Dynamical <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math>-Rényi Entropies of Local Hamiltonians Grow at Most Linearly in Time

We consider a generic one-dimensional spin system of length <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>L</a:mi></a:math>, arbitrarily large, with strictly local interactions, for example, nearest neighbor, and prove that the dynamical <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>α</c:mi></c:math>-Rényi entropies, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mn>0</e:mn><e:mo>&lt;</e:mo><e:mi>α</e:mi><e:mo>≤</e:mo><e:mn>1</e:mn></e:math>, of an initial product state grow at most linearly in time. This result arises from a general relation among dynamical <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>α</g:mi></g:math>-Rényi entropies and Lieb-Robinson bounds. We extend our bound on the dynamical generation of entropy to systems with exponential decay of interactions, for values of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>α</i:mi></i:math> close enough to 1, and, moreover, to initial pure states with low entanglement, of the order of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>log</k:mi><k:mtext> </k:mtext><k:mi>L</k:mi></k:math>, that are typically represented by critical states. We establish that low entanglement states have an efficient matrix product state representation that persists at least up to times of the order of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>log</m:mi><m:mtext> </m:mtext><m:mi>L</m:mi></m:math>. The main technical tools are the Lieb-Robinson bounds, to locally approximate the dynamics of the spin chain, a strict upper bound of Audenaert on <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>α</o:mi></o:math>-Rényi entropies, and a bound on their concavity. Such a bound, that we provide in an appendix, can be of independent interest.