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arXiv:2601.00824

Defect Cocycles and the Structure of Finite Process Monoids

Paolo Vella

Published Dec 23, 2025• 0 citations
arXivHTMLPDF

Abstract

We study positive subunital maps on ordered effect spaces and introduce the defect $d(T) = u - T(u)$, which satisfies a cocycle identity under composition. Using only this identity and elementary order-theoretic arguments -- requiring no spectral decomposition or dimension-dependent techniques -- we prove that in any finite composition-closed family of positive subunital maps, defects are eventually annihilated under iteration (Theorem 4.1), with an explicit bound linear in the family size. Under a persistence hypothesis (nonzero positive elements map to nonzero positive elements), we establish that all maps in such families must be unital. For completely positive maps on finite-dimensional matrix algebras, we then prove a sharp dimension-dependent bound: the stabilization index satisfies $n_T \le d$ where $d$ is the Hilbert space dimension, independent of the family size. This bound is achieved by a shift channel construction. These results provide a structural explanation for why finite operational repertoires in process theories cannot sustain systematic information loss, with applications to quantum foundations and categorical probability.

DOI
10.48550/arxiv.2601.00824

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We study positive subunital maps on ordered effect spaces and introduce the defect d(T)=u−T(u)d(T) = u - T(u)d(T)=u−T(u), which satisfies a cocycle identity under composition. Using only this identity and elementary order-theoretic arguments -- requiring no spectral decomposition or dimension-dependent techniques -- we prove that in any finite composition-closed family of positive subunital maps, defects are eventually annihilated under iteration (Theorem 4.1), with an explicit bound linear in the family size. Under a persistence hypothesis (nonzero positive elements map to nonzero positive elements), we establish that all maps in such families must be unital. For completely positive maps on finite-dimensional matrix algebras, we then prove a sharp dimension-dependent bound: the stabilization index satisfies nT≤dn_T \le dnT​≤d where ddd is the Hilbert space dimension, independent of the family size. This bound is achieved by a shift channel construction. These results provide a structural explanation for why finite operational repertoires in process theories cannot sustain systematic information loss, with applications to quantum foundations and categorical probability.
We study what happens when an exactly flat contextual operator-order transport is reducededgewise by a unital completely positive (UCP) map. The relevant obstruction is not nonunitarityby itself but failure of multiplicativity. For a UCP map Φ we use the standard multiplicativitydefect∆Φ(X, Y ) = Φ(XY ) − Φ(X)Φ(Y )and prove an exact decomposition of every reduced flat-loop defect into transported multiplicativity defects. The same defect obeys an exact composition law under nested reductions and hasa Stinespring leakage representation. For a uniformly continuous UCP semigroup Φt = etL, themultiplicativity defect satisfies an exact Duhamel formula, and for a bounded GKSL generatorthe corresponding Leibniz defect isΓL(X, Y ) = Xα[V∗α , X][Y, Vα].This yields an exact integral representation of reduced loop holonomy and explicit qubit braid-loopformulas for dephasing, depolarization, and amplitude damping.We then study the inverse problem. Closed-loop finite-channel data need not identify anarbitrary UCP channel, whereas the full infinitesimal Leibniz defect determines a boundedfinite-dimensional generator modulo Hamiltonian derivations. In Md(C) the dissipative quotienthas real dimension (d2 − 1)2, giving the face-count lower bound LCoxd ≥ ⌊d2/2⌋. Using blockextension, centered scalar-one tangent geometry, binder-compatible transversality, a transversenative-tilt argument, and a determinant-one odd-to-even transfer, we prove the matching upperbound. Thus, in the declared bounded finite-dimensional matrix-valued first-order Coxeter-facemeasurement model,LCoxd =d22, d ≥ 3.This sharp count is not a universal lower bound for arbitrary process tomography or arbitraryLindbladian-learning protocols. Conditioning, sample complexity, noise robustness, and finitetime channel reconstruction remain separate questions.
Can an external environment be absorbed universally into a quantum process? This article identifies trace normalization as the obstruction and tracks it through higher orders. Exact ranks are derived for sequential-comb normalization at every order. Two concrete families of higher-order maps are defined, and their transformation-space dimensions are computed. In the family stable under arbitrary auxiliary combs, adding an environment to an input has neither a left nor a right adjoint from order three onward. Together with the channel and superchannel cases, this yields an all-order family of no-go theorems. The claim is restricted to specified process types: broader normalized type systems may still admit internal homs. Without normalization, a two-sided adjunction returns for normal completely positive maps. Exactly the finite-dimensional Hilbert spaces are dualizable, but trace scaling and rank-one evaluation prevent restriction to deterministic channels. The continuous scalar laws underlying composition- and tensor-additive dynamics are proved to be logarithmic boundary terms. Nonadditive quadratic actions are analyzed through exact stationarity equations and several exact qubit phase classifications. Finally, continuous channel encoding below the full channel-space dimension has worst-case diamond-norm error at least 1/(dAdB ), while normalized weighted cups have sharp defect 1 − 1/δ2. The results identify three ways around the obstruction: enlarge the type universe, remove normalization, or accept controlled approximation
