Finite Speed of Quantum Scrambling with Long Range Interactions

In a locally interacting many-body system, two isolated qubits, separated by a large distance $r$, become correlated and entangled with each other at a time $t\ge r/v$. This finite speed $v$ of quantum information scrambling limits quantum information processing, thermalization, and even equilibrium correlations. Yet most experimental systems contain long range power-law interactions—qubits separated by $r$ have potential energy $V(r)\propto r^{-\alpha }$. Examples include the long range Coulomb interactions in plasma ($\alpha =1$) and dipolar interactions between spins ($\alpha =3$). In one spatial dimension, we prove that the speed of quantum scrambling remains finite for sufficiently large $\alpha$. This result parametrically improves previous bounds, compares favorably with recent numerical simulations, and can be realized in quantum simulators with dipolar interactions. Our new mathematical methods lead to improved algorithms for classically simulating quantum systems, and improve bounds on environmental decoherence in experimental quantum information processors.

Figure 1

Couplings ${\mathcal{L}}_{ij}$ ($i<j$) can be broken up into the scale on which the coupling acts in a unique way. Intuitively, the scale $q$ of a coupling is approximately $\lfloor {\log }_2(j-i)\rfloor$. Each scale with different values of $q$ is denoted with a different color: from large (orange) to short (purple). In this example, we study $\parallel [A_1(t),B_{16}]\parallel$, and sites $n$ obeying $n<1$ or $n>16$ are grouped in with these end sites when combining couplings. (a) Each scale or color would form a chain; (b) a more precise presentation. This $L$-shaped tiling ensures that the sum of each $q$ block scales as $2^{-q(\alpha -2)}$ and can be extended to arbitrary large $q$.

Figure 2

Any sequence of $\mathcal{L}$ that grows $A_1(t)$ to the final site 16 must have a long sequence of couplings on at least one scale. For the particular sequence shown, there are three scales with sufficiently long sequences (no shorter than $16/\log 16=4$), and we bound the contribution of this sequence to $\parallel [A_1(t),B_{16}]\parallel$ by summing over the weight of all possible paths that contain the solid colored couplings (corresponding to ${\mathcal{L}}_j^{\mathrm{\Gamma }}$) in a precise order. The lightly shaded couplings (corresponding to ${\tilde{\mathcal{L}}}_j^{\mathrm{\Gamma }}$) do not contribute to Eq. (9).