Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.13951

Predictive Holonomy and Minimal Completion of Coarse Dynamics

Maxim V. Churilov

quant-phmath-phSubmitted Aug 7, 2026• Announced Aug 10, 2026
arXivHTMLPDF

Abstract

A coarse description of a dynamical system generally fails to evolve autonomously: evolving and coarse-graining in different orders gives different predictions, and the compressed propagator fails the semigroup law. We show that this path dependence is the observable boundary of missing state. For a finite continuous-time Markov chain with generator $L$, coarse channel $C$, and stochastic section $R$ satisfying $CR=I$, define $P=RC$, $Q=I-P$, and $K=CLR$. The exact coarse propagator $T_t=C\mathrm{e}^{tL}R$ obeys a Volterra equation with memory kernel $\mathcal{M}(t)=CLQ\,\mathrm{e}^{tQLQ}\,QLR$. Every memory jet $\mathcal{M}^{(k)}(0)$ is recursively recoverable from the short-time derivatives of $T_t$ alone. The rank of the block Hankel operator $\mathbb{H}_{\infty}=(\mathcal{M}^{(i+j)}(0))_{i,j\geq 0}$ is exactly the smallest number of auxiliary linear state variables required to turn the coarse dynamics into an exact autonomous, time-homogeneous system. A minimal completion is constructible by Ho--Kalman factorization and is unique up to similarity. If $d=\min\{k:\mathcal{M}^{(k)}(0)\neq 0\}$, then every smooth classical $f$-divergence generically scales as $t^{2d+4}$. Exact Markov examples realize exponents $4,6,8$ and closure ranks $1,2,3$. We also derive finite-sample singular-value certificates for hidden predictive coordinates, including a matrix-Bernstein bound with intrinsic-variance control. Finally, histories are identified when they induce the same future law under every admissible intervention, yielding the minimal counterfactual predictive state. The results turn coarse-graining into a constructive state-completion problem and separate ordinary linear realization from the harder requirement of a positive or Markov realization.

Comments
13 pages, 3 figures; ancillary symbolic and numerical verification code included

Related papers

Finite Quantum Histories: Holonomy Spectra, Minimal Clocks, and Exact Clock-Change Covariance

Maxim V. Churilov

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 $M=U_{L-1}\cdots U_0$. Its spectrum is $λ_{a,k}=1-\cos((2πk-θ_a)/L)$, where $e^{iθ_a}\in\mathrm{spec}(M)$. Thus the exact history sector is isomorphic to $\mathrm{Fix}(M)$, 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 $U$, the minimal number of clock events is the projective order of $U$. We distinguish the normalizer of the clock algebra from transformations preserving the coherent history code and classify oriented exact sharp clock changes by $U(r)\times\mathbb{Z}_L$ on a rank-$r$ 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.

quant-pharXiv:2608.05748•Aug 7, 2026•0 citations

Path Dependence and Confluence in Quantum Coarse-Graining

Yoshiki Ueoka

Does a sequence of quantum coarse-grainings depend only on its initial and final resolutions, or can the intermediate route remain operationally observable? We study finite-dimensional closed systems whose coarse descriptions are partitions of a fixed orthogonal fine decomposition. Each partition defines a non-selective L\"uders channel, and a monotone coarse-graining path alternates unitary evolution with nested L\"uders maps. Although all projectors commute at equal time, we exhibit a four-state example in which two paths with identical endpoints give final coarse probabilities $3/8$ and $5/8$. We show that arbitrary saturated merge paths are connected by two rank-two local moves, so local channel equality is sufficient for global monotone confluence. For each local comparison we derive an all-time necessary-and-sufficient microscopic criterion: the path defect vanishes exactly when Hamiltonian evolution never generates a coherence sector selected by the difference of the intermediate L\"uders maps. Equivalently, no local source block can dynamically feed both ends of a selected coherence pair. The first nonzero order is determined by a two-arm Hamiltonian--Krylov distance. Finally, we distinguish hereditary confluence from fixed-source confluence, where a nonzero local defect can remain inaccessible from the chosen global source. These results characterize when the history of changing coarse resolutions may be removed from an endpoint-based operational description.