A well-known theorem of W. Arveson states that a completely positive (CP) map dominated (difference is CP) by a given CP map is described through a positive contraction in the commutant of the homomorphism of Stinespring representation. This contraction plays the role of Radon-Nikodym derivative of measure theory. We look at implications of this to one parameter semigroups of unital completely positive maps, known as quantum Markov semigroups. We focus our attention to quantum Markov semigroups of B(H), the algebra of all bounded operators on a Hilbert space. These semigroups can be dilated to semigroups of unital endomorphisms (E₀-semigroups) of B(K) for some other Hilbert space K. Here we see that dominated CP semigroups dilate to dominated semigroups of the dilation. Moreover at the level of Eo-semigroups the dominated CP semigroups are described through positive contractive local cocycles, which are really families of contractions of 'Radon-Nikodym derivatives' mentioned above. We analyze the structure for quantum semigroups with bounded generators.
In this paper we study quantum dynamical mappings called also quantum processes.The set of values of such mapping is a one-parameter family of completely positive trace-preserving linear operators defined on a finite-dimensional Hilbert space.In quantum information theory such operators are referred to as quantum channels.An important concept for quantum dynamical mappings is their divisibility.There are different types of this concept.The present paper deals with so-called completely positive divisible quantum processes.For two such processes, which are bijective and satisfy a commutativity condition, we construct a compound quantum process.It is shown that this compound quantum process is also completely positive divisible.Endowing a set of quantum channels with the norm topology, we consider continuous quantum processes and continuous completely positive evolutions.The latter are defined as two-parameter families of quantum channels satisfying additional properties.We prove that a continuous bijective completely positive divisible quantum process generates a continuous completely positive evolution.In order to illustrate the considered concepts and the results on them, we provide examples of quantum dynamical mappings with values in the set of qubit channels.In particular, a completely positive divisible compound quantum process is constructed for two bijective commuting quantum processes.Geometric and physical interpretations of this compound quantum process are given.
Abstract. We construct explicit bistochastic entanglement-breaking maps on Mn whose completely bounded idempotence defects tend to zero, while their distances from all unital completely positive idempotents on Mn have lower bounds tending to one. This answers negatively Kitaev’s question on dimension-independent approximation by idempotent unital completely positive maps. The construction uses a regular simplex with n + 1 vertices, and the defect is exactly 2(n2 − n− 1)/n3. The obstruction persists under arbitrary finite matrix amplifications, with an asymptotic lower bound of 1/4 uniform in the auxiliary dimension. The proof combines a rank obstruction with the Choi–Effros theorem and an elementary rigidity result: two finite-dimensional C∗-algebras admitting unital completely positive approximate inverses with completely bounded error below 1/4 are ∗-isomorphic.
抄録 Unital completely positive maps preserve effects but need not preserve projections or commutative observable algebras. We systematize a sharp-context calculus around this obstruction. For composable maps we derive an exact kernel-corrected formula for the multiplicative domain of the composite. It yields the maximal partial transport among contexts on which the first map is multiplicative, and identifies how a later coarse-graining can erase an intermediate sharpness defect. In finite dimension every strict composition defect is detected by a binary context generated by one projection. Unitary detection extends to arbitrary commutative C*-contexts, while projection witnesses persist for von Neumann contexts. For normal maps, the von Neumann context transport is Scott-continuous on a directed-complete domain. We lift the partial transport to spans of covariant presheaf topoi carrying tautological internal commutative algebras and constructive Gelfand spectra. The lift is an identity-normal oplax functor into a bicategory of presented spans; its comparison cells record exactly the same sequential-domain inclusion as the operator-algebraic transport. These results give a precise separation between effects, which transport globally, and sharp classical contexts, which transport only on the multiplicative domain. They also make apparent recovery under nonfaithful coarse-graining visible at the level of contextual semantics. AI利用の開示 OpenAI Codex assisted with manuscript inventory, bibliographic extraction and preparation of this deposit. The uploaded PDF is an unchanged copy of the identified author manuscript. Authorship-stage AI assistance, where stated by the author, is described in the manuscript. No AI system is credited as an author. 検証範囲(Lean で形式検証した範囲と未検証部分) This deposit does not establish proof-assistant verification of the full manuscript. No new Lean or other proof-assistant verification, independent mathematical review, or empirical study was performed as part of this deposit preparation. Mathematical arguments, computational checks, educational design claims and any unresolved issues retain the scope stated in the manuscript; DOI assignment does not certify them.