Updates references (#384)

* updates references

* Updates biblio

---------

Co-authored-by: Nicolas Quesada <nquesada@pop-os.localdomain>

docs/references.bib

@@ -642,3 +642,17 @@ @article{bulmer2022threshold
   journal={arXiv preprint arXiv:2202.04600},
   year={2022}
 }
+
+@article{houde2024matrix,
+  title={Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson},
+  author={Houde, Martin and McCutcheon, Will and Quesada, Nicol{\'a}s},
+  journal={arXiv preprint arXiv:2403.04596},
+  year={2024}
+}
+
+@article{cardin2022photon,
+  title={Photon-number moments and cumulants of Gaussian states},
+  author={Cardin, Yanic and Quesada, Nicol{\'a}s},
+  journal={arXiv preprint arXiv:2212.06067},
+  year={2022}
+}

thewalrus/_montrealer.py

@@ -1,8 +1,8 @@
 """
 Montrealer Python interface
 
-* Yanic Cardin and Nicolás Quesada. "Photon-number moments and cumulants of Gaussian states"
-  `arxiv:12212.06067 (2023) <https://arxiv.org/abs/2212.06067>`_
+* See Yanic Cardin and Nicolás Quesada. "Photon-number moments and cumulants of Gaussian states"
+  `arxiv:12212.06067 (2023) <https://arxiv.org/abs/2212.06067>`_ :cite:`cardin2022photon`
 """
 import numpy as np
 import numba

thewalrus/decompositions.py

@@ -22,6 +22,10 @@
 This module implements common shared matrix decompositions that are
 used to perform gate decompositions.
 
+For mathematical details of these decompositions see
+
+`Houde et al. Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson <https://arxiv.org/abs/2403.04596>`_
+:cite:`houde2024matrix`
 Summary
 -------
 
@@ -46,8 +50,7 @@
 def williamson(V, rtol=1e-05, atol=1e-08):
     r"""Williamson decomposition of positive-definite (real) symmetric matrix.
 
-    See `this thread <https://math.stackexchange.com/questions/1171842/finding-the-symplectic-matrix-in-williamsons-theorem/2682630#2682630>`_
-    and the `Williamson decomposition documentation <https://strawberryfields.ai/photonics/conventions/decompositions.html#williamson-decomposition>`_
+    See `Houde et al. Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson <https://arxiv.org/abs/2403.04596>`_
 
     Args:
         V (array[float]): positive definite symmetric (real) matrix
@@ -116,7 +119,10 @@ def symplectic_eigenvals(cov):
 def blochmessiah(S):
     """Returns the Bloch-Messiah decomposition of a symplectic matrix S = O @ D @ Q
        where O and Q are orthogonal symplectic matrices and D is a positive-definite diagonal matrix
-       of the form diag(d1,d2,...,dn,d1^-1, d2^-1,...,dn^-1),
+       of the form diag(d1,d2,...,dn,d1^-1, d2^-1,...,dn^-1).
+
+       See `Houde et al. Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson <https://arxiv.org/abs/2403.04596>`_
+
 
     Args:
         S (array[float]): 2N x 2N real symplectic matrix
@@ -150,7 +156,8 @@ def takagi(A, svd_order=True):
     r"""Autonne-Takagi decomposition of a complex symmetric (not Hermitian!) matrix.
     Note that the input matrix is internally symmetrized by taking its upper triangular part.
     If the input matrix is indeed symmetric this leaves it unchanged.
-    See `Carl Caves note. <http://info.phys.unm.edu/~caves/courses/qinfo-s17/lectures/polarsingularAutonne.pdf>`_
+
+    See `Houde et al. Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson <https://arxiv.org/abs/2403.04596>`_
 
     Args:
         A (array): square, symmetric matrix
@@ -205,6 +212,7 @@ def takagi(A, svd_order=True):
 def pre_iwasawa(S):
     """Pre-Iwasawa decomposition of a symplectic matrix.
     See `Arvind et al. The Real Symplectic Groups in Quantum Mechanics and Optics <https://arxiv.org/pdf/quant-ph/9509002.pdf>`_
+    and `Houde et al. Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson <https://arxiv.org/abs/2403.04596>`_
 
 
     Args:
@@ -242,6 +250,7 @@ def pre_iwasawa(S):
 def iwasawa(S):
     """Iwasawa decomposition of a symplectic matrix.
     See `Arvind et al. The Real Symplectic Groups in Quantum Mechanics and Optics <https://arxiv.org/pdf/quant-ph/9509002.pdf>`_
+    and `Houde et al. Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson <https://arxiv.org/abs/2403.04596>`_
 
 
     Args: