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:2607.08568

Renormalization flows for 1D mixed states and a quantum Goursat lemma

Léo Le-Nestour, David Pérez-García, Alberto Ruiz-de-Alarcón

math-phcond-mat.str-elmath.QAquant-phSubmitted Jul 9, 2026• Announced Jul 10, 2026• 0 citations
arXivHTMLPDF

Abstract

Renormalization provides a framework for relating microscopic models of physical systems to effective descriptions at larger length scales. This procedure is studied for the boundary states of non-chiral two-dimensional topologically ordered models. The initial data consist of renormalization fixed points built from representations of finite-dimensional $C^*$-Hopf algebras, which are then perturbed by uniform on-site noise quantum channels and repeatedly coarse-grained. The resulting flows admit an intrinsic algebraic description in terms of completely positive maps on the $C^*$-Hopf algebra or, equivalently, positive linear functionals on its enveloping $C^*$-Hopf algebra. Their iteration is governed by convolution powers, and convergent trajectories yield new matrix product density operator fixed points, described by finite $*$-quantum hypergroups. This provides a concrete physical interpretation of such structures. For finite group algebras and their duals, we provide explicit classifications via Goursat's lemma for groups. Finally, we formulate and prove a quantum generalization of Goursat's lemma for finite-dimensional $C^*$-Hopf algebras, a result of independent interest, which gives an explicit structural description of all convergent renormalization trajectories.

Comments
49 pages, 6 figures

Related papers

Matrix Product Operator Algebras II: Phases of Matter for 1D Mixed States

Alberto Ruiz-de-Alarcón, José Garre-Rubio, András Molnár +1 more

The classification of topological phases of matter is fundamental to understand and characterize the properties of quantum materials. In this paper we study phases of matter in one-dimensional open quantum systems. We define two mixed states to be in the same phase if both states can be transformed into the other by a shallow circuit of local quantum channels. We aim to understand the phase diagram of matrix product density operators that are renormalization fixed points. These states arise, for example, as boundaries of two-dimensional topologically ordered states. We first construct families of such states based on C*-weak Hopf algebras, the algebras whose representations form a fusion category. More concretely, we provide explicit local fine-graining and local coarse-graining quantum channels for the renormalization procedure of these states. Finally, we prove that those arising from C*-Hopf algebras are in the trivial phase.

arXiv:2204.06295•Apr 13, 2022•0 citations

Informational and algebraic renormalization group

Takato Mori, Teruaki Nagasawa

Renormalization group (RG) is a core concept from statistical mechanics to quantum field theory, yet its schemes differ widely across literature. We formulate Wilsonian RG for generic, including finite-dimensional, quantum systems as a quantum channel combining coarse graining with isometric rescaling, whose Kraus representation yields pure conditional trajectories for pure inputs and recovers mixed-state flows upon averaging. We then develop an algebraic extension without assuming tensor-product factorization: the retained low-energy algebra is constructed from the one-particle spectrum, and coarse graining is a conditional expectation onto it. Identifying the channel's Cesàro mean with a resource-destroying map onto fixed states establishes a resource theory of RG and yields a relative-entropy monotone without variational optimization. For thermal states in the zero-temperature limit, the resulting Bogoliubov--Kubo--Mori metric per unit spacetime volume is proportional to the Zamolodchikov metric, directly relating the resource density to $c-c_{\rm IR}$ near a two-dimensional IR fixed point. We also introduce the relative entropy of coarse graining to measure discarded UV information, prove its monotonicity along successive coarse grainings, and evaluate it holographically in global $\mathrm{AdS}_3$.

hep-thCross-listedarXiv:2609.30362•Sep 28, 2026•0 citations

Exact renormalization group flow for matrix product density operators

Kohtaro Kato

Matrix product density operator (MPDO) provides an efficient tensor network representation of mixed states on one-dimensional quantum many-body systems. We study a real-space renormalization group transformation of MPDOs represented by a circuit of local quantum channels. We require that the renormalization group flow is exact, in the sense that it exactly preserves the correlation between the coarse-grained sites and is therefore invertible by another circuit of local quantum channels. Unlike matrix product states (MPS), which always have a well-defined isometric renormalization transformation, we show that general MPDOs do not necessarily admit a converging exact renormalization group flow. We then introduce a subclass of MPDOs with a well-defined renormalization group flow, and show the structure of the MPDOs in the subclass as a representation of a pre-bialgebra structure. As a result, such MPDOs obey generalized symmetry represented by matrix product operator algebras associated with the pre-bialgebra. We also discuss implications with the classification of mixed-state quantum phases.

arXiv:2410.22696•Oct 30, 2024•1 citation

Joint State-Law Fixed Points in Quantum Theory: Recursive Dynamics, Bifurcations, and Renormalization-Group Deformations

