Mixed-State Quantum Anomaly and Multipartite Entanglement

Abstract

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.