The Inverse Born Rule Fallacy: On the Informational Limits of Phase-Locked Amplitude Encoding
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Quantum Physics arXiv:2602.21350 (quant-ph) [Submitted on 24 Feb 2026] Title:The Inverse Born Rule Fallacy: On the Informational Limits of Phase-Locked Amplitude Encoding Authors:Sebastian Zając, Jacob L. Cybulski, Bartosz Dziewit, Tomasz Kulpa View a PDF of the paper titled The Inverse Born Rule Fallacy: On the Informational Limits of Phase-Locked Amplitude Encoding, by Sebastian Zaj\k{a}c and 3 other authors View PDF HTML (experimental) Abstract:In Quantum Machine Learning (QML) and Quantum Finance, amplitude encoding is often motivated by its logarithmic storage capacity arXiv:1307.0411. This paradigm typically relies on the mapping $\psi = \sqrt{P}$, treating the quantum state as a derivative of a classical probability distribution $P$. By restricting the data manifold to the positive real orthant $\mathcal{S}^+$, the accessible Hilbert space is effectively abelianized, rendering the representation ``phase-deaf''. We rigorously establish that while $P$ is a projection of $|\psi|^2$, the simple square-root mapping fails to recover the non-commutative structure necessary for genuine quantum advantage in classification tasks. Furthermore, we clarify why applying basis changes (like Hadamard gates) to these states fails to replicate the computational power of active phase-kickback mechanisms. Finally, we advocate for Dynamical Hamiltonian Encoding (based on QIFT), where data generates non-commutative evolution rather than serving as a static, phase-locked vector. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2602.21350 [quant-ph] (or arXiv:2602.21350v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2602.21350 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Bartosz Dziewit [view email] [v1] Tue, 24 Feb 2026 20:37:13 UTC (9 KB) Full-text links: Access Paper: View a PDF of the paper titled The Inverse Born Rule Fallacy: On the Informational Limits of Phase-Locked Amplitude Encoding, by Sebastian Zaj\k{a}c and 3 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-02 References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)