Symplectic matrix

In mathematics, a symplectic matrix is a $2n\times 2n$ matrix $M$ with real entries that satisfies the condition

$$M^{\text{T}}\Omega M=\Omega ,$$ 1

where $M^{\text{T}}$ denotes the transpose of $M$ and $\Omega$ is a fixed $2n\times 2n$ nonsingular, skew-symmetric matrix. This definition can be extended to $2n\times 2n$ matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.

Typically $\Omega$ is chosen to be the block matrix $$\Omega =\left[\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right],$$ where $I_n$ is the $n\times n$ identity matrix. The matrix $\Omega$ has determinant $+1$ and its inverse is ${\Omega }^{-1}={\Omega }^{\text{T}}=-\Omega$.

Properties

Generators for symplectic matrices

Every symplectic matrix has determinant $+1$, and the $2n\times 2n$ symplectic matrices with real entries form a subgroup of the general linear group $\mathrm{G}\mathrm{L}(2n;ℝ)$ under matrix multiplication since being symplectic is a property stable under matrix multiplication. Topologically, this symplectic group is a connected noncompact real Lie group of real dimension $n(2n+1)$, and is denoted $\mathrm{S}\mathrm{p}(2n;ℝ)$. 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 set of generators, which can be used to find all possible symplectic matrices. This includes the following sets $$\begin{array}{cc}D(n)=& \{\left(\begin{array}{cc}A& 0\\ 0& (A^T{)}^{-1}\end{array}\right):A\in \text{GL}(n;ℝ)\}\\ N(n)=& \{\left(\begin{array}{cc}{I}_n& B\\ 0& I_n\end{array}\right):B\in \text{Sym}(n;ℝ)\}\end{array}$$ where $\text{Sym}(n;ℝ)$ is the set of $n\times n$ symmetric matrices. Then, $\mathrm{S}\mathrm{p}(2n;ℝ)$ is generated by the set^{p. 2} $$\{\Omega \}\cup D(n)\cup N(n)$$ of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in $D(n)$ and $N(n)$ together, along with some power of $\Omega$.

Inverse matrix

Every symplectic matrix is invertible with the inverse matrix given by $$M^{-1}={\Omega }^{-1}{M}^{\text{T}}\Omega .$$ Furthermore, the product of two symplectic matrices is, again, a symplectic matrix. This gives the set of all symplectic matrices the structure of a 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 $$\text{Pf}(M^{\text{T}}\Omega M)=\det (M)\text{Pf}(\Omega ).$$ Since $M^{\text{T}}\Omega M=\Omega$ and $\text{Pf}(\Omega )\ne 0$ we have that $\det (M)=1$.

When the underlying field is real or complex, one can also show this by factoring the inequality $\det (M^{\text{T}}M+I)\ge 1$.

Block form of symplectic matrices

Suppose Ω is given in the standard form and let $M$ be a $2n\times 2n$ block matrix given by $$M=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$$

where

$A,B,C,D$

are

$n\times n$

matrices. The condition for

$M$

to be symplectic is equivalent to the two following equivalent conditions

$A^{\text{T}}C,B^{\text{T}}D$

symmetric, and

$A^{\text{T}}D-C^{\text{T}}B=I$

$A{B}^{\text{T}},C{D}^{\text{T}}$

symmetric, and

$A{D}^{\text{T}}-B{C}^{\text{T}}=I$

The second condition comes from the fact that if

$M$

is symplectic, then

$M^T$

is also symplectic. When

$n=1$

these conditions reduce to the single condition

$\det (M)=1$

. Thus a

$2\times 2$

matrix is symplectic iff it has unit determinant.

Inverse matrix of block matrix

With $\Omega$ in standard form, the inverse of $M$ is given by $$M^{-1}={\Omega }^{-1}{M}^{\text{T}}\Omega =\left(\begin{array}{cc}{D}^{\text{T}}& -B^{\text{T}}\\ -C^{\text{T}}& A^{\text{T}}\end{array}\right).$$ The group has dimension $n(2n+1)$. This can be seen by noting that $(M^{\text{T}}\Omega M{)}^{\text{T}}=-M^{\text{T}}\Omega M$ is anti-symmetric. Since the space of anti-symmetric matrices has dimension $(\genfrac{}{}{0ex}{}{2n}{2}),$ the identity $M^{\text{T}}\Omega M=\Omega$ imposes $(\genfrac{}{}{0ex}{}{2n}{2})$ constraints on the $(2n{)}^2$ coefficients of $M$ and leaves $M$ with $n(2n+1)$ independent coefficients.

Symplectic transformations

In the abstract formulation of linear algebra, matrices are replaced with linear transformations 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 $(V,\omega )$ is a $2n$-dimensional vector space $V$ equipped with a nondegenerate, skew-symmetric bilinear form $\omega$ called the symplectic form.

