Editing Symplectic matrix

{{Short description|Mathematical concept}}
In mathematics, a '''symplectic matrix''' is a <math>2n\times 2n</math> [[matrix (mathematics)|matrix]] <math>M</math> with [[real number|real]] entries that satisfies the condition

{{NumBlk||<math display="block">M^\text{T} \Omega M = \Omega,</math>|{{EquationRef|1}}}}

where <math>M^\text{T}</math> denotes the [[transpose]] of <math>M</math> and <math>\Omega</math> is a fixed <math>2n\times 2n</math> [[nonsingular matrix|nonsingular]], [[skew-symmetric matrix]]. This definition can be extended to <math>2n\times 2n</math> matrices with entries in other [[field (mathematics)|field]]s, such as the [[complex number]]s, [[finite field]]s, [[p-adic number|''p''-adic number]]s, and [[Function field of an algebraic variety|function fields]].

Typically <math>\Omega</math> is chosen to be the [[block matrix]]
<math display="block"> \Omega = \begin{bmatrix}
0 & I_n \\
-I_n & 0 \\
\end{bmatrix}, </math>
where <math>I_n</math> is the <math>n\times n</math> [[identity matrix]]. The matrix <math>\Omega</math> has [[determinant]] <math>+1</math> and its inverse is <math>\Omega^{-1} = \Omega^\text{T} = -\Omega</math>.

==Properties==

=== Generators for symplectic matrices ===
Every symplectic matrix has determinant <math>+1</math>, and the <math>2n\times 2n</math> symplectic matrices with real entries form a [[subgroup]] of the [[general linear group]] <math>\mathrm{GL}(2n;\mathbb{R})</math> under [[matrix multiplication]] since being symplectic is a property stable under matrix multiplication. [[Topology|Topologically]], this [[symplectic group]] is a [[connected space|connected]] [[compact space|noncompact]] [[real Lie group]] of real dimension <math>n(2n+1)</math>, and is denoted <math>\mathrm{Sp}(2n;\mathbb{R})</math>. The symplectic group can be defined as the set of [[linear transformations]] that preserve the symplectic form of a real [[symplectic vector space]].

This symplectic group has a distinguished [[Generating set of a group|set of generators]], which can be used to find all possible symplectic matrices. This includes the following sets
<math display="block">\begin{align}
D(n) =& \left\{
\begin{pmatrix}
A & 0 \\
0 & (A^T)^{-1}
\end{pmatrix} : A \in \text{GL}(n;\mathbb{R})
\right\} \\
N(n) =& \left\{
\begin{pmatrix}
I_n & B \\
0 & I_n
\end{pmatrix} : B \in \text{Sym}(n;\mathbb{R})
\right\}
\end{align}</math>
where <math>\text{Sym}(n;\mathbb{R})</math> is the set of <math>n\times n</math> [[Symmetric matrix|symmetric matrices]]. Then, <math>\mathrm{Sp}(2n;\mathbb{R})</math> is generated by the set<ref>{{Cite book|last=Habermann, Katharina, 1966-|url=http://worldcat.org/oclc/262692314|title=Introduction to symplectic Dirac operators|date=2006|publisher=Springer|isbn=978-3-540-33421-7|oclc=262692314}}</ref><sup>p. 2</sup>
<math display="block">\{\Omega \} \cup D(n) \cup N(n)</math>
of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in <math>D(n)</math> and <math>N(n)</math> together, along with some power of <math>\Omega</math>.

=== Inverse matrix ===
Every symplectic matrix is invertible with the [[inverse matrix]] given by
<math display="block">
M^{-1} = \Omega^{-1} M^\text{T} \Omega.
</math>
Furthermore, the [[matrix multiplication|product]] of two symplectic matrices is, again, a symplectic matrix. This gives the set of all symplectic matrices the structure of a [[group (mathematics)|group]]. There exists a natural [[manifold]] structure on this group which makes it into a (real or complex) [[Lie group]] called the [[symplectic group]].

=== Determinantal properties ===
It follows easily from the definition that the [[determinant]] of any symplectic matrix is ±1. Actually, it turns out that the determinant is always +1 for any field. One way to see this is through the use of the [[Pfaffian]] and the identity
<math display="block">\mbox{Pf}(M^\text{T} \Omega M) = \det(M)\mbox{Pf}(\Omega).</math>
Since <math>M^\text{T} \Omega M = \Omega</math> and <math>\mbox{Pf}(\Omega) \neq 0</math> we have that <math>\det(M) = 1</math>.

