De Finetti theorems convert permutation symmetry into approximate mixtures of product states and thereby justify a wide range of reductions in classical and quantum statistics. In this work we study de Finetti hierarchies in which the feasible states are additionally constrained to be fixed points of quantum channels, a condition that subsumes invariance under arbitrary compact symmetry groups. Combining the mean-ergodic theorem with the structure theory of conditional expectations, we prove a tight bound on the entanglement-assisted classical capacity of the dual of a conditional expectation, block-wise distortion bounds for informationally complete measurements adapted to fixed point algebras, and an exact type-based refinement of the chain rule for permutation-invariant states. From these tools we derive several de Finetti theorems: a double-sided extension theorem with $O\left(\sqrt{\log n}/n\right)$ convergence, an interpolation theorem whose dimension dependence is governed solely by the block structure of the fixed point algebras and which recovers the dimension-independent classical behavior for maximal tori, and a Bose-symmetric variant. Exploiting Schur-Weyl duality and Gelfand-Tsetlin bases, we further show that the rounding scheme producing certifiably good separable inner approximations for (constrained) separability problems can be implemented in time polynomial in $1/\epsilon$ for fixed local dimensions, complementing the known efficient outer hierarchies. Applications to bilinear optimization under symmetries and to approximate quantum error correction are discussed.
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.
L'eo Le-Nestour, David Pérez-García, Alberto Ruiz-de-Alarcón· 0 citations
We prove an exponential variant of the Gaussian de Finetti theorem: the subsystems of permutation-invariant, free-fermionic Gaussian states are well-approximated by convex combinations of almost-i.i.d. states that are Gaussian on subsets of their parts. Our result provides an error bound between the original state and its approximants that decays exponentially in the number of unconstrained parts, becoming super-exponential when the subsystem under consideration is small. The dimensional penalty of our bound is polylogarithmic in the local Hilbert space dimension, an exponential improvement over the standard de Finetti theorem of [Nat. Phys. 3, 645-649]. In the fully i.i.d. limit, our bound recovers the Gaussian de Finetti theorem of [arXiv:2603.12392]. Previous works considered Gaussian-symmetric states, which are supported on the trivial irrep of the tensor matchgate representation. We extend these to a broader class of Gaussian-invariant states containing, for example, i.i.d. copies of single-replica mixed Gaussian states. We show that Gaussian-invariant states are precisely the partial traces of Gaussian-symmetric states on locally enlarged replicas, and always admit a purification into a larger Gaussian-symmetric state. This extends de Finetti theorems to the full set of Gaussian-invariant states, with only a polynomial overhead in the dimensional penalty of the error bound.
Jędrzej Burkat, Michal Studzi'nski, Sergii Strelchuk· 0 citations
In this paper, we investigate the underlying algebraic and combinatorial structures governing the coset representatives (shift vectors) for Construction A of lattices, with a particular focus on equations (14) and (15) presented in Paragraph 3, Chapter 8 of the seminal work by J.H. Conway and N.J.A. Sloane,"Sphere Packings, Lattices and Groups". These dual equations define the boundary conditions for the analytical generation of the 64-dimensional Quebbemann lattice (Q64). We prove that seeking non-zero solutions constrained by arithmetic or geometric progressions yields a structural collapse to the trivial zero vector due to the transcendental nature of {\pi}. Conversely, by relaxing these bounds to unique coordinate configurations, we uncover an exact, closed-form algebraic core governed by the alternating coefficients of the Pascal triangle. Furthermore, we implement an energy-minimization model via continuous-to-discrete projection that yields an optimal shift vector with a strictly integer Euclidean norm ||z||^2 = 20.000000. Finally, we bridge this formulation to the Repeated Differences paradigm of Craig's lattices Anm, showing how the spectral components of the {\Theta}-series are naturally filtered by these binomial structures.
We develop a dynamical framework for non-commutative de Finetti theory. We first establish the non-commutative Hewitt-Savage 0-1 law for quantum stochastic processes. We identify the factorization condition of the distribution of a spreadable process which characterizes tail-triviality, which is also characterized dynamically in terms of a certain attractivity property of the distribution. These three equivalent ergodic conditions identify a distinguished level in the non-commutative hierarchy of ergodic properties, which all collapse to ergodicity in the classical probability setting. To pass from the ergodic to the general case, we construct the conditional expectation onto the tail algebra for a spreadable process, in the GNS representation of the distribution of the canonical bilateral extension of the process. In this representation we establish the non-commutative Olshen Theorem identifying the tail algebra with the stationary algebra, and with the exchangeable algebra when the process is exchangeable. The resulting conditional expectation in particular inherits the same type of factorization property as distributions satisfying the Hewitt-Savage 0-1 law. This factorization strengthens conditional independence conditions arising in previous literature, while collapsing to the same notion in the classical case. This dynamical viewpoint leads to the identification of the minimal distributional symmetry underlying non-commutative de Finetti Theory, which we call weak spreadability, and which in the classical setting is equivalent to exchangeability. We prove that a stationary process is weakly spreadable if and only if its tail algebra admits a unique normal conditional expectation satisfying Hewitt-Savage type factorization, thereby establishing a general non-commutative de Finetti Theorem.
This paper establishes a rigorous, unified geometric framework for quantum state spaces by constructing smooth, regular embeddings into higher-order dual number algebras $\mathcal{Q}_N \equiv \mathbb{R}[\varepsilon]/(\varepsilon^{N^2-1})$, wherein every quantum state is faithfully represented as a non-reduced scheme-theoretic point. We show that under this unified family of truncated rings, the non-linear matrix commutators governing the Liouville-von Neumann dynamics map globally onto flat, linear, and rigid algebraic flows, establishing nilpotent dual algebras as a pristine geometric landscape for higher-dimensional quantum kinematics. As $N \to \infty$, this family converges to a Cauchy-complete power series ring $\mathcal{Q}_\infty$, where non-Archimedean completion linearizes the phase space, aligning the Fubini-Study geometry with the classical Fisher-Rao manifold.
Deterministic exact inversion of an arbitrary $d$-dimensional unitary requires {$\Theta(d^2)$} coherent forward calls in the worst case. We ask how this cost changes for Hamiltonian evolution $U(x)=\exp(i\sum_j x_jH_j)$ when the generators are known but the parameters are hidden. For one-parameter families with a fixed eigenbasis, we show that additive relations among the distinct eigenvalues determine the optimal query number exactly, and we construct the corresponding inversion protocol. For general families, we prove that repeated symmetry sectors do not affect the exact query complexity and give an automatic construction for combining inverses from inequivalent active sectors. We also give a sufficient phase-alignment condition under which family-specific structure can reduce the query number. These results establish structure-dependent bounds for reversing the unknown dynamics arising in Tavis-Cummings out-of-time-order correlator protocols, collective-spin echo verification, and passive multimode links, without requiring prior knowledge or explicit estimation of the underlying coupling strengths.
Jizhe Lai, M. Jing, Erdong Huang et al.· 0 citations