•Sep 12, 2026•0 citations

The PIHR Master Theorem: Minimal Realizations and Finite Recurrence of Recurrent Physical Architectures From Weighted Physical Histories to Canonical Structural Equivalence

Borros Arneth

Many physical systems generate an arbitrarily large or infinite family of histories from a finite set of elementary processes. Multiple scattering, repeated reflections, Green-function insertions, network traversals, resonant tunneling, and repeated loop windings are familiar examples. This work develops a structural description of such systems based on propagation, interaction, holonomy or closed-path monodromy, and recursive closure, collectively denoted PIHR. The central step is to replace a particular factorization of the physical equations by the complete weighted behaviour of their recurrent histories. Primitive return processes generate words in a history alphabet, and the corresponding physical amplitudes define a noncommutative weighted behaviour series. Its Hankel matrix encodes the response of every admissible past to every admissible continuation. Established realization theory then implies that finite Hankel rank is equivalent to the existence of a finite-dimensional linear realization, that the minimal realization dimension equals this rank, and that minimal realizations are unique up to similarity. These classical results are formulated here as a PIHR Master Theorem for recurrent physical architectures. A further consequence follows from the Cayley–Hamilton theorem: recursion-depth amplitudes generated by a finite minimal return kernel necessarily satisfy a finite linear recurrence. Thus an exponentially proliferating family of physical histories can possess finite architectural memory. A three-arm quantum network provides an explicit example in which the number of histories grows by a factor of nine at each return while the coherent amplitude sequence is governed by a recurrence of order at most three. Structural equivalence of minimal PIHR realizations yields a corresponding transfer principle for realization dimension, kernel spectra, recurrence laws, and boundary generating functions. PIHR is therefore proposed not as a replacement for established physical theories, but as a framework for identifying when infinite recurrent physical behaviour is generated by a finite, canonical internal architecture.

•Aug 18, 2026•0 citations

META–CAUSAL: The Geometry of Predictive Memory and the Limits of Reconstructibility

Boris Batenin, Andrei Preece

META–CAUSAL: The Geometry of Predictive Memory and the Limits of Reconstructibility investigates a fundamental question: how much internal memory is required to reproduce an observable process, and how much of its hidden structure can be recovered from measurements? The monograph connects predictive memory, spectral invariants, and operational reconstruction across classical stochastic and quantum models. Its central perspective is that a simple observed signal need not admit a simple internal realization. Memory requirements depend on positivity, probability normalization, accessible measurements, and the distinction between exact reproduction and approximation. A principal result turns the approximation of a periodic spike into an exact fractional matching problem on a divisor hypergraph. For a spike of amplitude b, with 0 < b ≤ 1, repeating every n steps, the optimal uniform error of an m-state homogeneous Markov model, where 1 ≤ m < n, is Eₘ = b/(1 + νₘ). Here νₘ is the largest fractional matching mass attainable within the available cycle budget. This correspondence produces explicit stochastic constructions and matching lower bounds, revealing a staircase of memory thresholds. For period 210, the exact values include E₁₀₀ = 2b/7 and E₁₀₇ = b/4; period 1050 exhibits an additional plateau at 3b/8. A universal rational polytope organizes the possible error levels through determinant formulas, while a structural analysis identifies the final three thresholds for an arbitrary number of distinct prime divisors without enumerating the polytope’s vertices. The broader investigation examines what observations preserve, what they conceal, and what remains certifiable under noise. Spectral witnesses, positive realizations, finite-sample topological certificates, and geometric models involving S³, Hopf structures, and Berger metrics address complementary aspects of this problem. In open quantum systems, the distinction between a damping gap and a purity gap separates dynamical relaxation from the stability of stationary-state topology. The resulting framework combines exact mathematical statements, rational computational certificates, and explicitly conditional physical models. Its contribution is a concrete account of how algebraic, combinatorial, and geometric structure governs both the cost of prediction and the limits of reconstruction.

