{"id":"palomar/PALOMAR-2026-10-30-000005","url":"https://github.com/stalex444/hardy-ladder/tree/5764ec4a239fcb14f8de263372fc906bd217f349","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-30-000005-v1.json","repo":"https://github.com/stalex444/hardy-ladder","tags":["quant-ph","math.NT","math-ph","msc:81P40","msc:81P13","msc:11R16","msc:26C10"],"score":0,"title":"The two-qubit Hardy ladder: the closed form P_K*(q) and its unique maximiser for every K, Hardy's optimum at K = 1, and the optimum (8 rho^2 + rho - 11)/25 in the cubic field Q(rho) at K = 2","trust":"high","domain":"quant-ph","source":"palomar","authors":["Stephanie Alexander"],"reviews":0,"subject":"quant-ph","summary":"Kernel-checked optimum of Hardy's K-step ladder for two qubits (Hardy 1993; Hardy 1997; Boschi, Branca, De Martini and Hardy 1997), for a real Schmidt state c0|00> + c1|11> and real projective settings, the probabilities written out as squared real amplitudes. For every K: under the ladder constraints P(A_K = 1, B_K = 1) 0 and every large enough K some state and settings exceed 1/2 - eps (the \"(50 - delta)%\" of Hardy 1997 and Boschi et al. 1997). For every K >= 1 the maximiser of the closed form on (0, 1) is the unique zero there of Cereceda's polynomial m_K (Cereceda 2002, eq. (20), stated…","category":"research","comments":0,"disputed":false,"division":"physics","mentions":0,"verified":true,"discussed":0,"published":"2026-10-30","reactions":{},"formalized":true,"formalizes":[{"title":"Nonlocality for two particles without inequalities for almost all entangled states","identifier":"https://doi.org/10.1103/PhysRevLett.71.1665; Physical Review Letters 71 (1993) 1665-1668","relationship":"formalizes"},{"title":"Ladder Proof of Nonlocality without Inequalities: Theoretical and Experimental Results","identifier":"https://doi.org/10.1103/PhysRevLett.79.2755; Physical Review Letters 79 (1997) 2755-2758","relationship":"formalizes"},{"title":"Ladder proof of nonlocality for two spin-half particles revisited","identifier":"https://doi.org/10.1088/0305-4470/35/43/309; Journal of Physics A: Mathematical and General 35 (2002) 9105-9111; arXiv:quant-ph/0206015","relationship":"formalizes"},{"title":"A Bigger Contradiction Between Quantum Theory and Locality for Two Particles Without Inequalities","identifier":"https://doi.org/10.1007/978-94-011-5886-2_22; In: New Developments on Fundamental Problems in Quantum Physics (M. Ferrero, A. van der Merwe, eds.), Kluwer (1997), pp. 163-170","relationship":"formalizes"}]}