Quantum mechanics reveals a reality where uncertainty is not merely epistemic but ontological, challenging classical logics that rely on determinacy. We introduce neutrosophic logic as a novel framework to formalise this intrinsic indeterminacy. Unlike probability or fuzzy systems, neutrosophy treats truth (T), falsity (F), and indeterminacy (I) as independent dimensions, enabling a richer representation of quantum superposition, entanglement, and measurement collapse. We show that neutrosophic triplets capture the undefined yet real state of qubits before observation and provide a coherent description of entangled correlations beyond probabilistic models. This approach offers a unifying formalism that accommodates both structural indeterminacy and outcome probabilities, bridging foundational debates in quantum theory with practical advances in quantum computation and information science. By explicitly integrating indeterminacy into mathematical modelling, neutrosophy advances our understanding of quantum reality and suggests new avenues for experimental validation and quantum technology design.
Quantum mechanics reveals a reality where uncertainty is not merely epistemic but ontological, challenging classical logics that rely on determinacy. We introduce neutrosophic logic as a novel framework to formalise this intrinsic indeterminacy. Unlike probability or fuzzy systems, neutrosophy treats truth (T), falsity (F), and indeterminacy (I) as independent dimensions, enabling a richer representation of quantum superposition, entanglement, and measurement collapse. We show that neutrosophic triplets capture the undefined yet real state of qubits before observation and provide a coherent description of entangled correlations beyond probabilistic models. This approach offers a unifying formalism that accommodates both structural indeterminacy and outcome probabilities, bridging foundational debates in quantum theory with practical advances in quantum computation and information science. By explicitly integrating indeterminacy into mathematical modelling, neutrosophy advances our understanding of quantum reality and suggests new avenues for experimental validation and quantum technology design.
Quantum computing exploits superposition and entanglement to perform computations beyond the capabilities of classical machines. However, practical quantum systems operate under substantial environmental noise, measurement uncertainty, and decoherence, leading to ambiguous or partially determined quantum states. Conventional probabilistic descriptions capture stochastic behaviour but do not explicitly represent indeterminacy arising from incomplete information or measurement ambiguity. In this work, we introduce a neutrosophic representation of qubit states that integrates principles of Neutrosophic Logic with the mathematical framework of Quantum Computing. In the proposed model, a qubit is characterised by three independent parameters representing truth, indeterminacy, and falsity. This formulation enables explicit representation of ambiguous quantum states and uncertain measurement outcomes. We extend the neutrosophic formalism to quantum gate operations and analyse how the model captures uncertainty propagation in quantum circuits. Through theoretical examples and graphical analysis, we demonstrate that neutrosophic representation provides a richer description of uncertainty compared with conventional probabilistic approaches. The proposed framework offers a conceptual pathway for improved modelling of noisy intermediate-scale quantum devices and may contribute to the development of uncertainty-aware quantum computing architectures.
Neutrosophic information processing extends conventional information processing by explicitly representing three logically independent components: truth, indeterminacy, and falsity. While neutrosophic models have been extensively developed at the mathematical level, their physical implementation remains comparatively underdeveloped. This paper investigates possible physical architectures for neutrosophic information processing, with particular emphasis on quantum systems. We distinguish three levels of implementation. At the first level, ordinary physical devices are used to represent neutrosophic information as a classical data structure. At the second level, physical systems such as photonic, spin, superconducting, atomic, or solid-state platforms encode the truth, indeterminacy, and falsity components in different physical degrees of freedom. At the third level, a genuinely quantum-neutrosophic processor is considered, in which quantum superposition, measurement uncertainty, noise, and explicitly represented indeterminacy are integrated into a single computational model. A mathematical framework based on neutrosophic triples (T, I, F) is introduced together with neutrosophic bits, density operators, measurement maps, and elementary transformation operators. We then discuss possible implementations using multilevel quantum systems, photonic polarization, superconducting circuits, trapped ions, and hybrid quantum-classical architectures. The paper emphasizes the distinction between physical uncertainty already present in quantum mechanics and an explicit logical indeterminacy variable. This distinction is essential for designing experimental systems capable of processing neutrosophic information. The paper concludes with a research program for experimental neutrosophic processors and identifies several open problems concerning noise, calibration, readout, error correction, and complexity.
