 4ν, where ν is a state-dependent spectral quantity of the three-bodycorrelation tensor (the largest operator norm of its complex slice combinations), andthe identity maxF σmax(Z) = 2ν over all legal frames. The two ingredients of thebound—a complex-product encoding of the Svetlichny terms and a spectral norm ofthe correlation tensor—are well represented in the literature, and the bound is thereforeused here as the anchor of the attainability analysis rather than claimed as an isolatednovelty. The boundary condition νR = ν holds if and only if some boundary frameattains the global maximum of that spectral norm; under this condition an explicitmeasurement construction gives S(ρ) = 4ν, without assuming that the top singularvalue is simple. On a fixed frame, κ(s, d) = 0 decides whether the inner bound isattained; when ν > 0 and at least one S(ρ)-optimal frame has a positive simple topsingular value, the global criterion reads S(ρ) = 4ν if and only if there exists a framethat is both spectrally maximal and has κ = 0. The GHZ state shows that νR = ν isnot necessary. The same equality framework is then applied to the singly-excited Wstates: from the amplitudes a, b, c ≥ 0 we compute the three correlation slices termby term, reduce the maximization over complex directions to a two-variable functionF(p, r) depending only on p, r, and obtain the threshold polynomial E(A, B), withE(A, B) ≥ 0 if and only if ν = 1; written in the three two-body concurrences thesame polynomial is 1 − (C_212 + C_213 + C_223) − 2C_12 C_13 C_23, so the criterion is manifestlysymmetric under permutations of the three parties. In the violating region E < 0, ν isgiven as a radical closed form in the amplitudes, and the same maximizing point yieldssix real measurement directions that attain S(ρ) = 4ν. All closed forms and equalitydirections are confined to the singly-excited W family defined in this paper.
The violation of the Svetlichny's inequality (SI) [Phys. Rev. D 35, 3066 (1987)] is sufficient but not necessary for genuine tripartite nonlocal correlations. Here we quantify the relationship between tripartite entanglement and the maximum expectation value of the Svetlichny operator (which is bounded from above by the inequality) for the two inequivalent subclasses of pure three-qubit states: the Greenberger-Horne-Zeilinger (GHZ) class and the WWW class. We show that the maximum for the GHZ-class states reduces to Mermin's inequality [Phys. Rev. Lett. 65, 1838 (1990)] modulo a constant factor, and although it is a function of the three tangle and the residual concurrence, large numbers of states do not violate the inequality. We further show that by design SI is more suitable as a measure of genuine tripartite nonlocality between the three qubits in the WWW-class states, and the maximum is a certain function of the bipartite entanglement (the concurrence) of the three reduced states, and only when their sum attains a certain threshold value do they violate the inequality.
1. Exact planar GHZ nonlocality threshold for all n ≥ 4. We determine the sharp supremum of violations of full-correlation Bell inequalities by the n-qubit GHZ state under planar (XY) measurements, over arbitrary finite numbers of measurement settings. The result is the exact formula κₙ = (π/2)ⁿ ζ(n) / [2(1 + 2⁻ⁿ) ζ(2n)], n ≥ 4. In particular, κ₄ = 105/34, corresponding to the critical visibility v₄ = 34/105. The result replaces bounds for particular finite measurement configurations with an exact analytic expression for the stated class. 2. Substantially improved rigorous bounds for the open three-party GHZ problem. We establish 2.0349681535… ≤ κ₃ ≤ 105 ζ(3)/(2π³) = 2.0353294291… . Both bounds improve the previously published 2024 interval. The lower bound comes from a finite-harmonic Bell witness supported by 272 reproducible exact inequalities; the upper bound follows from an explicit square-wave/Möbius local model. The conjecture that the upper bound is attained remains open. The paper discusses an obstruction involving the Möbius function and zeros of the Riemann zeta function. 3. Sharp covariance inequalities under strong mixing. We establish the optimal coefficient 4√2 for bounded complex-valued observations, improving the classical bound 2π. More generally, the optimal Hilbert-space coefficient is 4K_G(k), where K_G(k) is the real Grothendieck constant of order k. For real scalar observations, the sharp coefficient is 4. 4. Sharp multilinear dependence inequalities. For n-linear forms that are real-valued on real inputs, we identify the optimal real and complex comparison factors 2ⁿ and 2ⁿ κₙ. These results strengthen classical estimates in a natural restricted class. They do not settle the more general problem for arbitrary complex multilinear forms considered by Bradley and Bryc (1985). 5. Asymptotically sharp growth of higher-order interaction measures. For the centred Fourier–Lancaster interaction functional, we derive explicit finite-dimensional bounds and establish L_d = 2ᵈ/(ed) · (1 + O(d⁻¹)). Thus, unlike the universal characteristic-function factorization defect, this higher-order interaction functional admits no dimension-independent bound. 6. Further structural results. The paper also develops restrictions to discrete phase groups, centred-interaction comparisons and quantitative links between universal Fourier factorization ceilings, strong mixing and multilinear dependence. Updated version, 18 October 2026 — two-PDF proof supplement. This record contains the original 14-page P04 main preprint (P04_CF_V6_S8.pdf), reproduced without alteration, and a 47-page English mathematical proof supplement (P04_K2_K7_R13_PROOF_SUPPLEMENT_V2.pdf). The supplement expands the analytical proof of the exact planar GHZ formula for n ≥ 4 and develops the three-party K7 lower bound using geometric reductions and 272 reproducible exact-rational inequalities. The exact three-party constant remains open.
