Breaking global symmetries with locality-preserving operations

In the general framework of quantum resource theories, one typically only distinguishes between operations that can or cannot generate the resource of interest. In many-body settings, one can further characterize quantum operations based on underlying geometrical constraints, and a natural question is to understand the power of resource-generating operations that preserve locality. In this work, we address this question within the resource theory of asymmetry, which has recently found applications in the study of many-body symmetry-breaking and symmetry-restoration phenomena. We consider symmetries corresponding to both Abelian and non-Abelian compact groups with a homogeneous action on the space of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>N</a:mi> </a:math> qubits, focusing on the prototypical examples of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mtext>U</b:mtext> <b:mo>(</b:mo> <b:mn>1</b:mn> <b:mo>)</b:mo> </b:mrow> </b:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mtext>SU</c:mtext> <c:mo>(</c:mo> <c:mn>2</c:mn> <c:mo>)</c:mo> </c:mrow> </c:math> . We study the so-called <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>G</d:mi> </d:math> asymmetry <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi mathvariant="normal">Δ</e:mi> <e:msubsup> <e:mi>S</e:mi> <e:mi>N</e:mi> <e:mi>G</e:mi> </e:msubsup> </e:mrow> </e:math> , and present two main results. First, we derive a general bound on the asymmetry that can be generated by locality-preserving operations acting on product states. We prove that, in any spatial dimension, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi mathvariant="normal">Δ</g:mi> <g:msubsup> <g:mi>S</g:mi> <g:mi>N</g:mi> <g:mi>G</g:mi> </g:msubsup> <g:mo>≤</g:mo> <g:mrow> <g:mo>(</g:mo> <g:mn>1</g:mn> <g:mo>/</g:mo> <g:mn>2</g:mn> <g:mo>)</g:mo> </g:mrow> <g:mi mathvariant="normal">Δ</g:mi> <g:msubsup> <g:mi>S</g:mi> <g:mi>N</g:mi> <g:mrow> <g:mi>G</g:mi> <g:mo>,</g:mo> <g:mi>max</g:mi> </g:mrow> </g:msubsup> <g:mrow> <g:mo>[</g:mo> <g:mn>1</g:mn> <g:mo>+</g:mo> <g:mi>o</g:mi> <g:mrow> <g:mo>(</g:mo> <g:mn>1</g:mn> <g:mo>)</g:mo> </g:mrow> <g:mo>]</g:mo> </g:mrow> </g:mrow> </g:math> , where <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mi mathvariant="normal">Δ</j:mi> <j:msubsup> <j:mi>S</j:mi> <j:mi>N</j:mi> <j:mrow> <j:mi>G</j:mi> <j:mo>,</j:mo> <j:mi>max</j:mi> </j:mrow> </j:msubsup> </j:mrow> </j:math> is the maximum value of the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mi>G</l:mi> </l:math> asymmetry in the full many-body Hilbert space. Second, we show that locality-preserving operations can generate maximal asymmetry, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:msubsup> <m:mi>S</m:mi> <m:mi>N</m:mi> <m:mi>G</m:mi> </m:msubsup> <m:mo>∼</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:msubsup> <m:mi>S</m:mi> <m:mi>N</m:mi> <m:mrow> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>max</m:mi> </m:mrow> </m:msubsup> </m:mrow> </m:math> , when applied to symmetric states featuring long-range entanglement. Our results provide a unified perspective on recent studies of asymmetry in many-body physics, highlighting a nontrivial interplay between asymmetry, locality, and entanglement.