The presented paper introduces the complex probabilities (multi-models) being represented in spectral area (modulus and phases) with similar attributes as, for example, Fourier series. Each probabilistic model component is newly composed of a phase parameter that characterises the composition rules of universal models and must be quantised to fulfill the single-valuedness condition. The algorithm for identification of modulus and phase parameters is also presented together with its interpretation with respect to complementarity principle.
The presented paper introduces the complex probabilities (multi-models) being represented in spectral area (modulus and phases) with similar attributes as, for example, Fourier series. Each probabilistic model component is newly composed of a phase parameter that characterises the composition rules of universal models and must be quantised to fulfill the single-valuedness condition. The algorithm for identification of modulus and phase parameters is also presented together with its interpretation with respect to complementarity principle.
The paper presents the new approach to wave composition rules for advanced modeling of soft systems in quantum system theory.Firstly, the interpretation of phase parameters is given.The phase parameters are essential to specify the mathematical operations assigned to different relations among subsystems, e.g.co-operation, connection, co-existence, competition.Using wave composition rules, we are able to create more complex and sophisticated quantum circuits.We present the application of methodology on three illustrative examples.
Born's rule is the recipe for calculating probabilities from quantum mechanical amplitudes. There is no generally accepted derivation of Born's rule from first principles. In this paper, it is motivated from assumptions that link the ontological content of a proper physical model to the epistemic conditions of the experimental context. More precisely, it is assumed that all knowable distinctions should correspond to distinctions in a proper model. This principle of "ontological completeness" means, for example, that the probabilistic treatment of the double slit experiment with and without path information should differ. Further, it is assumed that the model should rely only on knowable ontological elements, and that failure to fulfill this principle of "ontological minimalism" gives rise to wrong predictions. Consequently, probabilities should be assigned only to observable experimental outcomes. Also, the method to calculate such probabilities should not rely on the existence of a precise path of the observed object if this path is not knowable. A similar principle was promoted by Born, even though he did not apply it to probability. Another crucial assumption is that the proper rule to calculate probabilities should be generally valid. It should be applicable in all experimental contexts, regardless the setup that determines which attributes of the studied object are observed, together with the probability to observe each of the associated attribute values. There is no need to refer to the Hilbert space structure of quantum mechanics in the present treatment. Rather, some elements of this structure emerge from the analysis.
In the paper is discussed complete probabilistic description of quantum systems with application to multiqubit quantum computations. In simplest case it is a set of probabilities of transitions to some fixed set of states. The probabilities in the set may be represented linearly via coefficients of density matrix and it is very similar with description using mixed states, but also may give some alternative view on specific properties of quantum circuits due to possibility of direct comparison with classical statistical paradigm.
This paper addresses the foundational question deferred by Paper VII: why does the multiplicity Omega of U-field microstates combine as the squared modulus of a complex amplitude? We argue that if the coarse-grained U-field microstate sum admits a path-integral representation over field configurations with phase weights exp(iS/hbar_eff), then a Complex Multiplicity Amplitude Z(Y) is associated with each outcome Y. Via a Constraint Surface Geometry argument, we show that the effective multiplicity must take the form Omega(Y) = |Z(Y)|^n by phase invariance and multiplicativity, and the empirically observed bilinear interference structure selects n=2, yielding Omega(Y) = |Z(Y)|^2. The Born rule P proportional to |psi|^2 then follows from the multiplicity principle P proportional to Omega together with the identification psi proportional to Z. Interference arises from linear addition of complex amplitudes before squaring; classical probability addition is recovered via decoherence. The outcome-selection rule (the Probabilistic Bias Functional B) is grounded in the ergodic/mixing dynamics of the Tier-1 lattice, conditional on mixing within the accessible phase-space sector. The construction of the path-integral measure from U-field microstate counting remains an open problem. This is the eighth paper in the 18-paper USSFT technical series; for the U-field Lagrangian foundations see Papers I-VI, for the quantum mechanics operator formalism see Paper VII (DOI: 10.5281/zenodo.19675374), and for spin structure and Bell correlations see Paper IX.
The aim of the paper is to seek and identify mathematical models designated to understand better the quantum scale phenomena, models which could permit to link them with the real world phenomena. This shall be made by analyzing the common features exhibited by both, normal scale and quantum scale phenomena and then, by creating mathematical models that are describing both, the quantum and the real world. One main idea of the paper, on which the involved models are built is that observed physical quantities such as position and momentum of a quantum particle, but also the laws of their transformation (of that physical quantities) from one state into another such as for example the probability wave function described by the Schrödinger’s equation, are signals transmitted out of the quantum world into the real physical world by the a certain real, not yet detected wave function, wave function, underlying the visible measurable reality and which is describing this more complex reality underlying the observable one, and unavailable not only to our limited senses but also not available, even though using the new discovered technologies. The two observed physical quantities, position and momentum, are describing a quantum system, that is to say the quantum system can be represented either as a wave function of the position of the particle or as a wave function of the momentum of the particle, this two different functions being linked one to another by means of the Fourier Transform. The paper is furthermore describing the way in which the theory of Signal Processing, a powerful and useful tool, is to be used in order to model both, quantum and large scale phenomena.
The paper presents the theory of wave probabilistic models together with their features that represent basic principles of quantum informatics. These features are mathematically described and the illustrative examples are shown to demonstrate the possible applications of the presented theory. The proposed methodology represents the original approach which comes from defined mathematical assumptions like single-valuedness, resonance principle, etc. From these simple assumptions a lot of principle of quantum informatics can be derived like quantization principle, spin, entanglement, etc.