Asymptotically independent fluctuations of stabilizer Rényi entropy and entanglement in random unitary circuits

We investigate numerically the joint distribution of magic (as described by the stabilizer 2-Rényi entropy <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>M</a:mi><a:mn>2</a:mn></a:msub></a:math>) and entanglement (as described by the von Neumann entropy <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>S</b:mi></b:math>) in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>N</c:mi></c:math>-qubit Haar-random quantum states. The distribution <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msub><d:mi>P</d:mi><d:mi>N</d:mi></d:msub><d:mrow><d:mo>(</d:mo><d:msub><d:mi>M</d:mi><d:mn>2</d:mn></d:msub><d:mo>,</d:mo><d:mi>S</d:mi><d:mo>)</d:mo></d:mrow></d:mrow></d:math> and the marginals become exponentially localized, and centered around the typical values, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mover accent="true"><e:msub><e:mi>M</e:mi><e:mn>2</e:mn></e:msub><e:mo>̃</e:mo></e:mover><e:mo>→</e:mo><e:mi>N</e:mi><e:mo>−</e:mo><e:mn>2</e:mn></e:mrow></e:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mover accent="true"><g:mi>S</g:mi><g:mo>̃</g:mo></g:mover><g:mo>→</g:mo><g:mi>N</g:mi><g:mo>/</g:mo><g:mn>2</g:mn></g:mrow></g:math> as <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi>N</i:mi><i:mo>→</i:mo><i:mi>∞</i:mi></i:mrow></i:math>. Magic and entanglement fluctuations are, however, found to become exponentially uncorrelated. While the set of stabilizer states grows exponentially with <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>N</j:mi></j:math>, they nevertheless form a measure-zero subset of Haar-random states and thus occur with vanishing probability. Typical quantum states are characterized by large magic and entanglement entropy, and uncorrelated magic and entanglement fluctuations. The mutual information of the joint distribution vanishes exponentially as <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mi>N</k:mi><k:mo>→</k:mo><k:mi>∞</k:mi></k:mrow></k:math>, implying that fluctuations of these two resources are not only uncorrelated but asymptotically independent.