When the underlying field is real or complex, one can also show this by factoring the inequality <math>\det(M^\text{T} M + I) \ge 1</math>.<ref>{{cite journal |last=Rim |first=Donsub |date=2017 |title=An elementary proof that symplectic matrices have determinant one |journal=Adv. Dyn. Syst. Appl. |volume=12 |issue=1 |pages=15–20 |doi=10.37622/ADSA/12.1.2017.15-20 |arxiv=1505.04240  |s2cid=119595767 }}</ref>

=== Block form of symplectic matrices ===
Suppose Ω is given in the standard form and let <math>M</math> be a <math>2n\times 2n</math> [[block matrix]] given by
<math display="block">M = \begin{pmatrix}A & B \\ C & D\end{pmatrix}</math>
where <math>A,B,C,D</math> are <math>n\times n</math> matrices. The condition for <math>M</math> to be symplectic is equivalent to the two following equivalent conditions<ref>{{cite web|last1=de Gosson|first1=Maurice|title=Introduction to Symplectic Mechanics: Lectures I-II-III|url=https://www.ime.usp.br/~piccione/Downloads/LecturesIME.pdf}}</ref><blockquote><math>A^\text{T}C,B^\text{T}D</math> symmetric, and <math>A^\text{T} D - C^\text{T} B = I</math></blockquote><blockquote><math>AB^\text{T},CD^\text{T}</math> symmetric, and <math>AD^\text{T} - BC^\text{T} = I</math></blockquote>The second condition comes from the fact that if <math>M</math> is symplectic, then <math>M^T</math> is also symplectic. When <math>n=1</math> these conditions reduce to the single condition <math>\det(M)=1</math>. Thus a <math>2\times 2</math> matrix is symplectic [[iff]] it has unit determinant.

==== Inverse matrix of block matrix ====
With <math>\Omega</math> in standard form, the inverse of <math>M</math> is given by
<math display="block">
M^{-1} = \Omega^{-1} M^\text{T} \Omega=\begin{pmatrix}D^\text{T} & -B^\text{T} \\-C^\text{T} & A^\text{T}\end{pmatrix}.</math>
The group has dimension <math>n(2n+1)</math>. This can be seen by noting that <math>( M^\text{T} \Omega M)^\text{T} = -M^\text{T} \Omega M</math> is anti-symmetric. Since the space of anti-symmetric matrices has dimension <math>\binom{2n}{2},</math> the identity <math> M^\text{T} \Omega M = \Omega</math> imposes <math>2n \choose 2</math> constraints on the <math>(2n)^2</math> coefficients of <math>M</math> and leaves <math>M</math> with <math>n(2n+1)</math> independent coefficients.

==Symplectic transformations==

In the abstract formulation of [[linear algebra]], matrices are replaced with [[linear transformation]]s of [[finite-dimensional]] [[vector spaces]]. The abstract analog of a symplectic matrix is a '''symplectic transformation''' of a [[symplectic vector space]]. Briefly, a symplectic vector space <math>(V,\omega)</math> is a <math>2n</math>-dimensional vector space <math>V</math> equipped with a [[nondegenerate form|nondegenerate]], [[skew-symmetric matrix|skew-symmetric]] [[bilinear form]] <math>\omega</math> called the [[symplectic form]].

A symplectic transformation is then a linear transformation <math>L:V\to V</math> which preserves <math>\omega</math>, i.e.
<math display="block">\omega(Lu, Lv) = \omega(u, v).</math>
Fixing a [[basis (linear algebra)|basis]] for <math>V</math>, <math>\omega</math> can be written as a matrix <math>\Omega</math> and <math>L</math> as a matrix <math>M</math>. The condition that <math>L</math> be a symplectic transformation is precisely the condition that ''M'' be a symplectic matrix:
<math display="block">M^\text{T} \Omega M = \Omega.</math>

Under a [[change of basis]], represented by a matrix ''A'', we have
<math display="block">\Omega \mapsto A^\text{T} \Omega A</math>
<math display="block">M \mapsto A^{-1} M A.</math>
One can always bring <math>\Omega</math> to either the standard form given in the introduction or the block diagonal form described below by a suitable choice of ''A''.

==The matrix Ω==
Symplectic matrices are defined relative to a fixed [[nonsingular matrix|nonsingular]], [[skew-symmetric matrix]] <math>\Omega</math>. As explained in the previous section, <math>\Omega</math> can be thought of as the coordinate representation of a [[nondegenerate form|nondegenerate]] [[skew-symmetric bilinear form]]. It is a basic result in [[linear algebra]] that any two such matrices differ from each other by a [[change of basis]].

The most common alternative to the standard <math>\Omega</math> given above is the [[block diagonal]] form
<math display="block">\Omega = \begin{bmatrix}
\begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\
 & \ddots & \\
0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix}
\end{bmatrix}.</math>
This choice differs from the previous one by a [[permutation]] of [[basis vectors]].

