Quantum Fisher Information Deficit as a Geometric Signature of Strong-to-Weak Spontaneous Symmetry Breaking

Technical Summary: Quantum Fisher Information Deficit as a Geometric Signature of Strong-to-Weak Spontaneous Symmetry Breaking

Problem Statement
Recent developments have established Strong-to-Weak Spontaneous Symmetry Breaking (SW-SSB) as a framework for describing quantum phases in open systems, where strong symmetry (invariance of the density matrix support under a group representation) is spontaneously broken while weak symmetry (invariance under the adjoint action) remains intact. However, characterizing SW-SSB presents significant challenges. In mixed states, long-range charged correlations often cancel out upon averaging, causing conventional two-point correlators to vanish. While fidelity correlators ($F_O$) and Rényi-1 (Wightman) correlators ($R_1$) serve as robust diagnostics for this hidden order, they are difficult to measure in generic mixed states. Conversely, the accessible Rényi-2 correlator ($R_2$) lacks stability under local quantum channels and may not align with fidelity phase boundaries. This work addresses the need for a complementary, measurable approach to probe SW-SSB through quantum response, specifically investigating whether Quantum Fisher Information (QFI) can effectively detect this phenomenon.

Methodology
The authors propose a geometric framework centered on the Quantum Fisher Information Deficit ($D_s$). The methodology involves:

1. Symmetry-Neutral Pair Rotations: For a strongly $Z_2$-symmetric state $\rho$, a local charged Hermitian involution $\hat{O}$ (where $\hat{O}^2 = \hat{I}$ and $\hat{U}_g \hat{O} \hat{U}_g^\dagger = -\hat{O}$) is defined. A neutral generator $\hat{G} = \hat{O}_i \hat{O}_j^\dagger$ is constructed from disjoint endpoints $i$ and $j$. This generates a unitary orbit $\rho_\theta = e^{-i\theta \hat{G}} \rho e^{i\theta \hat{G}}$.
2. Defining the Deficit: The QFI $Q[\rho; \hat{G}]$ quantifies the distinguishability of $\rho$ under infinitesimal rotation. Since $\hat{G}^2 = \hat{I}$, the maximal QFI is 4. The authors define the normalized deficit as $D_s \equiv 1 - Q/4$.
3. Inequality Hierarchy: Using ensemble decompositions (convex-roof constructions) and spectral representations, the authors derive a rigorous inequality chain connecting $D_s$ to established non-linear correlators: the root-fidelity correlator $F_O$ and the Rényi-1 correlator $R_1$.
4. Thermal Analysis: The framework is extended to symmetry-projected Gibbs states ($\rho_{\beta, P}$). The authors utilize the Lehmann probability measure and energy-transfer spectra to relate the deficit to the mean energy injected by the pair operation ($\Delta E_G$).
5. Numerical Validation: The theoretical bounds are tested in two models: a dephased open ZXZ spin chain (where inequalities are saturated) and a two-dimensional edge-dephasing model (where inequalities are strict).

Key Contributions and Results

• Geometric Hierarchy of SW-SSB Diagnostics:
  The paper establishes a model-independent inequality hierarchy valid for any density matrix $\rho$ and Hermitian involution $\hat{G}$:
  $$F_O^2 \leq D_s \leq R_1 \leq F_O$$
  This hierarchy proves that the QFI deficit $D_s$ shares the same asymptotic long-range order (LRO) criteria as $F_O$ and $R_1$. Specifically, $L[F_O] > 0 \iff L[D_s] > 0 \iff L[R_1] > 0$, where $L[\cdot]$ denotes the long-distance limit. Unlike $R_2$, $D_s$ inherits the stability of $F_O$ and $R_1$ under strongly symmetric finite-depth local quantum channels.

• Thermal Energy Bounds:
  For symmetry-projected Gibbs states, the authors derive a lower bound on the deficit governed by the mean energy injected by the pair operation:
  $$D_s(\rho_{\beta, P}, \hat{G}) \geq e^{-\beta \Delta E_G}$$
  Crucially, for local Hamiltonians, the energy cost $\Delta E_G$ is uniformly bounded by the commutator norm of the Hamiltonian with the local endpoint operators ($2J_O$), independent of system size and endpoint separation. This yields a strictly positive lower bound:
  $$D_s \geq e^{-2\beta J_O} > 0$$
  This result implies that at any fixed finite temperature, the QFI remains uniformly bounded away from its algebraic maximum, establishing a finite geometric gap that persists at arbitrary distances.

• Operational Certification:
  The paper proposes a "three-angle readout protocol" to access the QFI. By combining the SLD variational functional with the parameter-shift identity for involutions, one can compute a lower bound $J(\hat{M})$ using measurements at $\theta = 0, \pm \pi/4$. The condition $J(\hat{M}) + \Delta_c < 4$ provides an operational certification of a nonzero deficit $D_s > 0$, as this inequality ensures the QFI is strictly less than its algebraic maximum ($Q < 4$).

• Model Realizations:

  • ZXZ Chain: In a dephased ZXZ chain, the hierarchy saturates ($F_O^2 = D_s = R_1$), confirming the tightness of the bounds.
  • 2D Edge Dephasing: In a 2D model with edge dephasing, the inequalities are strict ($F_O^2 < D_s < F_O$) at finite distances, yet all diagnostics converge to the same critical point in the thermodynamic limit, validating $D_s$ as a robust diagnostic for SW-SSB.

Significance and Claims
The paper claims to provide a quantitative information-geometric characterization of thermal SW-SSB. By demonstrating that SW-SSB manifests not as an enhancement of QFI (as in traditional symmetry breaking) but as a persistent geometric deficit relative to the maximal QFI, the authors offer a new diagnostic tool that is:

1. Robust: Equivalent to fidelity and Rényi-1 correlators in detecting long-range order.
2. Stable: Invariant under local quantum channels that preserve strong symmetry.
3. Accessible: Potentially measurable via three-angle readout protocols, unlike the difficult-to-measure fidelity correlators.
4. Thermally Stable: Capable of detecting SW-SSB at finite temperatures where conventional charged long-range order is absent, provided the system is governed by a local Hamiltonian.

The work connects finite-angle state overlaps to constraints on the infinitesimal Bures geometry, offering a unified framework to diagnose hidden order in mixed quantum states where traditional correlators fail.