Mixed-State Quantum Anomaly and Multipartite Entanglement

Quantum entanglement measures of many-body states have been increasingly useful to characterize phases of matter. Here, we explore a surprising connection between mixed-state entanglement and ’t Hooft anomaly. More specifically, we consider lattice systems in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>d</a:mi></a:math> space dimensions with anomalous symmetry <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>G</c:mi></c:math> where the anomaly is characterized by an invariant in the group cohomology <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>H</e:mi><e:mrow><e:mi>d</e:mi><e:mo>+</e:mo><e:mn>2</e:mn></e:mrow></e:msup><e:mo stretchy="false">[</e:mo><e:mi>G</e:mi><e:mo>,</e:mo><e:mi>U</e:mi><e:mo stretchy="false">(</e:mo><e:mn>1</e:mn><e:mo stretchy="false">)</e:mo><e:mo stretchy="false">]</e:mo></e:math>. We show that any mixed state <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>ρ</k:mi></k:math> that is strongly symmetric under <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>G</m:mi></m:math>, in the sense that <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>G</o:mi><o:mi>ρ</o:mi><o:mo>∝</o:mo><o:mi>ρ</o:mi></o:math> is necessarily (<q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mrow><q:mi>d</q:mi><q:mo>+</q:mo><q:mn>2</q:mn></q:mrow></q:math>)-nonseparable, i.e., is not the mixture of tensor products of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>d</s:mi><s:mo>+</s:mo><s:mn>2</s:mn></s:math> states in the Hilbert space. Furthermore, such states cannot be prepared from any (<u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mi>d</u:mi><u:mo>+</u:mo><u:mn>2</u:mn></u:mrow></u:math>)-separable states using finite-depth local quantum channels, so the nonseparability is long-ranged in nature. We provide proof of these results in <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>d</w:mi><w:mo>≤</w:mo><w:mn>1</w:mn></w:math> and plausibility arguments in <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>d</y:mi><y:mo>&gt;</y:mo><y:mn>1</y:mn></y:math>. The anomaly-nonseparability connection, thus, allows us to generate simple examples of mixed states with nontrivial long-ranged multipartite entanglement. In particular, in <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>d</ab:mi><ab:mo>=</ab:mo><ab:mn>1</ab:mn></ab:math> we find an example of quantum phase, in the sense that states in this phase cannot be two-way connected to any pure state through finite-depth local quantum channels. We also analyze a mixed anomaly involving both strong and weak symmetries, including systems constrained by the Lieb-Schultz-Mattis type of anomaly. We find that, while strong-weak mixed anomaly, in general, does not constrain quantum entanglement, it does constrain long-range correlations of mixed states in nontrivial ways. Namely, such states are not symmetrically invertible and not gapped Markovian, generalizing familiar properties of anomalous pure states.