Probing Operator Spreading with Local Quantum Fluctuations

Abstract

Measuring out-of-time-ordered correlators (OTOCs) is experimentally challenging, often requiring effective reversal of many-body dynamics. Here, we propose an alternative approach to probing operator spreading through local quantum fluctuations, using only forward evolution and local measurements. Motivated by the Heisenberg uncertainty relation, we introduce a differential local variance and establish a rigorous bound connecting it to operator noncommutativity. For a ferromagnetic Heisenberg chain in the single-magnon regime, we derive an exact relation between the local variance and a squared-commutator OTOC. In integrable and nonintegrable Ising chains, the differential variance and OTOCs reveal qualitatively consistent propagation fronts. Our results establish local quantum fluctuations as an experimentally accessible, state-dependent, and complementary diagnostic of operator spreading.

The paper asks whether local fluctuations can reveal the propagation of a distant perturbation without measuring OTOCs or reversing many-body dynamics. It proposes comparing forward evolution with and without a local unitary perturbation $P_i$, using the differential probe variance $\mathcal V_{i\to j}(t)=|\operatorname{Var}_P(B_j)-\operatorname{Var}_0(B_j)|$. For a Pauli probe, it proves $\mathcal V_{i\to j}(t)\leq2\sqrt{C_{BP}(t;\rho_0)}$, where $C_{BP}=\operatorname{Tr}\{\rho_0[B_j(t),P_i]^\dagger[B_j(t),P_i]\}$. The derivation factors the variance difference into changes and sums of local expectation values, then applies Cauchy–Schwarz. The bound holds for arbitrary initial states, including correlated and nonstationary states: a nonzero differential variance witnesses noncommutativity, but the converse need not hold. For a fully polarized ferromagnetic Heisenberg chain following one spin flip, single-magnon dynamics gives the exact identities $C_{XZ}=4p_j$ and $\operatorname{Var}(\sigma_j^z)=4p_j(1-p_j)=C_{XZ}(1-C_{XZ}/4)$. Thus the diagnostics agree to leading order at a front where $p_j\ll1$; a 20-site example illustrates this. Nine-site Ising-chain calculations at transverse field $g=1$ and longitudinal fields $h=0,0.5,1$ show qualitatively similar distance-dependent onset delays in differential variance and OTOCs. The exact identity is restricted to the polarized single-excitation setting. The Ising comparisons concern qualitative propagation, use different commutators from the bound, and do not directly test it. Sensitivity depends on the state and probe; at infinite temperature every unitary perturbation leaves the state unchanged, so the signal vanishes. Noise robustness and optimal state/probe choices remain open.