We develop a mathematical framework for neutrosophic quantum channels by combining the operator-theoretic structure of quantum channels with single-valued neutrosophic information. A quantum channel is conventionally modeled as a completely positive tracepreserving (CPTP) linear map between operator spaces. We retain this physical core and augment it with a neutrosophic state N = (T, I, F), representing truth, indeterminacy, and falsity information associated with a quantum state or a channel description. We define neutrosophic quantum states, neutrosophic quantum channels, their composition, tensor products, Kraus representations, Choi representations, and complementary channels. Several structural results are proved, including closure under composition, a tensor-product construction, and a criterion for compatibility between the quantum and neutrosophic components. A distinction is made between the physical quantum channel and the additional neutrosophic information layer. In particular, an arbitrary neutrosophic scalar multiplying a density operator is not identified with a physical quantum operation. This distinction is essential for preserving positivity, trace normalization, and complete positivity. The resulting theory provides a rigorous mathematical framework for incorporating explicit truth, indeterminacy, and falsity information into quantum information models.
Quantum computation provides algorithms that exploit superposition, interference, and entanglement to solve certain computational problems more efficiently than known classical algorithms. Two fundamental examples are Shor’s factoring algorithm and Grover’s search algorithm. In practical computational settings, however, information associated with quantum states, oracle responses, and measurements may be uncertain or indeterminate. In this paper, we introduce a mathematical framework for neutrosophic quantum algorithms in which quantum states are supplemented by three independent information components representing truth, indeterminacy, and falsity. We define neutrosophic quantum states, neutrosophic quantum measurements, neutrosophic quantum oracles, and scalarization functions. Using these concepts, we formulate neutrosophic versions of Grover’s search algorithm and Shor’s factoring algorithm. For Grover’s algorithm, the classical binary oracle is generalized to an oracle whose response may carry truth, indeterminacy, and falsity degrees. For Shor’s algorithm, neutrosophic information is incorporated into modular exponentiation, quantum Fourier transformation, measurement, and period estimation. We prove that the proposed construction contains the ordinary quantum algorithms as a special case. We also discuss complexity and indeterminacy propagation. The proposed theory is an information-theoretic mathematical extension of quantum algorithms. It is not intended to modify the physical postulates of quantum mechanics.
Quantum search algorithms provide a quadratic speedup for a broad class of unstructured search problems. In practical settings, however, the classification of a candidate as “good” or “bad” may itself contain uncertainty, indeterminacy, or contradictory information. Neutrosophic sets provide a natural mathematical language for representing such information through independent truth, indeterminacy, and falsity degrees. In this paper we introduce a scalarization framework for neutrosophic quantum search. The central idea is to transform a neutrosophic evaluation N = (T, I, F) into a real-valued utility while preserving the distinction between truth, indeterminacy, and falsity. We introduce a parameterized family of weighted scalarization functions and establish conditions under which these functions are normalized, monotone, continuous, and order-preserving. We then define an optimization criterion for choosing scalarization parameters according to the requirements of a quantum search problem. A neutrosophic quantum oracle is modeled by assigning a neutrosophic evaluation to each computational basis state. The resulting scalarized oracle determines the effective marked-state amplitude. We derive an explicit relationship between scalarized utility and the number of amplitude-amplification iterations. Several scalarization functions, including linear, nonlinear, entropy-sensitive, and indeterminacy-penalized forms, are compared. A theoretical criterion based on discrimination, robustness, and indeterminacy sensitivity is proposed. The resulting framework provides a mathematically explicit bridge between neutrosophic decision theory and quantum amplitude amplification.
We develop a mathematical framework for neutrosophic amplitude amplification and establish convergence results for the associated iterative process. The classical amplitude amplification procedure provides a quadratic improvement over classical repetition for searching for a good state, and its mathematical structure is based on a two-dimensional invariant subspace generated by the good and bad components of a quantum state. We extend this framework by assigning to the success event three quantities representing truth, indeterminacy, and falsity. The resulting neutrosophic amplitude is represented by a triple (T, I, F), where the three components are not required to satisfy the classical probability normalization condition. We introduce neutrosophic success amplitudes, neutrosophic amplification operators, and several notions of convergence. In particular, we prove a convergence theorem for the truth component under contractive neutrosophic perturbations and establish corresponding bounds for the indeterminacy and falsity components. We also study the ideal case, the perturbed case, and the fixed-point regime. The results reduce to the ordinary amplitudeamplification formulas when I = F = 0 and T is identified with the ordinary success probability. Examples are given to illustrate the behavior of the three components during amplification.