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The Brown Joint Realizability Theorem (N = 3)

Matthew Rick Brown Brown

Published Sep 19, 2026• 0 citations
Publisher

Abstract

This paper solves the joint-realizability problem exactly for generic three-state (N=3) quantum probability histories. It proves that a generic 3\times3 unistochastic matrix has exactly two gauge-inequivalent unitary completions, shows that the reachable endpoint set is genuinely two-dimensional, and eliminates the hidden connecting phases to obtain a finite necessary-and-sufficient algebraic criterion for determining whether three probability matrices A,B,C can arise from one consistent unitary composition. The resulting decision procedure reduces the problem to at most four completion branches and degree-\leq8 polynomial equations, and its floating-point implementation was validated on 500 constructed-realizable and 500 unrelated triples with no errors in the reported tests. The theorem applies to the generic stratum; several nongeneric boundary and singular cases remain for a fully exhaustive N=3 decision procedure.

DOI
10.5281/zenodo.22849770

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This paper solves the joint-realizability problem exactly for generic three-state (N=3) quantum probability histories. It proves that a generic 3\times3 unistochastic matrix has exactly two gauge-inequivalent unitary completions, shows that the reachable endpoint set is genuinely two-dimensional, and eliminates the hidden connecting phases to obtain a finite necessary-and-sufficient algebraic criterion for determining whether three probability matrices A,B,C can arise from one consistent unitary composition. The resulting decision procedure reduces the problem to at most four completion branches and degree-\leq8 polynomial equations, and its floating-point implementation was validated on 500 constructed-realizable and 500 unrelated triples with no errors in the reported tests. The theorem applies to the generic stratum; several nongeneric boundary and singular cases remain for a fully exhaustive N=3 decision procedure.
The Brown Rules develops a framework for deciding whether a sequence of probability transitions can arise from one consistent underlying unitary history. The central idea is that individual probability tables can each be valid on their own while still being incompatible when required to describe successive stages of the same process. The paper identifies probability-only tests that can rule out incompatible histories, shows how hidden phase information constrains composition, and demonstrates that in important cases the set of compatible outcomes occupies only a restricted portion of the possible probability space. It also extends the framework to longer histories and outlines computational approaches for testing compatibility. The results apply specifically to finite-dimensional unitary models and provide consistency tests within that setting rather than proposing a modification of quantum mechanics.
We investigates the joint compatibility problem of the three pairwise concurrences, the three-tangle, and the Kempe invariant in arbitrary pure three-qubit states, and gives necessary and sufficient conditions for these five quantities to be simultaneously realizable by the same pure state. This five-dimensional compatibility region can be viewed as a fiber structure over the previously derived four-dimensional reachable region. We further find that the residual continuous local-unitary freedom depends on the position of the state within this region. In the generic interior, one continuous degree of freedom remains. It is fixed if the three-tangle vanishes or a pairwise concurrence becomes zero, and also at the boundary of the four-dimensional tangle region. The Kempe invariant gives the simplest algebraic parametrization of the fifth coordinate, but the same degree of freedom may be described by a known pure-state entanglement monotone. The compatibility conditions can also be expressed in terms of concurrence-based bipartite measures and multipartite quantities fixed by the pairwise concurrences and the three-tangle.
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 ddd-dimensional Hilbert space contains at most ddd 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.
We solve cyclic finite-dimensional quantum histories for arbitrary time-dependent unitary steps, without assuming that one step has finite order. The propagation Hamiltonian is a unitary connection Laplacian on a cycle; its complete gauge invariant is the monodromy M=UL−1⋯U0M=U_{L-1}\cdots U_0M=UL−1​⋯U0​. Its spectrum is λa,k=1−cos⁡((2πk−θa)/L)λ_{a,k}=1-\cos((2πk-θ_a)/L)λa,k​=1−cos((2πk−θa​)/L), where eiθa∈spec(M)e^{iθ_a}\in\mathrm{spec}(M)eiθa​∈spec(M). Thus the exact history sector is isomorphic to Fix(M)\mathrm{Fix}(M)Fix(M), while frustration, the gap above a nonempty zero-energy sector, the determinant, and the finite-temperature trace are obtained in closed form. Ordinary spectral data recover the multiset of monodromy phase cosines but not phase orientation; low energy certifies proximity to an exact relational history. We then define the predictive quotient of a sharp finite clock relative to an accessible operator system as the unique coarsest event alphabet preserving all conditional statistics on a history sector. A finite-error theorem shows that threshold clustering recovers this quotient when the minimum diamond separation of inequivalent event channels exceeds four times the estimation error, and proves an optimal record-count bound. With full matrix access and homogeneous step UUU, the minimal number of clock events is the projective order of UUU. We distinguish the normalizer of the clock algebra from transformations preserving the coherent history code and classify oriented exact sharp clock changes by U(r)×ZLU(r)\times\mathbb{Z}_LU(r)×ZL​ on a rank-rrr history sector; without orientation the cyclic factor becomes dihedral. Reversible changes of full-information clock fibers are necessarily unitary, so irreversible coarse-graining is not exact clock covariance. Minimal realizations of a complete history Gram kernel are uniquely unitarily equivalent, with a finite-data Procrustes bound. Independent finite-matrix code verifies the main results.
This paper investigates whether locally consistent quantum probability histories can always be combined into a single globally consistent unitary history. It demonstrates, through an explicit and independently verified counterexample in the simplest nontrivial setting, that every three-time subset of a multi-time history can be individually realizable even though no single sequence of unitary transformations can reproduce the complete collection of transition probabilities. The paper identifies shared connecting phases as the mechanism behind this local-to-global obstruction, develops a general parameter-counting condition for when global realization may be possible, and relates the result to broader quantum compatibility and marginal problems as well as the independently developed Quantum Composition Paradox.