Kwan Hong TAN

This article develops a joint state-law fixed-point method for self-consistent quantum models. A physical model is represented by a density operator and a finite-dimensional vector of effective law parameters; admissible dynamics consist of a state-update channel and a law-update map. A recursive solution is their joint fixed point. This formulation separates stationarity of a state-dependent Hamiltonian from genuine co-determination of states and laws and preserves the option of completely positive trace-preserving subsystem dynamics. Three general results are obtained: existence for continuous maps on compact convex state-law domains, uniqueness and iterative stability under a contraction condition, and persistence of non-degenerate fixed points under weak deformation. An exactly solvable two-state model displays a supercritical pitchfork bifurcation, critical exponent one-half, and a divergent zero-field susceptibility at the feedback threshold. In renormalization-group theory space, the same perturbative result yields first-order shifts of fixed-point coordinates and critical exponents. An action-level construction shows how recursive consistency operators can enter the standard functional flow without adding ad hoc terms to the Wetterich equation. Covariant gravity and laboratory phase-shift formulas are presented as conditional templates whose predictions require a specified kernel and normalization. The framework therefore contributes a broadly applicable method linking quantum channels, nonlinear dynamics, fixed-point theory, and renormalization-group deformations, while making no claim to a completed theory of quantum gravity.

•Jul 23, 2026•0 citations

Lindbladian for exact renormalization of density operators in QFT

Samuel Goldman, Nima Lashkari, Robert G. Leigh

Fliss [.] extended the exact renormalization group (ERG) to arbitrary wave functionals in quantum field theory. Applying this formalism, we show that a Lindblad master equation gives the ERG flow of density matrices. The Lindbladian consists of a “Hamiltonian” term, which is the sum of a scaling and a coarse-graining (disentangling) operator, and a dissipative term with absorption and emission rates for each momentum mode. We consider as examples the flow of Gaussian states and the perturbative ground state of λ ϕ 4 theory and highlight the role of the dissipative terms in generating accurate contributions to the flow of couplings at nontrivial order in λ . Integrating the Lindblad master equation, we find that a finite ERG flow of density matrices is described by a quantum channel. It follows from the data processing inequality that any distinguishability measure of states is an ERG monotone.

arXiv:2410.16582•Physical review. D/Physical review. D.•Aug 13, 2026•1 citation

Establishing Mixed-State Phase Equivalence beyond Renormalization Fixed Points

Yuhan Liu

Understanding mixed-state quantum phases is a central challenge in the era of quantum simulation, where many existing studies focus on renormalization fixed points. In this work, we move beyond the renormalization fixed-point paradigm by constructing a quantum phase transition connecting two distinct one-dimensional fixed points, both exhibiting finite conditional mutual information and one of which is intrinsically nontrivial. We analytically establish phase equivalence within each of the two phases by explicitly constructing low-depth, quasi-local channel circuits that connect states within each phase. Crucially, our approach leverages the parent Lindbladian construction to generate the desired channel circuits. We further demonstrate that this framework generalizes naturally to a broad class of intrinsically nontrivial mixed-state quantum phases. Our method establishes a framework for rigorously analyzing phase equivalence of intrinsically non-trivial mixed states beyond the renormalization fixed points.

