Wandering range of robust quantum symmetries

The gist: This paper introduces and analyzes the concept of the wandering range of a robust symmetry S of a Hamiltonian H, quantifying how much its perturbed time evolution deviates from its unperturbed counterpart, and identifies conditions under which this deviation scales linearly with the perturbation strength ε.

Introduction to Symmetries and Perturbations

Symmetries play a crucial role in describing physical systems, as they are identified with operators that commute with the Hamiltonian of the system in quantum systems [3]. In real physical applications, where the Hamiltonian is known only approximately, it is important to understand whether a symmetry remains stable under perturbations of the Hamiltonian [4, 5]. This stability is essential for analog quantum simulation and understanding experimental imperfections [6, 7, 8]. Symmetries are classified based on their long-time stability: robust symmetries remain close to their initial value throughout the entire evolution despite a perturbation, while fragile symmetries accumulate non-negligible deviations over time [4].

Defining the Wandering Range

The wandering range of a robust symmetry quantifies the deviation of the perturbed symmetry from its unperturbed evolution [2]. The goal of this paper is to analyze how this wandering range depends on the strength of the perturbation ε, specifically identifying conditions under which it exhibits linear scaling with ε [2].

Scaling Behavior for Specific Subspaces and Symmetries

The paper establishes that for states belonging to the dense subspace given by the linear span of the eigenvectors of the unperturbed Hamiltonian, or for finite-rank symmetries, the wandering range is indeed of order ε [2]. This result is established in Theorem 2.1 for admissible perturbations satisfying a regularity condition and in Theorem 2.3 for arbitrary linear perturbations of a compact-resolvent Hamiltonian [8].

Analysis of Completely Robust Symmetries

The second part of the paper focuses on completely robust symmetries, which are robust against any bounded perturbation [3]. In this case, the wandering range is proven to be uniformly of order ε, and an explicit bound in terms of the norm of the perturbation and the minimal spectral gap η is derived in Theorem 3.1 [10]. The core proof involves constructing a new perturbation that commutes with H and generates dynamics approximating the perturbed evolution uniformly in time—an eternal block-diagonal approximation [10].

Technical Framework: Quantum KAM Iteration

The technical framework relies on the construction of a unitary block-diagonalizing transformation, known as the Schrieffer-Wolff transformation [11–13], which is equivalent to a quantum Kolmogorov-Arnold-Moser (KAM) iteration scheme [14, 15]. The analysis utilizes the homological equation, also known as the commutator equation in matrix analysis [16] and extended to unbounded Hamiltonians via Theorem 4.1 [8]. The formal construction of bounded operators W(ε) and Vˆ (ε) satisfying the properties of Theorem 3.3 is achieved through quantum KAM iteration, where coefficients are determined by recursive conditions derived from comparing terms in the formal series expansions [20, 21].

Convergence and Explicit Bounds

The convergence of the formal expansion is established by exploiting combinatorial sequences like the Catalan numbers [18]. Lemma 4.4 provides explicit estimates for operators Bs(ε), showing that their norm is bounded by a term involving Catalan numbers and constants α, b, and η [23]. The final construction shows that for states in the dense subspace, there exists a constant Cψ > 0 such that the wandering range is of order ε [34]. For completely robust symmetries under bounded perturbations, an explicit norm bound is obtained: sup t∈R eitH(ε)Se−itH(ε) − S ≤ β v η Sε for 0 ≤ ε ≤ η/vρ [51].

Physical Applications

Example 4.1 illustrates the application of these results to the Hamiltonian of a Josephson junction in an inductive loop [26]. The paper demonstrates that KAM iteration can be applied when the perturbation strength satisfies conditions such as ε∥V∥ ≤ ηρ, which relates to physical parameters like pEJ /ECEL/EC or π2/2 EL/EC + EJ/4EC < 1ρ [30].

