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On the Logical Foundation of Quantum Superposition: The Eigenstate Logical Qualification Criterion and the Invalidation Mechanism of Superposition States

shuanglin wang

Published Mar 18, 2026• 0 citations
Publisher
DOI
10.5281/zenodo.19096234

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Traditional formal logic is built upon classical bivalent foundations, featuring the law of excluded middle and the dichotomy of truth values. However, these classical frameworks struggle to capture the non-classical features of quantum information processing, such as superposition, entanglement, and coherence. To bridge the gap between quantum mechanics and formal logic, this paper introduces a novel logical framework termed Quantum Proof Logic (QPL). The core idea of QPL is to transcend the single-state limitation of traditional proofs by elevating proof states to state vectors within a complex Hilbert space. In this framework, a logical proof can exist in a linear superposition of multiple foundational proof paths, denoted as |Psi_proof> = sum_{k=1}^{K} c_k |s_k>, allowing the system to concurrently explore multiple logical hypotheses. We systematically establish QPL's axiomatic system—including the superposition axiom, unitary evolution derivation axiom, non-local entangled proof axiom, and measurement collapse axiom—alongside its algebraic semantics using density operators and logical entropy. Furthermore, we explore QPL's profound implications for the foundations of mathematics, including potential perspectives on Godel's incompleteness theorems, as well as its practical applications in fault-tolerant quantum error correction and the formal verification of quantum programs. QPL provides a rigorous mathematical foundation and a comprehensive logical tool for reasoning about complex quantum systems.

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Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences, and distinguishability, and is formalized using the distinctions (`dits') of a partition (a pair of points distinguished by the partition). All the definitions of simple, joint, conditional, and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this paper is to give the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., qudits of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates (cohered together in the pure superposition state being measured) that are distinguished by the measurement (decohered in the post-measurement mixed state). Both the classical and quantum versions of logical entropy have simple interpretations as "two-draw" probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states.

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Prequel to the Logical Interpretation of Quantum Mechanics: A Classical Logic Edition—How to Check, Via Experiment , Whether Schrödinger's Cat Is in Superposition Before Opening the Box

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This preprint introduces a novel thought experiment—two sealed double-slit apparatuses, one with which-path detectors and one without—to resolve Schrödinger's cat paradox using only classical logic and standard quantum formalism. The interference pattern (or lack thereof) on each screen is determined by the experimental operational setup, not by later observation. Macroscopic objects like the cat reside in a quantum mixed state, not a quantum superposition state—which must be a pure state—even in a closed box. These concepts form the foundation of the Logical Interpretation of Quantum Mechanics, a minimalist approach that preserves Aristotle's laws of thought, requires no supplementary postulates, and resolves paradoxes (measurement problem, EPR, cat) as linguistic artifacts. The interpretation aligns fully with quantum predictions and recognizes indistinguishability as the primary source of quantum behavior.

•Dec 31, 2025•0 citations
Traditional formal logic is built upon classical bivalent foundations, featuring the law of excluded middle and the dichotomy of truth values. However, these classical frameworks struggle to capture the non-classical features of quantum information processing, such as superposition, entanglement, and coherence. To bridge the gap between quantum mechanics and formal logic, this paper introduces a novel logical framework termed Quantum Proof Logic (QPL). The core idea of QPL is to transcend the single-state limitation of traditional proofs by elevating proof states to state vectors within a complex Hilbert space. In this framework, a logical proof can exist in a linear superposition of multiple foundational proof paths, denoted as |Psi_proof> = sum_{k=1}^{K} c_k |s_k>, allowing the system to concurrently explore multiple logical hypotheses. We systematically establish QPL's axiomatic system—including the superposition axiom, unitary evolution derivation axiom, non-local entangled proof axiom, and measurement collapse axiom—alongside its algebraic semantics using density operators and logical entropy. Furthermore, we explore QPL's profound implications for the foundations of mathematics, including potential perspectives on Godel's incompleteness theorems, as well as its practical applications in fault-tolerant quantum error correction and the formal verification of quantum programs. QPL provides a rigorous mathematical foundation and a comprehensive logical tool for reasoning about complex quantum systems.
Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences, and distinguishability, and is formalized using the distinctions (`dits') of a partition (a pair of points distinguished by the partition). All the definitions of simple, joint, conditional, and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this paper is to give the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., qudits of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates (cohered together in the pure superposition state being measured) that are distinguished by the measurement (decohered in the post-measurement mixed state). Both the classical and quantum versions of logical entropy have simple interpretations as "two-draw" probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states.
This preprint introduces a novel thought experiment—two sealed double-slit apparatuses, one with which-path detectors and one without—to resolve Schrödinger's cat paradox using only classical logic and standard quantum formalism. The interference pattern (or lack thereof) on each screen is determined by the experimental operational setup, not by later observation. Macroscopic objects like the cat reside in a quantum mixed state, not a quantum superposition state—which must be a pure state—even in a closed box. These concepts form the foundation of the Logical Interpretation of Quantum Mechanics, a minimalist approach that preserves Aristotle's laws of thought, requires no supplementary postulates, and resolves paradoxes (measurement problem, EPR, cat) as linguistic artifacts. The interpretation aligns fully with quantum predictions and recognizes indistinguishability as the primary source of quantum behavior.