We determine the least Greenberger-Horne-Zeilinger (GHZ) extractability compatible with a Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell score for every n≥3n\geq3n≥3. Extractability is the largest squared overlap with a GHZ state obtainable by separate local quantum channels. Between the biseparable bound and the quantum maximum, its exact minimum is the affine interpolation from one half to one. This proves the bound conjectured for six or more parties. The result holds for arbitrary binary measurements and density operators on tensor products of finite- or infinite-dimensional local spaces. We also determine the exact minimum for qubits, which is strictly higher at every interior score. One qutrit and qubits elsewhere attain the unrestricted minimum with measurements fixed as the score varies and, at interior scores, realize the smallest product of ranks of the local reduced states among states attaining that minimum. For n≥4n\geq4n≥4, the attaining state with this rank product is unique at each interior score, up to local isometries and party permutations. On one qutrit and qubits elsewhere, for n≥4n\geq4n≥4 and scores a fixed fraction away from both endpoints, a sufficiently small excess of extractability above the minimum controls trace-norm distance to the attaining state of the same score, up to local unitaries. The dependence is of square-root order, with optimal exponent and explicit constants depending only on that fraction, uniformly in nnn. Equality and stability concern actual extractability; identical Bell statistics can accompany different extractabilities.
Magic -- the resource enabling quantum computational advantage beyond stabilizer circuits -- has a clean phase-space characterization in odd prime dimensions that qubits notoriously lack. We study C(rho), the l_1 distance from a state's discrete Wigner function to the stabilizer polytope, and determine its exact geometry. For a single qubit with Bloch vector r we prove the closed form C = max(0, ||r||_1 - 1)/2, so the maximum is (sqrt(3)-1)/2, attained at the eight face states. For products of single-qubit states, C(rho_1 x ... x rho_n) <= prod_i (1 + C(rho_i)) - 1, with equality whenever every factor has Bloch sign product r_x r_y r_z <= 0; this includes the exact tensor powers ((1+sqrt(3))/2)^n - 1 and the exact maximum of C over fully separable n-qubit states. Because no discrete Wigner function for qubits is Clifford covariant, any such measure is frame dependent, and we identify which of its features are not. For a single qubit there are exactly two valid frames up to phase-space translation and C is identical on both. Pauli-covariant two-qubit frames number 2048 up to translation, and across them the ratio C(rho_Rx)/C(rho_Ry) takes the values 1/2, 1 and 2. In every one of these frames some stabilizer state has negative Wigner entries, a consequence of the Mermin-Peres square, so the two-qubit bound Gamma >= 1 + C/2 against the robustness of magic Gamma is frame independent. Within the Wootters frame we prove integer values 1, 2, 1 of the tightness ratio kappa := (Gamma-1)/C for three families in the two-qubit repetition codespace, with explicit certificates. For n-qubit GHZ phase states we prove Gamma = |sin phi| + |cos phi| for all n and C >= 2^(floor(n/2)-1) (|sin phi| + |cos phi| - 1), tight for n = 2 and, by exact linear programming, for n = 3, 4, so kappa decays exponentially. Finally, C is not a magic monotone, so it cannot replace Gamma in distillation bounds.
It is a challenging task to detect genuine multipartite nonlocality (GMNL). In this paper, the problem is considered via computing the maximal quantum value of Svetlichny operators for three-qubit systems and a tight upper bound is obtained. The constraints on the quantum states for the tightness of the bound are also presented. The approach enables us to give the necessary and sufficient conditions of violating the Svetlichny inequality (SI) for several quantum states, including the white and color noised Greenberger-Horne-Zeilinger (GHZ) states. The relation between the genuine multipartite entanglement concurrence and the maximal quantum value of the Svetlichny operators for mixed GHZ class states is also discussed. As the SI is useful for the investigation of GMNL, our results give an effective and operational method to detect the GMNL for three-qubit mixed states.
Identifying the nonlocality of a multiparty quantum state is an important task in quantum mechanics. Seevinck and Svetlichny [Phys. Rev. Lett. 89, 060401 (2002)], and independently, Collins and co-workers [Phys. Rev. Lett. 88, 170405 (2002)] have generalized the tripartite notion of Svetlichny nonlocality to n-parties. Here we have developed a tight upper bound for genuine four party Svetlichny type nonlocality. The constraints on the quantum states for the tightness of the bound are also presented. The method enables us to provide necessary and sufficient conditions for violating the four qubit Svetlichny type inequality for several quantum states. The relations between the genuine multipartite entanglement and the maximal quantum value of the Seevinck and Svetlichny operators for pure four qubit states are also discussed. Consequently, we have exhibited genuine four qubit hidden nonlocality under local filtering. Our result provides an effective and operational method for further study of multipartite quantum nonlocality.