In conclusion, the work provides sufficient conditions for the wandering range of robust symmetries to be linear in ε for specific states and symmetries, and it establishes explicit norm bounds for completely robust symmetries under bounded perturbations by employing a Schrieffer-Wolff transformation based on a quantum KAM iteration scheme. The derivation of these bounds generalizes results from finite-dimensional systems to unbounded Hamiltonians while removing dependence on the number of distinct eigenvalues [31].

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Wandering range of robust quantum symmetries 1

How it works

1. The wandering range is defined as δH(ε)(S, ψ) = sup t∈R (eitH(ε)Se−itH(ε) − S)ψ [7]. A symmetry S is robust if this wandering range goes to zero as ε → 0 [8].

2. The paper proves that for states in the dense subspace given by the linear span of eigenvectors of H, or for finite-rank symmetries, the wandering range is indeed of order ε [2].

3. For completely robust symmetries under bounded perturbations, an explicit norm bound is derived: sup t∈R eitH(ε)Se−itH(ε) − S ≤ β v η Sε for 0 ≤ ε ≤ η/vρ [51].

4. The proof of this bound utilizes a Schrieffer-Wolff transformation constructed through a KAM iteration scheme, which requires the use of Catalan numbers to control the convergence of the formal series expansions [10, 26].

Key Results

(i) If ψ ∈ H belongs to the linear span D of the eigenvectors of H, then there exists a Cψ > 0 such that sup t∈R (eitH(ε)Se−itH(ε) − S)ψ ≤ Cψε [25].

(ii) If S is a finite-rank operator, then there exists a C > 0 such that sup t∈R eitH(ε)Se−itH(ε) − S ≤ Cε [26].

(iii) For completely robust symmetries under bounded perturbations, the wandering range is bounded by sup t∈R eitH(ε)Se−itH(ε) − S ≤ β v η Sε for 0 ≤ ε ≤ η/vρ [51].

Conclusion

The work establishes sufficient conditions for the wandering range of robust symmetries to scale linearly with the perturbation strength ε for specific states and symmetries, and it provides an explicit norm bound for completely robust symmetries under bounded perturbations by employing a Schrieffer-Wolff transformation based on a quantum KAM iteration scheme. The derivation of these bounds generalizes results from finite-dimensional systems to unbounded Hamiltonians while removing dependence on the number of distinct eigenvalues [31].

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Key Results

(i) If ψ ∈ H belongs to the linear span D of the eigenvectors of H, then there exists a Cψ > 0 such that sup t∈R (eitH(ε)Se−itH(ε) − S)ψ ≤ Cψε [25].

(ii) If S is a finite-rank operator, then there exists a C > 0 such that sup t∈R eitH(ε)Se−itH(ε) − S ≤ Cε [26].

**(iii) For completely robust symmetries under bounded perturbations, the wandering range is bounded by sup t∈R eitH(ε)Se−itH(ε) − S ≤ β v η Sε for 0 ≤ ε ≤ η/vρ [51].

1. The AI system can achieve precise quantification of symmetry stability by measuring the wandering range of a robust symmetry, which quantifies the deviation of the perturbed symmetry from its unperturbed evolution. This allows for distinguishing between robust symmetries and fragile symmetries based on whether the deviation is of order ε.

2. The system can establish explicit, nonperturbative bounds for the wandering range under specific conditions, as demonstrated by Theorem 3.1: its wandering range is bounded by sup t∈R e itH(ε)Se−itH(ε) − S ≤ β v η Sε, where β ≈ 15.3 and ρ ≈ 34.8 are defined in Table 1.

3. The system can perform a quantum KAM iteration scheme to construct an eternal block-diagonal approximation of the perturbed Hamiltonian, which allows for the study of dynamics that approximates the perturbed evolution uniformly in time.

4. The system can apply the derived bounds to complex physical models, such as a Josephson junction Hamiltonian, enabling predictive modeling when conditions like εV ≤ ηρ are met.

5. The AI can analyze spectral properties of Hamiltonians to determine robustness; specifically, a symmetry S is robust against an admissible perturbation H(ε) if [S, Pn(0)] = 0 for all n ≥ 1.