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

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.