Symmetry determines observable dimensionality in dynamical-decoupling filter maps.Mirror-preserving timing deformations enforce an even filter response, leading to a rank-deficient local Jacobian and a symmetry-protected collapse of observable directions. Breaking this symmetry generically restores a finite first-order direction, with the recovered singular value varying continuously with the symmetry-breaking strength. A phase diagram in deformation amplitude and mirror-breaking strength separates symmetry-protected collapse from multi-dimensional accessibility. The results establish that observable dimensionality is controlled by symmetry class rather than parameter count. The analysis is shown to be independent of the specific feature representation by reproducing the same hierarchy, phase-diagram geometry, and scaling structure using smooth feature functionals that exclude non-differentiable peak observables.
Symmetry determines observable dimensionality in dynamical-decoupling filter maps.Mirror-preserving timing deformations enforce an even filter response, leading to a rank-deficient local Jacobian and a symmetry-protected collapse of observable directions. Breaking this symmetry generically restores a finite first-order direction, with the recovered singular value varying continuously with the symmetry-breaking strength. A phase diagram in deformation amplitude and mirror-breaking strength separates symmetry-protected collapse from multi-dimensional accessibility. The results establish that observable dimensionality is controlled by symmetry class rather than parameter count. The analysis is shown to be independent of the specific feature representation by reproducing the same hierarchy, phase-diagram geometry, and scaling structure using smooth feature functionals that exclude non-differentiable peak observables.
The fate of average symmetry-protected topological (ASPT) states based on the one-dimensional cluster state is investigated by using the Choi mapping, namely, the doubled Hilbert-space formalism. By introducing two effective spins, τ and η, on each doubled rung, we obtain a transparent representation of the ASPT parent Hamiltonian and systematically identify the conserved stabilizers and the corresponding orders. We further study symmetry-preserving operations that deform the ASPT by combining qualitative effective-Hamiltonian and systematic parent-Hamiltonian analyses with numerical matrix-product-state filtering calculations. We find that although the ASPT remains robust over a broad range of decoherence strengths, strong-to-weak spontaneous symmetry breaking (SWSSB) becomes exact in the maximal-decoherence limit. In addition, we consider symmetry preserving no-click postselection, which removes the exact conservation law supporting the Rényi-2 string order of the ASPT and drives the system toward weak-symmetry spontaneous symmetry breaking (WSSB). Again, the ASPT remains robust over a broad range of postselection strengths, with WSSB becoming pronounced only near the projective limit and exact at the limit itself.
We generalize the hidden symmetry-breaking picture of symmetry-protected topological (SPT) order developed by Kennedy and Tasaki in the context of the Haldane phase. Our generalization applies to a wide class of SPT phases in one-dimensional spin chains, protected by an on-site representation of a finite Abelian group. This generalization takes the form of a nonlocal unitary map that relates local symmetry-respecting Hamiltonians in an SPT phase to local Hamiltonians in a symmetry-broken phase. Using this unitary, we establish a relation between the two-point correlation functions that characterize fully symmetry-broken phases with the string-order correlation functions that characterize the SPT phases, therefore establishing the perspective in these systems that SPT phases are characterized by hidden symmetry breaking. Our generalization is also applied to systems with continuous symmetries, including SO(2k+1) and SU(k).
We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization approach that removes local entanglement and produces a coarse-grained lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixed-point tensors Tinv plus the symmetry group Gsym of the tensors (i.e., the symmetry group of the Lagrangian) characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by two-dimensional (2D) statistical Ising model, 2D statistical loop-gas model, and 1+1D quantum spin-1/2 and spin-1 models. In particular, using such a (Gsym,Tinv) characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry-protected topological phase. The (Gsym,Tinv) characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.
We present a concise review and perspective on noise-induced synchronisation and coherence protection in open quantum systems, with emphasis on recent work involving coupled spins, oscillators, and anyons. When local environments exhibit internal correlations, the structure of the noise determines which collective modes become decoherence-protected. This leads to steady-state entanglement, phase locking, and exceptional points (EPs) in the Liouvillian spectrum, signalling a collapse of the mode basis and the emergence of non-dissipative stabilised dynamics. Using a Lindblad framework, we show that symmetry in the noise correlations acts as a control parameter–protecting symmetric or antisymmetric modes depending on the sign of the correlation. In the pure-dephasing limit, coherence decay mirrors the Anderson–Kubo model, where the effective fluctuation strength scales as σ2(1±ξ), and the dynamical regime (Gaussian vs. Lorentzian) is set by the ratio σ/γ. Thus, the environment not only drives decoherence but can also selectively suppress it through symmetry filtering. We also revisit historical and conceptual origins of this idea, beginning with Huygens' synchronised pendulum clocks and culminating in modern non-Hermitian dynamics. Correlated noise–though classically stochastic–can organise quantum dynamics and protect coherence without direct control over the system. These insights offer a unifying view of synchronisation in classical and quantum regimes, with implications for quantum sensing, engineered decoherence, and long-lived coherence in complex environments such as biological light-harvesting complexes or avian magnetoreception.
