.. _sphx_glr_api_gallery_rmt_wishart:

Wishart and Jacobi ensembles
----------------------------

Covariance-type ensembles: Wishart's original sample-covariance
construction, the Marchenko-Pastur law, the Wachter law for Jacobi (MANOVA)
ensembles, and Page's conjecture for the average entanglement entropy of a
random quantum state.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The Wishart (Laguerre) ensembles LOE (\beta=1), LUE (\beta=2), and LSE (\beta=4) are built from the sample covariance matrix W = X X^\dagger of an n \times m Gaussian data matrix X (n variables, m samples, m \geq n), with real, complex, or quaternionic entries respectively. The relevant control parameter is the aspect ratio \gamma = n/m \in (0, 1].">

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  .. image:: /api/gallery/rmt/wishart/images/thumb/sphx_glr_plot_marchenko_pastur_law_thumb.png
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  :doc:`/api/gallery/rmt/wishart/plot_marchenko_pastur_law`

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      <div class="sphx-glr-thumbnail-title">The Marchenko-Pastur Law</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="A Haar-random pure state on \mathbb{C}^n \otimes \mathbb{C}^k (n \leq k), traced out over the k-dimensional subsystem, leaves an n-dimensional reduced density matrix \rho = A A^\dagger / \mathrm{Tr}(A A^\dagger) built from an n \times k complex Ginibre matrix A (the &quot;induced measure&quot;; k=n is the flat Hilbert-Schmidt measure on density matrices). Page&#x27;s exact result gives the average von Neumann entropy S = -\sum_i \lambda_i \log \lambda_i (in nats, \lambda_i the eigenvalues of \rho) of this random reduced state, for any finite n, k:">

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  .. image:: /api/gallery/rmt/wishart/images/thumb/sphx_glr_plot_page_entanglement_entropy_thumb.png
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  :doc:`/api/gallery/rmt/wishart/plot_page_entanglement_entropy`

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      <div class="sphx-glr-thumbnail-title">Page's Conjecture: the Average Entanglement Entropy of a Random State</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="The Jacobi (MANOVA) ensemble is built from two independent Wishart-type matrices A and B, each n \times n and formed from m_1 and m_2 independent Gaussian &quot;samples&quot; of n variables respectively (m_1, m_2 \geq n). The double-Wishart ratio matrix">

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  .. image:: /api/gallery/rmt/wishart/images/thumb/sphx_glr_plot_wachter_law_thumb.png
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  :doc:`/api/gallery/rmt/wishart/plot_wachter_law`

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      <div class="sphx-glr-thumbnail-title">The Wachter Law for Jacobi Ensembles</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="For an m \times n data matrix X of m independent samples of n variables, the sample covariance (scatter) matrix">

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  .. image:: /api/gallery/rmt/wishart/images/thumb/sphx_glr_plot_wishart_sample_covariance_thumb.png
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  :doc:`/api/gallery/rmt/wishart/plot_wishart_sample_covariance`

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      <div class="sphx-glr-thumbnail-title">Wishart's Sample Covariance Matrix</div>
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.. toctree::
   :hidden:

   /api/gallery/rmt/wishart/plot_marchenko_pastur_law
   /api/gallery/rmt/wishart/plot_page_entanglement_entropy
   /api/gallery/rmt/wishart/plot_wachter_law
   /api/gallery/rmt/wishart/plot_wishart_sample_covariance