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

Reflection Positivity under Projection-Type Coarse Graining: Rigidity, Null Escape, and Exact Finite-Scale Obstructions

Tao Lin

Published Sep 2, 2026• 0 citations
Publisher

Abstract

Reflection positivity is a positivity condition on a reflected half-space, and it is not inherited automatically by covariance pieces produced by coarse graining. We isolate a finite-dimensional obstruction for a common projection-type construction. Let $\mathcal{H}=\mathcal{H}_+\oplus\mathcal{H}_-$ carry a unitary reflection $\Theta$ exchanging the two halves, let $C=C^*$ commute with $\Theta$, and assume Osterwalder--Schrader (OS) positivity on $\mathcal{H}_+$. For a reflection-symmetric orthogonal projection $\Pi$ that also preserves $\mathcal{H}_+$, we characterize exactly when the projection-sandwich residual $$\widetilde{C} = C - \Pi C \Pi$$ remains OS-positive: this holds if and only if the positive-half projection $\Pi_+$ commutes with the OS Gram operator $W=P_+\Theta C P_+$. Equivalently, the mixed positive/negative-time commutator block vanishes. This gives a direct algebraic test for projection-type coarse graining. We then record three consequences. First, in the scalar infinite-lattice setting a finite-range, fine-translation-invariant orthogonal projection is necessarily $0$ or $\mathrm{id}$, so nontrivial local projections must break fine translation invariance or leave this class. Second, a null-escape mechanism shows that a projection which sends an OS-null vector to a strictly positive OS-energy direction forces the residual to violate reflection positivity. For the massive free covariance with bond reflection we identify the OS null cone explicitly and obtain its dimension. Third, a complete $4\times4$ rational example gives a standalone exact failure certificate with positive original covariance and a negative residual OS direction. A larger audited $8\times6$ block-harmonic computation is retained only as supplementary provenance because the closure archive does not contain the complete projector table required for independent recomputation. The obstruction is deliberately narrow: it applies to projection-sandwich residuals and does not exclude non-projective Markov kernels, conditional expectations, completely positive maps, nonlinear coarse graining, or continuum constructions.

DOI
10.5281/zenodo.22248100

Related papers

Reconstructability of background-independent growth dynamics: Markovianity, cocycles, and reflection positivity without reflections

Mingdong He

Reflection positivity (RP) is the single positivity statement through which a Euclidean path integral is promoted to a unitary real-time quantum theory (Osterwalder–Schrader reconstruction). Its classical proofs consult a geometric reflection — an involutive automorphism fixing a hyperplane — which a randomly grown geometry generically lacks. This preprint shows that on ensembles of sequentially assembled geometries the positivity structure survives, and that it is carried entirely by the measure: RP holds if and only if the growth dynamics is memoryless on isomorphism classes. Characterization. Class-level Markovianity, class-local product form with terminal normalization, and history-independent Gram realizability of the conditional chain forms are equivalent; the multiplicative cocycle identity is derived as the Markov property in multiplicative form, not hypothesized separately. Explicit spectrum. Under these conditions the chain form is block-diagonal over terminal classes with rank-one blocks; the positive spectrum equals the arrival measure of the class process, independent of the phase rule — phases act only on the eigenbasis, identifying the chain form with the decoherent diagonal. Sharpness. Outside the class, failure is not gradual: a history-dependent weight family violates the class Markov property with exact rational witnesses (1/60 ≠ 9/260, 1/30 ≠ 3/130), and no Gram realization exists. Physical anchoring. The covariant Rideout–Sorkin family of classical sequential growth dynamics for causal sets satisfies the hypotheses, so all theorems apply verbatim: its chain forms are positive semidefinite at every depth. A dictionary instantiates the framework in causal dynamical triangulations, group field theory and spin foams, tensor/matrix models, and causal sets. Verification. All statements verified by exhaustive exact enumeration in rational arithmetic through graph size eight (isomorphism classes 1–3–14–86–709–8534; full depth-two assembly over all 85,393 paths), across six weight families and three degree bounds.