Crystalline topological insulators owe their topological character to the protection that certain boundary states acquire because of certain point-group symmetries. We first show that a Hermitian operator obeying supersymmetric quantum mechanisms (SUSY QM) delivers the entanglement spectrum. We then show that such an entanglement spectrum that is compatible with a certain point-group symmetry obeys a certain local spectral symmetry. The latter result is applied to the stability analysis of four fermionic non-interacting Hamiltonians, the last of which describes graphene with a Kekule distortion. All examples have the remarkable property that their entanglement spectra inherit a local spectral symmetry from either an inversion or reflection symmetry that guarantees the stability of gapless boundary entangling states, even though all examples fail to support protected gapless boundary states at their physical boundaries.
Multi-Ratio Identifiability and Local Completeness in Dynamical-Decoupling Spectroscopy This work studies how inverse identifiability changes when several independent protocol ratios are measured simultaneously in dynamical-decoupling spectroscopy. Building on earlier reduced-observation and single-ratio inverse frameworks, the paper formulates a multi-ratio map R(θ) ∈ R^m with Jacobian J ∈ R^{m×n} and Fisher matrix F = J^T Σ^{-1} J on an n-dimensional spectral family. The central question is when observable enrichment restores local identifiability, and what obstructions remain even after Fisher-rank recovery. The paper proves a generic local-completeness theorem: for an n-dimensional spectral family, fewer than n independent observables are generically insufficient for local identifiability, whereas n observables are generically sufficient whenever the Jacobian has full rank. This establishes m_min = n as the sharp threshold for generic local completeness and separates rank recovery from stronger notions such as global injectivity or practical conditioning. These general results are then demonstrated in an explicit three-parameter mixed spectral family S(ω; α1, α2, w) = wAω^{α1} + (1 − w)Aω^{α2}, using an octave-separated sequence of protocol ratios. In this family, one ratio generically gives rank one, two ratios generically give rank two, and three ratios generically give rank three, confirming the sharpness of the completeness bound in a concrete model. The paper also identifies a symmetry-protected obstruction. Along a distinguished symmetric locus, all exchange-symmetric ratios satisfy ∂w R = 0 and ∂{α1}R = ∂_{α2}R, forcing the Jacobian to remain rank one no matter how many symmetric ratios are added. Thus multi-ratio enrichment does not automatically remove local branching: some fold obstructions survive as a consequence of symmetry rather than insufficient observable number. To resolve this obstruction, the paper introduces asymmetric ratios that probe the two spectral components at different effective sampling scales. Replacing at least one symmetric ratio by an asymmetric one lifts the symmetry protection and restores full rank on the symmetric locus. The resulting picture yields a three-layer inverse structure: generic local completeness through rank recovery, symmetry-protected local branching on special loci, and residual global non-injectivity beyond local analysis. Overall, the work extends the inverse geometry of protocol ratios from the scalar setting to vector-valued observations and provides a precise observable hierarchy for spectral reconstruction in dynamical-decoupling spectroscopy. It clarifies that adding more observables can generically restore local identifiability, but that symmetry-protected obstructions require symmetry-breaking observables rather than mere multiplicity. Description (v2)We extend the inverse geometry of dynamical-decoupling protocol ratios from the scalar to a multi-ratio setting. For an nnn-parameter spectral family, we prove that fewer than nnn observables are generically insufficient for local identifiability, while nnn independent ratios achieve generic local completeness when the Jacobian has full rank. In a three-parameter mixed spectral model, we explicitly demonstrate rank recovery (1→2→3) with increasing ratios. We further identify a symmetry-protected obstruction: on a Z2\mathbb{Z}_2Z2-invariant locus, exchange-symmetric ratios enforce rank collapse and preserve fold branching. This obstruction is lifted only by asymmetric ratios that break the symmetry. The results establish a three-layer inverse structure: generic local completeness, symmetry-protected differential obstruction, and residual global non-injectivity.