•Oct 15, 2026•0 citations

Intervention-Relative Predictive Memory

Alexandre Dumas

This preprint develops a framework for intervention-relative predictive memory: the minimum information that must be retained about the past in order to predict every future experiment allowed by a declared intervention set.For finite-dimensional linear processes, predictive memory is characterized exactly by a reachable-observable quotient and the rank of the corresponding past-future Hankel response matrix. The paper derives finite certification procedures, monotonicity under richer control, a decomposition of memory growth into reachability expansion and visibility activation, symmetry-protected compression, and no-compression results under sufficiently rich intervention algebras.The framework is connected to process-tensor bond rank, hidden-memory realization, and physical memory costs. At a prepare-and-measure temporal cut, linear, classical, and quantum memory are distinguished through ordinary rank, nonnegative rank, and positive-semidefinite rank. A robust quantum-dimension witness is derived for an n-message decoding task.A proposed single-photon spatial-mode experiment provides a concrete test. Phase-coded histories have identical present path-intensity distributions but can be perfectly distinguished by permitted future phase and Fourier operations, giving an ideal response matrix equal to the identity. For 17 modes, sufficiently high decoding success directly certifies a 17-dimensional quantum memory.The paper distinguishes established realization, automata, process-tensor, quantum-control, and factorization results from the proposed synthesis and experimental interpretation. It concludes with two natural next directions: experimental implementation and a multitime completely-positive / PSD-rank theory of intervention-relative quantum memory.

•Sep 29, 2026•0 citations

Coarse-Graining and Long-Range Response in Generated Systems

Jinku Guo

We study when a reduced description preserves the response to an applied source. For systems defined by continuously differentiable state updates and observations a bounded surjective linear coarse map gives differentiable descent when the projected maps are constant on each fibre. Their common source response is then preserved on the invertible domain. For approximate descent an exact error identity separates defects in the update and source coupling from defects in the readout and direct term. Resolvent and contour estimates carry these errors and the displacement of the reference branch to visible poles and their residues at shifted critical roots. In a reciprocal Hilbert-space realisation an isolated finite-dimensional critical subspace with a positive crossing matrix has a positive-semidefinite finite-rank residue when the endpoint operator is self-adjoint and the complementary inverse is bounded. Source overlap determines its visible directions. Arithmetic and reflecting-heat examples separate stationary response preservation from memoryless dynamical closure. In the two-mode example pole displacement decays geometrically with the retained memory length. The residue error includes an additional factor of that length. For a matched vacuum covariance the residue is the zero-mass atom of its positive spectral measure. This identification uses the same physical projection and source pairing together with the continuation and subtraction conventions and endpoint matching.

