Entanglement of purification in random tensor networks

The entanglement of purification <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>E</a:mi><a:mi>P</a:mi></a:msub><a:mo stretchy="false">(</a:mo><a:mi>A</a:mi><a:mo>∶</a:mo><a:mi>B</a:mi><a:mo stretchy="false">)</a:mo></a:math> is a powerful correlation measure, but it is notoriously difficult to compute because it involves an optimization over all possible purifications. In this paper, we prove a new inequality: <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msub><e:mrow><e:mi>E</e:mi></e:mrow><e:mrow><e:mi>P</e:mi></e:mrow></e:msub><e:mo stretchy="false">(</e:mo><e:mi>A</e:mi><e:mo>∶</e:mo><e:mi>B</e:mi><e:mo stretchy="false">)</e:mo><e:mo>≥</e:mo><e:mfrac><e:mrow><e:mn>1</e:mn></e:mrow><e:mrow><e:mn>2</e:mn></e:mrow></e:mfrac><e:msubsup><e:mrow><e:mi>S</e:mi></e:mrow><e:mrow><e:mi>R</e:mi></e:mrow><e:mrow><e:mo stretchy="false">(</e:mo><e:mn>2</e:mn><e:mo stretchy="false">)</e:mo></e:mrow></e:msubsup><e:mo stretchy="false">(</e:mo><e:mi>A</e:mi><e:mo>∶</e:mo><e:mi>B</e:mi><e:mo stretchy="false">)</e:mo></e:mrow></e:math>, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:msubsup><m:mrow><m:mi>S</m:mi></m:mrow><m:mrow><m:mi>R</m:mi></m:mrow><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>n</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:msubsup><m:mo stretchy="false">(</m:mo><m:mi>A</m:mi><m:mo>∶</m:mo><m:mi>B</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:math> is the Renyi reflected entropy. Using this, we compute <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>E</s:mi><s:mi>P</s:mi></s:msub><s:mo stretchy="false">(</s:mo><s:mi>A</s:mi><s:mo>∶</s:mo><s:mi>B</s:mi><s:mo stretchy="false">)</s:mo></s:math> for a large class of random tensor networks at large bond dimension and show that it is equal to the entanglement wedge cross section <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>E</w:mi><w:mi>W</w:mi><w:mo stretchy="false">(</w:mo><w:mi>A</w:mi><w:mo>∶</w:mo><w:mi>B</w:mi><w:mo stretchy="false">)</w:mo></w:math>, proving a previous conjecture motivated from AdS/CFT.