Sometimes the notation <math>J</math> is used instead of <math>\Omega</math> for the skew-symmetric matrix. This is a particularly unfortunate choice as it leads to confusion with the notion of a [[linear complex structure|complex structure]], which often has the same coordinate expression as <math>\Omega</math> but represents a very different structure. A complex structure <math>J</math> is the coordinate representation of a linear transformation that squares to <math>-I_n</math>, whereas <math>\Omega</math> is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which <math>J</math> is not skew-symmetric or <math>\Omega</math> does not square to <math>-I_n</math>.

Given a [[hermitian structure]] on a vector space, <math>J</math> and <math>\Omega</math> are related via
<math display="block">\Omega_{ab} = -g_{ac}{J^c}_b</math>
where <math>g_{ac}</math> is the [[metric tensor|metric]]. That <math>J</math> and <math>\Omega</math> usually have the same coordinate expression (up to an overall sign) is simply a consequence of the fact that the metric ''g'' is usually the identity matrix.

==Diagonalization and decomposition==
{{bullet list
|For any [[Positive-definite matrix|positive definite]] symmetric <math>2n\times 2n</math> real symplectic matrix <math>S</math>, there is a symplectic unitary <math>U</math>, <math display="block">U \in \mathrm{U}(2n,\mathbb{R}) \cap \operatorname{Sp}(2n,\mathbb{R}) = \mathrm{O}(2n) \cap \operatorname{Sp}(2n,\mathbb{R}), </math>such that<math display="block">S = U^\text{T} D U \quad \text{for} \quad D = \operatorname{diag}(\lambda_1,\ldots,\lambda_n,\lambda_1^{-1},\ldots,\lambda_n^{-1}),</math>where the diagonal elements of <math>D</math> are the [[Eigenvalues and eigenvectors|eigenvalues]] of <math>S</math>.<ref name=":0">{{Cite book|title=Symplectic Methods in Harmonic Analysis and in Mathematical Physics - Springer|last=de Gosson|first=Maurice A.|language=en|doi=10.1007/978-3-7643-9992-4|year = 2011|isbn = 978-3-7643-9991-7}}</ref><ref>{{cite arXiv|first1=Martin|last1=Houde|first2=Will|last2=McCutcheon|first3=Nicolas|last3=Quesada|title=Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson|at= Sec. V, p. 5 |date=13 March 2024|eprint=2403.04596}}</ref>

|Any real symplectic matrix {{math|S}} has a [[polar decomposition]] of the form:<ref name=":0" /><math display="block">S = UR,</math>where<math display="block">U \in \operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{U}(2n,\mathbb{R}),</math> and<math display="block">R \in \operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{Sym}_+(2n,\mathbb{R}).</math>

|Any real symplectic matrix can be decomposed as a product of three matrices:<math display="block">S = O\begin{pmatrix}D & 0 \\ 0 & D^{-1}\end{pmatrix}O',</math>where <math>O</math> and <math>O'</math> are both symplectic and [[Orthogonal matrix|orthogonal]], and <math>D</math> is [[Positive-definite matrix|positive-definite]] and [[Diagonal matrix|diagonal]].<ref>{{cite arXiv|first1=Alessandro|last1=Ferraro|first2=Stefano|last2=Olivares|first3=Matteo G. A.|last3=Paris|title=Gaussian states in continuous variable quantum information|at= Sec. 1.3, p. 4 |date=31 March 2005|eprint=quant-ph/0503237}}</ref> This decomposition is closely related to the [[singular value decomposition]] of a matrix and is known as an 'Euler' or 'Bloch-Messiah' decomposition.

| The set of orthogonal symplectic matrices forms a (maximal) compact subgroup of the symplectic group.<ref name=":4">{{ cite book|title= Quantum Continuous Variables |last=Serafini|first=Alessio|language=en|doi=10.1201/9781003250975|year = 2023|isbn = 9781003250975}}</ref> This set is isomorphic to the set of unitary matrices of dimension <math> n </math>, <math>\mathrm{U}(2n,\mathbb{R}) \cap \operatorname{Sp}(2n,\mathbb{R}) = \mathrm{O}(2n) \cap \operatorname{Sp}(2n,\mathbb{R}) \cong \mathrm{U}(n,\mathbb{C})</math>. Every symplectic orthogonal matrix can be written as
{{NumBlk||<math display="block">
\begin{pmatrix}
\Re(V) & -\Im(V) \\
\Im(V) & \Re(V)
\end{pmatrix} = \left[\frac{1}{\sqrt{2}}\begin{pmatrix}
I_n & i  I_n \\
I_n & -i I_n
\end{pmatrix}  \right]^\dagger \begin{pmatrix}
V & 0 \\
0 & V^*
\end{pmatrix} \left[\frac{1}{\sqrt{2}}\begin{pmatrix}
I_n & i  I_n \\
I_n & -i I_n
\end{pmatrix}  \right],
</math><br/>|{{EquationRef|2}}}}
with <math> V \in \mathrm{U}(n,\mathbb{C})</math>.
This equation implies that every symplectic orthogonal matrix has determinant equal to +1 and thus that this is true for all symplectic matrices as its polar decomposition is itself given in terms symplectic matrices.
}}