•Aug 1, 2026•0 citations

DEPRECATED VERSION Persistent Noncommutativity of Coarse-Graining - A CPTP Obstruction in a Two-Mode Sector

Claudio Menéndez

This work presents a minimal and fully explicit result showing that quantum coarse-graining procedures need not commute, even in the limit of arbitrarily fine resolution. Working entirely within the framework of completely positive trace-preserving (CPTP) maps, we construct two natural pinching (dephasing) channels acting on a two-dimensional effective subspace of an infinite-dimensional Hilbert space. These channels represent distinct coarse-graining operations corresponding to incompatible measurement bases. We prove that the order of these coarse-graining operations affects the final quantum state in trace norm, compute the exact optimal incompatibility over all input states, and show that this noncommutativity persists under CPTP refinements. The result is quantitative, stable, and operationally meaningful. The construction is minimal in dimension, requires no interpretational assumptions, and highlights a structural obstruction to the commutativity of coarse-graining procedures in quantum theory. It provides a concrete benchmark for discussions of decoherence, measurement order, and the emergence of effective classical descriptions from quantum systems. This result is part of the broader research program of Finite-Closure Geometry (FCG), which investigates how finite-resolution closure procedures give rise to effective descriptions and incompatible classical limits. The present work isolates a concrete and fully operational instance of this phenomenon within standard quantum information theory. Keywords quantum coarse-graining, CPTP maps, quantum channels, noncommutativity, pinching channels, trace norm, decoherence, quantum measurement, Bloch sphere, quantum foundations, operator algebras, information loss, dephasing, effective descriptions, quantum state space, incompatibility, refinement stability, mathematical physics, quantum information theory, finite-resolution limits

•Dec 26, 2025•0 citations

Sharp-Context Transport and Defect Erasure under Completely Positive Maps

Joaquim Reizi Higuchi

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.

•Oct 10, 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 scalar–Haar dichotomy for consistent covariant quantum projections

Dmitry Lubenskiy

Preprint. This manuscript has not undergone external peer review. We characterize exact projection channels on all tensor powers of a finite-dimensional system. Fix d ≥ 2. Suppose the channels are unital, completely positive, trace preserving and idempotent, covariant under collective SU(d) transformations and tensor permutations, consistent with identity embeddings, and completely depolarizing on one site. Then the entire family is either complete depolarization or collective Haar twirling. The proof combines standard conditional-expectation theory, the two-site Schur–Weyl commutant and an elementary consequence of infinite exchangeability. It requires no general classification of invariant subalgebras. For qubits, any nontrivial family necessarily preserves a logical qubit on three spins, excluding a nontrivial purely classical retained algebra under these assumptions. We illustrate the obstruction with a total-spin measurement channel and give sharp three-spin disturbance bounds. The entanglement-breaking bound is an application of the established trine state-reconstruction benchmark. An explicit example separates small normalized-Choi consistency defects from uniform operational closeness, clarifying a limitation of approximate extensions.

•Sep 27, 2026•0 citations

Predictive Holonomy and Minimal Completion of Coarse Dynamics

Maxim V. Churilov

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.