A symplectic transformation is then a linear transformation $L:V\to V$ which preserves $\omega$, i.e. $$\omega (Lu,Lv)=\omega (u,v).$$ Fixing a basis for $V$, $\omega$ can be written as a matrix $\Omega$ and $L$ as a matrix $M$. The condition that $L$ be a symplectic transformation is precisely the condition that M be a symplectic matrix: $$M^{\text{T}}\Omega M=\Omega .$$

Under a change of basis, represented by a matrix A, we have $$\Omega \mapsto A^{\text{T}}\Omega A$$ $$M\mapsto A^{-1}MA.$$ One can always bring $\Omega$ 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, skew-symmetric matrix $\Omega$. As explained in the previous section, $\Omega$ can be thought of as the coordinate representation of a 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 $\Omega$ given above is the block diagonal form $$\Omega =\left[\begin{array}{ccc}\begin{array}{cc}0& 1\\ -1& 0\end{array}& & 0\\ & \ddots & \\ 0& & \begin{array}{cc}0& 1\\ -1& 0\end{array}\end{array}\right].$$ This choice differs from the previous one by a permutation of basis vectors.

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

Given a hermitian structure on a vector space, $J$ and $\Omega$ are related via $${\Omega }_{ab}=-g_{ac}{J^c}_b$$ where $g_{ac}$ is the metric. That $J$ and $\Omega$ 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

• For any positive definite symmetric $2n\times 2n$ real symplectic matrix $S$, there is a symplectic unitary $U$, $$U\in \mathrm{U}(2n,ℝ)\cap \mathrm{Sp}(2n,ℝ)=\mathrm{O}(2n)\cap \mathrm{Sp}(2n,ℝ),$$such that$$S=U^{\text{T}}DU\text{for}D=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n,{\lambda }_1^{-1},\dots ,{\lambda }_n^{-1}),$$where the diagonal elements of $D$ are the eigenvalues of $S$.
• Any real symplectic matrix S has a polar decomposition of the form:$$S=UR,$$where$$U\in \mathrm{Sp}(2n,ℝ)\cap \mathrm{U}(2n,ℝ),$$ and$$R\in \mathrm{Sp}(2n,ℝ)\cap {\mathrm{Sym}}_{+}(2n,ℝ).$$
• Any real symplectic matrix can be decomposed as a product of three matrices:$$S=O\left(\begin{array}{cc}D& 0\\ 0& D^{-1}\end{array}\right)O^{\prime },$$where $O$ and $O^{\prime }$ are both symplectic and orthogonal, and $D$ is positive-definite and diagonal. 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. This set is isomorphic to the set of unitary matrices of dimension $n$, $\mathrm{U}(2n,ℝ)\cap \mathrm{Sp}(2n,ℝ)=\mathrm{O}(2n)\cap \mathrm{Sp}(2n,ℝ)\cong \mathrm{U}(n,ℂ)$. Every symplectic orthogonal matrix can be written as
  $$\left(\begin{array}{cc}\mathrm{\Re }(V)& -\mathrm{\Im }(V)\\ \mathrm{\Im }(V)& \mathrm{\Re }(V)\end{array}\right)={\left[\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{I}_n& i{I}_n\\ I_n& -i{I}_n\end{array}\right)\right]}^{†}\left(\begin{array}{cc}V& 0\\ 0& V^{*}\end{array}\right)\left[\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{I}_n& i{I}_n\\ I_n& -i{I}_n\end{array}\right)\right],$$ 2

  with $V\in \mathrm{U}(n,ℂ)$.

  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 2n × 2n matrix with complex entries, the definition is not standard throughout the literature. Many authors adjust the definition above to

$$M^{*}\Omega M=\Omega .$$ 3

where M^* 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 retain the definition (1) for complex matrices and call matrices satisfying (3) conjugate symplectic.

Applications

Transformations described by symplectic matrices play an important role in quantum optics and in continuous-variable quantum information theory. For instance, symplectic matrices can be used to describe Gaussian (Bogoliubov) transformations of a quantum state of light. In turn, the Bloch-Messiah decomposition (2) means that such an arbitrary Gaussian transformation can be represented as a set of two passive linear-optical interferometers (corresponding to orthogonal matrices O and O' ) intermitted by a layer of active non-linear squeezing transformations (given in terms of the matrix D). In fact, one can circumvent the need for such in-line active squeezing transformations if two-mode squeezed vacuum states are available as a prior resource only.

See also

• Symplectic vector space
• Symplectic group
• Symplectic representation
• Orthogonal matrix
• Unitary matrix
• Hamiltonian mechanics
• Linear complex structure
• Williamson theorem
• Hamiltonian matrix