==Complex matrices==

If instead ''M'' is a {{nowrap|2''n'' × 2''n''}} [[matrix (mathematics)|matrix]] with [[complex number|complex]] entries, the definition is not standard throughout the literature.  Many authors <ref>{{cite journal|last = Xu|first= H. G.|title= An SVD-like matrix decomposition and its applications|journal= Linear Algebra and Its Applications|date= July 15, 2003|volume= 368|pages=1–24|doi = 10.1016/S0024-3795(03)00370-7|hdl= 1808/374|hdl-access= free}}</ref> adjust the definition above to
{{NumBlk||<math display="block">M^* \Omega M = \Omega\,.</math>|{{EquationRef|3}}}}
where ''M<sup>*</sup>'' denotes the [[conjugate transpose]] of ''M''. In this case, the determinant may not be 1, but will have [[absolute value]] 1. In the 2×2 case (''n''=1), ''M'' will be the product of a real symplectic matrix and a complex number of absolute value 1.

Other authors <ref>{{Cite report|last1=Mackey |last2= Mackey|first1= D. S. |first2= N.|title=  On the Determinant of Symplectic Matrices|year= 2003
|type=Numerical Analysis Report 422|publisher=Manchester Centre for Computational Mathematics|location=Manchester, England
}}</ref> retain the definition ({{EquationNote|1}}) for complex matrices and call matrices satisfying ({{EquationNote|3}}) ''conjugate symplectic''.

==Applications==

Transformations described by symplectic matrices play an important role in [[Quantum Optics|quantum optics]] and in [[Continuous-variable quantum information|continuous-variable quantum information theory]]. For instance, symplectic matrices can be used to describe [[Bogoliubov transformation|Gaussian (Bogoliubov) transformations]] of a quantum state of light.<ref>{{Cite journal|last1=Weedbrook|first1=Christian|last2=Pirandola|first2=Stefano|last3=García-Patrón|first3=Raúl|last4=Cerf|first4=Nicolas J.|last5=Ralph|first5=Timothy C.|last6=Shapiro|first6=Jeffrey H.|last7=Lloyd|first7=Seth|date=2012|title=Gaussian quantum information|journal=Reviews of Modern Physics|volume=84|issue=2|pages=621–669|arxiv=1110.3234|doi=10.1103/RevModPhys.84.621|bibcode=2012RvMP...84..621W|s2cid=119250535 }}</ref> In turn, the Bloch-Messiah decomposition ({{EquationNote|2}}) means that such an arbitrary Gaussian transformation can be represented as a set of two passive [[Linear optics|linear-optical]] interferometers (corresponding to orthogonal matrices ''O'' and ''O' '') intermitted by a layer of active non-linear [[Squeezed coherent state#Operator representation|squeezing]] transformations (given in terms of the matrix ''D'').<ref>{{Cite journal|last=Braunstein|first=Samuel L.|title=Squeezing as an irreducible resource|date=2005|journal=Physical Review A|volume=71|issue=5|pages=055801|doi=10.1103/PhysRevA.71.055801|arxiv=quant-ph/9904002|bibcode=2005PhRvA..71e5801B|s2cid=16714223 }}</ref> In fact, one can circumvent the need for such ''in-line'' active squeezing transformations if [[Squeezed coherent state|two-mode squeezed vacuum states]] are available as a prior resource only.<ref>{{cite journal|last1=Chakhmakhchyan|first1=Levon|last2=Cerf|first2=Nicolas|title= Simulating arbitrary Gaussian circuits with linear optics|journal=Physical Review A|date=2018|volume=98|issue=6|page=062314|doi=10.1103/PhysRevA.98.062314|arxiv=1803.11534|bibcode=2018PhRvA..98f2314C|s2cid=119227039 }}</ref>

==See also==
{{Portal|Mathematics}}
* [[Symplectic vector space]]
* [[Symplectic group]]
* [[Symplectic representation]]
* [[Orthogonal matrix]]
* [[Unitary matrix]]
* [[Hamiltonian mechanics]]
* [[Linear complex structure]]
* [[Williamson theorem]]

==References==
{{reflist}}

{{Matrix classes}}

[[Category:Matrices (mathematics)]]
[[Category:Symplectic geometry]]