quant-pharXiv:2610.13951•Aug 10, 2026
Reflection positivity is a positivity condition on a reflected half-space, and it is not inherited automatically by covariance pieces produced by coarse graining. We isolate a finite-dimensional obstruction for a common projection-type construction. Let H=H+⊕H−\mathcal{H}=\mathcal{H}_+\oplus\mathcal{H}_-H=H+​⊕H−​ carry a unitary reflection Θ\ThetaΘ exchanging the two halves, let C=C∗C=C^*C=C∗ commute with Θ\ThetaΘ, and assume Osterwalder--Schrader (OS) positivity on H+\mathcal{H}_+H+​. For a reflection-symmetric orthogonal projection Π\PiΠ that also preserves H+\mathcal{H}_+H+​, we characterize exactly when the projection-sandwich residual C~=C−ΠCΠ\widetilde{C} = C - \Pi C \PiC=C−ΠCΠ remains OS-positive: this holds if and only if the positive-half projection Π+\Pi_+Π+​ commutes with the OS Gram operator W=P+ΘCP+W=P_+\Theta C P_+W=P+​ΘCP+​. Equivalently, the mixed positive/negative-time commutator block vanishes. This gives a direct algebraic test for projection-type coarse graining. We then record three consequences. First, in the scalar infinite-lattice setting a finite-range, fine-translation-invariant orthogonal projection is necessarily 000 or id\mathrm{id}id, so nontrivial local projections must break fine translation invariance or leave this class. Second, a null-escape mechanism shows that a projection which sends an OS-null vector to a strictly positive OS-energy direction forces the residual to violate reflection positivity. For the massive free covariance with bond reflection we identify the OS null cone explicitly and obtain its dimension. Third, a complete 4×44\times44×4 rational example gives a standalone exact failure certificate with positive original covariance and a negative residual OS direction. A larger audited 8×68\times68×6 block-harmonic computation is retained only as supplementary provenance because the closure archive does not contain the complete projector table required for independent recomputation. The obstruction is deliberately narrow: it applies to projection-sandwich residuals and does not exclude non-projective Markov kernels, conditional expectations, completely positive maps, nonlinear coarse graining, or continuum constructions.
Reflection positivity (RP) is the single positivity statement through which a Euclidean path integral is promoted to a unitary real-time quantum theory (Osterwalder–Schrader reconstruction). Its classical proofs consult a geometric reflection — an involutive automorphism fixing a hyperplane — which a randomly grown geometry generically lacks. This preprint shows that on ensembles of sequentially assembled geometries the positivity structure survives, and that it is carried entirely by the measure: RP holds if and only if the growth dynamics is memoryless on isomorphism classes. Characterization. Class-level Markovianity, class-local product form with terminal normalization, and history-independent Gram realizability of the conditional chain forms are equivalent; the multiplicative cocycle identity is derived as the Markov property in multiplicative form, not hypothesized separately. Explicit spectrum. Under these conditions the chain form is block-diagonal over terminal classes with rank-one blocks; the positive spectrum equals the arrival measure of the class process, independent of the phase rule — phases act only on the eigenbasis, identifying the chain form with the decoherent diagonal. Sharpness. Outside the class, failure is not gradual: a history-dependent weight family violates the class Markov property with exact rational witnesses (1/60 ≠ 9/260, 1/30 ≠ 3/130), and no Gram realization exists. Physical anchoring. The covariant Rideout–Sorkin family of classical sequential growth dynamics for causal sets satisfies the hypotheses, so all theorems apply verbatim: its chain forms are positive semidefinite at every depth. A dictionary instantiates the framework in causal dynamical triangulations, group field theory and spin foams, tensor/matrix models, and causal sets. Verification. All statements verified by exhaustive exact enumeration in rational arithmetic through graph size eight (isomorphism classes 1–3–14–86–709–8534; full depth-two assembly over all 85,393 paths), across six weight families and three degree bounds.
This work presents a minimal and fully explicit result showing that quantum coarse-graining procedures need not commute, even in the limit of arbitrarily fine resolution. Working entirely within the framework of completely positive trace-preserving (CPTP) maps, we construct two natural pinching (dephasing) channels acting on a two-dimensional effective subspace of an infinite-dimensional Hilbert space. These channels represent distinct coarse-graining operations corresponding to incompatible measurement bases. We prove that the order of these coarse-graining operations affects the final quantum state in trace norm, compute the exact optimal incompatibility over all input states, and show that this noncommutativity persists under CPTP refinements. The result is quantitative, stable, and operationally meaningful. The construction is minimal in dimension, requires no interpretational assumptions, and highlights a structural obstruction to the commutativity of coarse-graining procedures in quantum theory. It provides a concrete benchmark for discussions of decoherence, measurement order, and the emergence of effective classical descriptions from quantum systems. This result is part of the broader research program of Finite-Closure Geometry (FCG), which investigates how finite-resolution closure procedures give rise to effective descriptions and incompatible classical limits. The present work isolates a concrete and fully operational instance of this phenomenon within standard quantum information theory. Keywords quantum coarse-graining, CPTP maps, quantum channels, noncommutativity, pinching channels, trace norm, decoherence, quantum measurement, Bloch sphere, quantum foundations, operator algebras, information loss, dephasing, effective descriptions, quantum state space, incompatibility, refinement stability, mathematical physics, quantum information theory, finite-resolution limits
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.
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.
Preprint. This manuscript has not undergone external peer review. We characterize exact projection channels on all tensor powers of a finite-dimensional system. Fix d ≥ 2. Suppose the channels are unital, completely positive, trace preserving and idempotent, covariant under collective SU(d) transformations and tensor permutations, consistent with identity embeddings, and completely depolarizing on one site. Then the entire family is either complete depolarization or collective Haar twirling. The proof combines standard conditional-expectation theory, the two-site Schur–Weyl commutant and an elementary consequence of infinite exchangeability. It requires no general classification of invariant subalgebras. For qubits, any nontrivial family necessarily preserves a logical qubit on three spins, excluding a nontrivial purely classical retained algebra under these assumptions. We illustrate the obstruction with a total-spin measurement channel and give sharp three-spin disturbance bounds. The entanglement-breaking bound is an application of the established trine state-reconstruction benchmark. An explicit example separates small normalized-Choi consistency defects from uniform operational closeness, clarifying a limitation of approximate extensions.
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.