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.