hep-thCross-listedarXiv:2608.01989•Aug 4, 2026
A coarse description of a dynamical system generally fails to evolve autonomously: evolving and coarse-graining in different orders gives different predictions, and the compressed propagator fails the semigroup law. We show that this path dependence is the observable boundary of missing state. For a finite continuous-time Markov chain with generator LLL, coarse channel CCC, and stochastic section RRR satisfying CR=ICR=ICR=I, define P=RCP=RCP=RC, Q=I−PQ=I-PQ=I−P, and K=CLRK=CLRK=CLR. The exact coarse propagator Tt=CetLRT_t=C\mathrm{e}^{tL}RTt​=CetLR obeys a Volterra equation with memory kernel M(t)=CLQ etQLQ QLR\mathcal{M}(t)=CLQ\,\mathrm{e}^{tQLQ}\,QLRM(t)=CLQetQLQQLR. Every memory jet M(k)(0)\mathcal{M}^{(k)}(0)M(k)(0) is recursively recoverable from the short-time derivatives of TtT_tTt​ alone. The rank of the block Hankel operator H∞=(M(i+j)(0))i,j≥0\mathbb{H}_{\infty}=(\mathcal{M}^{(i+j)}(0))_{i,j\geq 0}H∞​=(M(i+j)(0))i,j≥0​ is exactly the smallest number of auxiliary linear state variables required to turn the coarse dynamics into an exact autonomous, time-homogeneous system. A minimal completion is constructible by Ho--Kalman factorization and is unique up to similarity. If d=min⁡{k:M(k)(0)≠0}d=\min\{k:\mathcal{M}^{(k)}(0)\neq 0\}d=min{k:M(k)(0)=0}, then every smooth classical fff-divergence generically scales as t2d+4t^{2d+4}t2d+4. Exact Markov examples realize exponents 4,6,84,6,84,6,8 and closure ranks 1,2,31,2,31,2,3. We also derive finite-sample singular-value certificates for hidden predictive coordinates, including a matrix-Bernstein bound with intrinsic-variance control. Finally, histories are identified when they induce the same future law under every admissible intervention, yielding the minimal counterfactual predictive state. The results turn coarse-graining into a constructive state-completion problem and separate ordinary linear realization from the harder requirement of a positive or Markov realization.
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.
Does a sequence of quantum coarse-grainings depend only on its initial and final resolutions, or can the intermediate route remain operationally observable? We study finite-dimensional closed systems whose coarse descriptions are partitions of a fixed orthogonal fine decomposition. Each partition defines a non-selective L\"uders channel, and a monotone coarse-graining path alternates unitary evolution with nested L\"uders maps. Although all projectors commute at equal time, we exhibit a four-state example in which two paths with identical endpoints give final coarse probabilities 3/83/83/8 and 5/85/85/8. We show that arbitrary saturated merge paths are connected by two rank-two local moves, so local channel equality is sufficient for global monotone confluence. For each local comparison we derive an all-time necessary-and-sufficient microscopic criterion: the path defect vanishes exactly when Hamiltonian evolution never generates a coherence sector selected by the difference of the intermediate L\"uders maps. Equivalently, no local source block can dynamically feed both ends of a selected coherence pair. The first nonzero order is determined by a two-arm Hamiltonian--Krylov distance. Finally, we distinguish hereditary confluence from fixed-source confluence, where a nonzero local defect can remain inaccessible from the chosen global source. These results characterize when the history of changing coarse resolutions may be removed from an endpoint-based operational description.
Many physical systems generate an arbitrarily large or infinite family of histories from a finite set of elementary processes. Multiple scattering, repeated reflections, Green-function insertions, network traversals, resonant tunneling, and repeated loop windings are familiar examples. This work develops a structural description of such systems based on propagation, interaction, holonomy or closed-path monodromy, and recursive closure, collectively denoted PIHR. The central step is to replace a particular factorization of the physical equations by the complete weighted behaviour of their recurrent histories. Primitive return processes generate words in a history alphabet, and the corresponding physical amplitudes define a noncommutative weighted behaviour series. Its Hankel matrix encodes the response of every admissible past to every admissible continuation. Established realization theory then implies that finite Hankel rank is equivalent to the existence of a finite-dimensional linear realization, that the minimal realization dimension equals this rank, and that minimal realizations are unique up to similarity. These classical results are formulated here as a PIHR Master Theorem for recurrent physical architectures. A further consequence follows from the Cayley–Hamilton theorem: recursion-depth amplitudes generated by a finite minimal return kernel necessarily satisfy a finite linear recurrence. Thus an exponentially proliferating family of physical histories can possess finite architectural memory. A three-arm quantum network provides an explicit example in which the number of histories grows by a factor of nine at each return while the coherent amplitude sequence is governed by a recurrence of order at most three. Structural equivalence of minimal PIHR realizations yields a corresponding transfer principle for realization dimension, kernel spectra, recurrence laws, and boundary generating functions. PIHR is therefore proposed not as a replacement for established physical theories, but as a framework for identifying when infinite recurrent physical behaviour is generated by a finite, canonical internal architecture.
META–CAUSAL: The Geometry of Predictive Memory and the Limits of Reconstructibility investigates a fundamental question: how much internal memory is required to reproduce an observable process, and how much of its hidden structure can be recovered from measurements? The monograph connects predictive memory, spectral invariants, and operational reconstruction across classical stochastic and quantum models. Its central perspective is that a simple observed signal need not admit a simple internal realization. Memory requirements depend on positivity, probability normalization, accessible measurements, and the distinction between exact reproduction and approximation. A principal result turns the approximation of a periodic spike into an exact fractional matching problem on a divisor hypergraph. For a spike of amplitude b, with 0 < b ≤ 1, repeating every n steps, the optimal uniform error of an m-state homogeneous Markov model, where 1 ≤ m < n, is Eₘ = b/(1 + νₘ). Here νₘ is the largest fractional matching mass attainable within the available cycle budget. This correspondence produces explicit stochastic constructions and matching lower bounds, revealing a staircase of memory thresholds. For period 210, the exact values include E₁₀₀ = 2b/7 and E₁₀₇ = b/4; period 1050 exhibits an additional plateau at 3b/8. A universal rational polytope organizes the possible error levels through determinant formulas, while a structural analysis identifies the final three thresholds for an arbitrary number of distinct prime divisors without enumerating the polytope’s vertices. The broader investigation examines what observations preserve, what they conceal, and what remains certifiable under noise. Spectral witnesses, positive realizations, finite-sample topological certificates, and geometric models involving S³, Hopf structures, and Berger metrics address complementary aspects of this problem. In open quantum systems, the distinction between a damping gap and a purity gap separates dynamical relaxation from the stability of stationary-state topology. The resulting framework combines exact mathematical statements, rational computational certificates, and explicitly conditional physical models. Its contribution is a concrete account of how algebraic, combinatorial, and geometric structure governs both the cost of prediction and the limits of reconstruction.
This preprint develops a framework for intervention-relative predictive memory: the minimum information that must be retained about the past in order to predict every future experiment allowed by a declared intervention set.For finite-dimensional linear processes, predictive memory is characterized exactly by a reachable-observable quotient and the rank of the corresponding past-future Hankel response matrix. The paper derives finite certification procedures, monotonicity under richer control, a decomposition of memory growth into reachability expansion and visibility activation, symmetry-protected compression, and no-compression results under sufficiently rich intervention algebras.The framework is connected to process-tensor bond rank, hidden-memory realization, and physical memory costs. At a prepare-and-measure temporal cut, linear, classical, and quantum memory are distinguished through ordinary rank, nonnegative rank, and positive-semidefinite rank. A robust quantum-dimension witness is derived for an n-message decoding task.A proposed single-photon spatial-mode experiment provides a concrete test. Phase-coded histories have identical present path-intensity distributions but can be perfectly distinguished by permitted future phase and Fourier operations, giving an ideal response matrix equal to the identity. For 17 modes, sufficiently high decoding success directly certifies a 17-dimensional quantum memory.The paper distinguishes established realization, automata, process-tensor, quantum-control, and factorization results from the proposed synthesis and experimental interpretation. It concludes with two natural next directions: experimental implementation and a multitime completely-positive / PSD-rank theory of intervention-relative quantum memory.
We study when a reduced description preserves the response to an applied source. For systems defined by continuously differentiable state updates and observations a bounded surjective linear coarse map gives differentiable descent when the projected maps are constant on each fibre. Their common source response is then preserved on the invertible domain. For approximate descent an exact error identity separates defects in the update and source coupling from defects in the readout and direct term. Resolvent and contour estimates carry these errors and the displacement of the reference branch to visible poles and their residues at shifted critical roots. In a reciprocal Hilbert-space realisation an isolated finite-dimensional critical subspace with a positive crossing matrix has a positive-semidefinite finite-rank residue when the endpoint operator is self-adjoint and the complementary inverse is bounded. Source overlap determines its visible directions. Arithmetic and reflecting-heat examples separate stationary response preservation from memoryless dynamical closure. In the two-mode example pole displacement decays geometrically with the retained memory length. The residue error includes an additional factor of that length. For a matched vacuum covariance the residue is the zero-mass atom of its positive spectral measure. This identification uses the same physical projection and source pairing together with the continuation and subtraction conventions and endpoint matching.