quant-pharXiv:2605.04159•May 7, 2026•0 citations
Renormalization provides a framework for relating microscopic models of physical systems to effective descriptions at larger length scales. This procedure is studied for the boundary states of non-chiral two-dimensional topologically ordered models. The initial data consist of renormalization fixed points built from representations of finite-dimensional C∗C^*C∗-Hopf algebras, which are then perturbed by uniform on-site noise quantum channels and repeatedly coarse-grained. The resulting flows admit an intrinsic algebraic description in terms of completely positive maps on the C∗C^*C∗-Hopf algebra or, equivalently, positive linear functionals on its enveloping C∗C^*C∗-Hopf algebra. Their iteration is governed by convolution powers, and convergent trajectories yield new matrix product density operator fixed points, described by finite ∗*∗-quantum hypergroups. This provides a concrete physical interpretation of such structures. For finite group algebras and their duals, we provide explicit classifications via Goursat's lemma for groups. Finally, we formulate and prove a quantum generalization of Goursat's lemma for finite-dimensional C∗C^*C∗-Hopf algebras, a result of independent interest, which gives an explicit structural description of all convergent renormalization trajectories.
The classification of topological phases of matter is fundamental to understand and characterize the properties of quantum materials. In this paper we study phases of matter in one-dimensional open quantum systems. We define two mixed states to be in the same phase if both states can be transformed into the other by a shallow circuit of local quantum channels. We aim to understand the phase diagram of matrix product density operators that are renormalization fixed points. These states arise, for example, as boundaries of two-dimensional topologically ordered states. We first construct families of such states based on C*-weak Hopf algebras, the algebras whose representations form a fusion category. More concretely, we provide explicit local fine-graining and local coarse-graining quantum channels for the renormalization procedure of these states. Finally, we prove that those arising from C*-Hopf algebras are in the trivial phase.
Renormalization group (RG) is a core concept from statistical mechanics to quantum field theory, yet its schemes differ widely across literature. We formulate Wilsonian RG for generic, including finite-dimensional, quantum systems as a quantum channel combining coarse graining with isometric rescaling, whose Kraus representation yields pure conditional trajectories for pure inputs and recovers mixed-state flows upon averaging. We then develop an algebraic extension without assuming tensor-product factorization: the retained low-energy algebra is constructed from the one-particle spectrum, and coarse graining is a conditional expectation onto it. Identifying the channel's Cesàro mean with a resource-destroying map onto fixed states establishes a resource theory of RG and yields a relative-entropy monotone without variational optimization. For thermal states in the zero-temperature limit, the resulting Bogoliubov--Kubo--Mori metric per unit spacetime volume is proportional to the Zamolodchikov metric, directly relating the resource density to c−cIRc-c_{\rm IR}c−cIR​ near a two-dimensional IR fixed point. We also introduce the relative entropy of coarse graining to measure discarded UV information, prove its monotonicity along successive coarse grainings, and evaluate it holographically in global AdS3\mathrm{AdS}_3AdS3​.
Matrix product density operator (MPDO) provides an efficient tensor network representation of mixed states on one-dimensional quantum many-body systems. We study a real-space renormalization group transformation of MPDOs represented by a circuit of local quantum channels. We require that the renormalization group flow is exact, in the sense that it exactly preserves the correlation between the coarse-grained sites and is therefore invertible by another circuit of local quantum channels. Unlike matrix product states (MPS), which always have a well-defined isometric renormalization transformation, we show that general MPDOs do not necessarily admit a converging exact renormalization group flow. We then introduce a subclass of MPDOs with a well-defined renormalization group flow, and show the structure of the MPDOs in the subclass as a representation of a pre-bialgebra structure. As a result, such MPDOs obey generalized symmetry represented by matrix product operator algebras associated with the pre-bialgebra. We also discuss implications with the classification of mixed-state quantum phases.
This article develops a joint state-law fixed-point method for self-consistent quantum models. A physical model is represented by a density operator and a finite-dimensional vector of effective law parameters; admissible dynamics consist of a state-update channel and a law-update map. A recursive solution is their joint fixed point. This formulation separates stationarity of a state-dependent Hamiltonian from genuine co-determination of states and laws and preserves the option of completely positive trace-preserving subsystem dynamics. Three general results are obtained: existence for continuous maps on compact convex state-law domains, uniqueness and iterative stability under a contraction condition, and persistence of non-degenerate fixed points under weak deformation. An exactly solvable two-state model displays a supercritical pitchfork bifurcation, critical exponent one-half, and a divergent zero-field susceptibility at the feedback threshold. In renormalization-group theory space, the same perturbative result yields first-order shifts of fixed-point coordinates and critical exponents. An action-level construction shows how recursive consistency operators can enter the standard functional flow without adding ad hoc terms to the Wetterich equation. Covariant gravity and laboratory phase-shift formulas are presented as conditional templates whose predictions require a specified kernel and normalization. The framework therefore contributes a broadly applicable method linking quantum channels, nonlinear dynamics, fixed-point theory, and renormalization-group deformations, while making no claim to a completed theory of quantum gravity.
Fliss [.] extended the exact renormalization group (ERG) to arbitrary wave functionals in quantum field theory. Applying this formalism, we show that a Lindblad master equation gives the ERG flow of density matrices. The Lindbladian consists of a “Hamiltonian” term, which is the sum of a scaling and a coarse-graining (disentangling) operator, and a dissipative term with absorption and emission rates for each momentum mode. We consider as examples the flow of Gaussian states and the perturbative ground state of λ ϕ 4 theory and highlight the role of the dissipative terms in generating accurate contributions to the flow of couplings at nontrivial order in λ . Integrating the Lindblad master equation, we find that a finite ERG flow of density matrices is described by a quantum channel. It follows from the data processing inequality that any distinguishability measure of states is an ERG monotone.
Understanding mixed-state quantum phases is a central challenge in the era of quantum simulation, where many existing studies focus on renormalization fixed points. In this work, we move beyond the renormalization fixed-point paradigm by constructing a quantum phase transition connecting two distinct one-dimensional fixed points, both exhibiting finite conditional mutual information and one of which is intrinsically nontrivial. We analytically establish phase equivalence within each of the two phases by explicitly constructing low-depth, quasi-local channel circuits that connect states within each phase. Crucially, our approach leverages the parent Lindbladian construction to generate the desired channel circuits. We further demonstrate that this framework generalizes naturally to a broad class of intrinsically nontrivial mixed-state quantum phases. Our method establishes a framework for rigorously analyzing phase equivalence of intrinsically non-trivial mixed states beyond the renormalization fixed points.