An observable on a quantum structure is any
σ-homomorphism of quantum structures from the Borel
σ-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form
(−∞,t) is sufficient to determine uniquely the whole observable defined on quantum structures like
σ-MV-algebras,
σ-effect algebras, Boolean
σ-algebras, monotone
σ-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.
With high precision and sensitivity, quantum gyroscope has become a research hotspot of inertial technology in recent years. Because the Bose-Einstein Condensate (BEC) in atomic gas has more condensate atoms, atomic gas is pure and thin, and there is no friction in atoms, the matter–wave gyroscopy with counter-rotating vortex superpositions has great potential of higher sensitivity, higher precision, and miniaturization. The basic mechanism and sensitivity of the matter– wave gyroscopy with counter-rotating vortex superpositions were analyzed, and the interaction between superposition of vortex beams and BEC atomic gases were studied, it is Stimulated Raman Adiabatic Passage (STIRAP). In the study, the outstanding issue that the gyroscopy is difficult to obtain stable vortices with high order orbital angular momentum was clarified by theoretical derivation on the base of Gross-Pitaevskii equation which is the basic equation in BEC. For solving the above problems, the scheme which is easy to achieve based on Sombrero potential trap was proposed.