Topological Lower Bound on Quantum Chaos by Entanglement Growth

A fundamental result in modern quantum chaos theory is the Maldacena-Shenker-Stanford upper bound on the growth of out-of-time-order correlators, whose infinite-temperature limit is related to the operator-space entanglement entropy of the evolution operator. Here we show that, for one-dimensional quantum cellular automata (QCA), there exists a lower bound on quantum chaos quantified by such entanglement entropy. This lower bound is equal to twice the index of the QCA, which is a topological invariant that measures the chirality of information flow, and holds for all the Rényi entropies, with its strongest Rényi-$\infty$ version being tight. The rigorous bound rules out the possibility of any sublinear entanglement growth behavior, showing in particular that many-body localization is forbidden for unitary evolutions displaying nonzero index. Since the Rényi entropy is measurable, our findings have direct experimental relevance. Our result is robust against exponential tails which naturally appear in quantum dynamics generated by local Hamiltonians.

Figure 1

(a) A range-$r$ 1D QCA $U$ transforms an operator supported on the $j$th site into one supported on $[j-r,j+r]$. (b) Bipartition $A\bigsqcup B$ and vectorization of $U$ into the CJS $|U⟩$ on a doubled Hilbert space [cf. Eq. (4)]. The operator entanglement of $U$ with respect to $A\bigsqcup B$ is defined as the entanglement entropy of $|U⟩$ with respect to $A{A}^{\prime }\bigsqcup B{B}^{\prime }$, where $A^{\prime }$, $B^{\prime }$ are the ancillary qudits associated with $A$, $B$, respectively. (c) AKLT-like range-1 QCA with local Hilbert space ${\mathbb{C}}^{pq}$ (dashed rectangles) and $\mathrm{ind}=\ln (p/q)$ generated by disjoint unitaries $u$. Here the thin and thick legs correspond to ${\mathbb{C}}^p$ and ${\mathbb{C}}^q$, respectively. (d) Operator entanglement Rényi entropy $S_{\alpha }$ of a single $u$ in (c) with $p=2$ and $q=3$ approaches $|\mathrm{ind}|=\log (3/2)$ when $\alpha \to \infty$, implying the saturation of Eq. (1) for $S_{\infty }$.

Figure 2

(a) Representation of a QCA as a bilayer product of nearest neighbor unitaries, with possibly different Hilbert-space bipartition for their inputs and outputs. Here the thick and dotted legs refer to virtual local Hilbert spaces at even and odd sites in the hidden layer, respectively. (b) CJS $|U⟩$ defined in Eq. (4) and the entanglement bipartition related to the index. (c) Relevant CJS used to derive the entropy expression (5) and the main result (1). (d) Demonstration of Eq. (5) for a general representative QCA with index $\log (p/q)$ consisting of the right and left translations of local Hilbert spaces ${\mathbb{C}}^p$ (red circles) and ${\mathbb{C}}^q$ (blue circles), respectively.

Figure 3

(a) Graphic representation of $\mathrm{Tr}[{\rho }_A^n]=\mathrm{Tr}[{\mathbb{T}}_A{\rho }_{AB}^{\otimes n}]$ with ${\mathbb{T}}_A={\mathbb{S}}_A^{[n-1,n]}\dots {\mathbb{S}}_A^{[2,3]}{\mathbb{S}}_A^{[1,2]}$, where ${\mathbb{S}}_A^{[j,k]}$ is a swap over the $j$th and $k$th copies of subsystem $A$. (b) Experimental setup for measuring the operator entanglement Rényi-$n$ entropy as well as the index. Here ${\mathbb{S}}_A$’s are controlled by the ancilla qubit, $H=(X+Z)/\sqrt{2}$ is the Hadamard gate, $|I⟩$’s are global maximally entangled states which become $|U⟩$ upon the action of $U$, and $A$ is a subsystem of interest, whose choice depends on which quantity (index or operator entanglement) we would like to measure.