Finite complexity of the de Sitter vacuum

The <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>Einstein</a:mi><a:mtext>−</a:mtext><a:mi>Rosen</a:mi><a:mo>=</a:mo><a:mi>Einstein</a:mi><a:mtext>−</a:mtext><a:mi>Podolsky</a:mi><a:mtext>−</a:mtext><a:mi>Rosen</a:mi></a:mrow></a:math> (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mi>ER</c:mi><c:mo>=</c:mo><c:mi>EPR</c:mi></c:mrow></c:math>) conjecture states that quantum entanglement between boundary degrees of freedom leads to the emergence of bulk spacetime itself. Although this has been tested extensively in string theory for asymptotically anti–de Sitter spacetimes, its implications for an accelerating universe, such as our own, remain less explored. Assuming a cosmic version of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mi>ER</e:mi><e:mo>=</e:mo><e:mi>EPR</e:mi></e:mrow></e:math> for de Sitter space, we explore computational complexity corresponding to long-range entanglement responsible for bulk states on spacelike hypersurfaces. Rather remarkably, we find that the complexity (per unit volume) of the Euclidean vacuum, as an entangled state over two boundary conformal field theory vacua, is finite both in the UV and the IR, which provides additional evidence for “cosmic <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:mi>ER</g:mi><g:mo>=</g:mo><g:mi>EPR</g:mi></g:mrow></g:math>.” Our result seems to be a universal feature of spacetimes with horizons and, moreover, hints at new features of the thermofield double state for studying thermalization of any quantum system.