Alternative to purification in conformal field theory

<jats:p>In conformal field theories, in contrast to some auxiliary states into the bipartite mixed state <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>ρ</a:mi><a:mrow><a:mi>A</a:mi><a:mi>B</a:mi></a:mrow></a:msub></a:math> as the usual purifications do, we show a pure entangled state <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>ψ</c:mi><c:mrow><c:mi>A</c:mi><c:mi>B</c:mi></c:mrow></c:msub></c:math> can be constructed by the undetectable regions. In this pure state <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>ψ</e:mi><e:mrow><e:mi>A</e:mi><e:mi>B</e:mi></e:mrow></e:msub></e:math>, the von Neumann entropy <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>S</g:mi><g:mi>vN</g:mi></g:msub><g:mo stretchy="false">(</g:mo><g:mi>A</g:mi><g:mo>:</g:mo><g:mi>B</g:mi><g:mo stretchy="false">)</g:mo></g:math> naturally captures quantum entanglement between <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>A</k:mi></k:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>B</m:mi></m:math>. We verify that <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>S</o:mi><o:mi>vN</o:mi></o:msub><o:mo stretchy="false">(</o:mo><o:mi>A</o:mi><o:mo>:</o:mo><o:mi>B</o:mi><o:mo stretchy="false">)</o:mo></o:math> is equal to the entanglement wedge cross section <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>E</s:mi><s:mi>W</s:mi></s:msub></s:math> in anti–de Sitter spacetime, which is conjectured to be the holographic dual of the entanglement of purification.</jats:p> <jats:sec> <jats:title/> <jats:supplementary-material> <jats:permissions> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material> </jats:sec>