The Quantum Composition Paradox
Abstract
Quantum theory does not generally permit the probability laws obtained from individual unitary steps by the Born rule to be sewn into a consistent genealogy; we classify the exceptions and show that faithful composition can hold from an initial boundary yet fail after an internal restart. For finite-dimensional, composition-closed unitary families, universal composition holds exactly for unitary monomials, the phase-dressed permutations whose Born kernels realize reversible deterministic state machines. Prescribed sequences evade this obstruction. We classify all pairs of qubit unitary steps, give a necessary-and-sufficient criterion for pairs of qutrit unitary steps, and prove that a boundary-stable unitary sequence on a d-dimensional Hilbert space contains at most d fully mixing steps, with equality in every prime dimension. We also construct arbitrarily long genuinely mixing sequences that compose from their initial boundary but fail after an internal restart, and a qutrit-controlled two-qubit realization with active interference. We define a Born--Chapman--Kolmogorov current that vanishes exactly when coherent and stepwise-checked endpoint laws agree, together with an associated measure of how many bits the endpoint reveals about the intermediate checking schedule. This state-machine connection provides a foundation for a quantum theory of music, in which unitary operations are notes and temporal boundaries are cues. Musical-transition prediction is proved PromiseBQP-complete, and the stepwise patterns that preserve a genealogy specify rules for rhythm, whereas the exceptional failure of that genealogy from an internal cue is the quantum music paradox.
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BibTeX
@misc{biamonte2026quantum,
title = {{The Quantum Composition Paradox}},
author = {Jacob Biamonte},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11402},
eprint = {2609.11402},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11402},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Quantum Composition Paradox
AU - Jacob Biamonte
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11402
UR - https://arxiv.org/abs/2609.11402
AB - Quantum theory does not generally permit the probability laws obtained from individual unitary steps by the Born rule to be sewn into a consistent genealogy; we classify the exceptions and show that faithful composition can hold from an initial boundary yet fail after an internal restart. For finite-dimensional, composition-closed unitary families, universal composition holds exactly for unitary monomials, the phase-dressed permutations whose Born kernels realize reversible deterministic state machines. Prescribed sequences evade this obstruction. We classify all pairs of qubit unitary steps, give a necessary-and-sufficient criterion for pairs of qutrit unitary steps, and prove that a boundary-stable unitary sequence on a d-dimensional Hilbert space contains at most d fully mixing steps, with equality in every prime dimension. We also construct arbitrarily long genuinely mixing sequences that compose from their initial boundary but fail after an internal restart, and a qutrit-controlled two-qubit realization with active interference. We define a Born--Chapman--Kolmogorov current that vanishes exactly when coherent and stepwise-checked endpoint laws agree, together with an associated measure of how many bits the endpoint reveals about the intermediate checking schedule. This state-machine connection provides a foundation for a quantum theory of music, in which unitary operations are notes and temporal boundaries are cues. Musical-transition prediction is proved PromiseBQP-complete, and the stepwise patterns that preserve a genealogy specify rules for rhythm, whereas the exceptional failure of that genealogy from an internal cue is the quantum music paradox.
ER -