Abstract
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-∞ 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.
Highlights
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 operatorspace entanglement entropy of the evolution operator
For one-dimensional quantum cellular automata (QCA), there exists a lower bound on quantum chaos quantified by such entanglement entropy
Equivalent formulation of the index.—As a starting point, we show that the index (3) can be expressed exactly as an entanglement entropy difference between two reduced states of the vectorized operator jUi, technically known as the Choi-Jamiołkowski state (CJS) [40]: jUi ≡ ðU ⊗ IÞjIi; ð4Þ
Summary
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 operatorspace entanglement entropy of the evolution operator. 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 Renyi entropies, with its strongest Renyi-∞ version being tight. The main difficulty to answer this question lies perhaps in the original definition of the index [22], which was given by the rank of certain operator algebras, lacking an immediate physical interpretation
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