Symplectic matrix

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

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

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} $$\{\mathrm{\Omega }\}\cup D(n)\cup N(n)$$ 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 $\mathrm{\Omega }$.

Inverse matrix

Every symplectic matrix is invertible with the inverse matrix given by $$M^{-1}={\mathrm{\Omega }}^{-1}{M}^{\text{T}}\mathrm{\Omega }.$$ 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 $$\text{Pf}(M^{\text{T}}\mathrm{\Omega }M)=\det (M)\text{Pf}(\mathrm{\Omega }).$$ Since $M^{\text{T}}\mathrm{\Omega }M=\mathrm{\Omega }$ and $\text{Pf}(\mathrm{\Omega })\ne 0$ 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 $\mathrm{\Omega }$ in standard form, the inverse of $M$ is given by $$M^{-1}={\mathrm{\Omega }}^{-1}{M}^{\text{T}}\mathrm{\Omega }=\left(\begin{array}{cc}{D}^{\text{T}}& -B^{\text{T}}\\ -C^{\text{T}}& A^{\text{T}}\end{array}\right).$$ The group has dimension $n(2n+1)$. This can be seen by noting that $(M^{\text{T}}\mathrm{\Omega }M{)}^{\text{T}}=-M^{\text{T}}\mathrm{\Omega }M$ is anti-symmetric. Since the space of anti-symmetric matrices has dimension $(\genfrac{}{}{0ex}{}{2n}{2}),$ the identity $M^{\text{T}}\mathrm{\Omega }M=\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ and $L$ as a matrix $M$. The condition that $L$ be a symplectic transformation is precisely the condition that M be a symplectic matrix:

$M^{\text{T}}\mathrm{\Omega }M=\mathrm{\Omega }.$

Under a change of basis, represented by a matrix A, we have

$\mathrm{\Omega }\mapsto A^{\text{T}}\mathrm{\Omega }A$

$M\mapsto A^{-1}MA.$

One can always bring $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$. As explained in the previous section, $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ given above is the block diagonal form

$\mathrm{\Omega }=\left[\begin{array}{ccc}\begin{array}{cc}0& 1\\ -1& 0\end{array}& & 0\\ & \ddots & \\ 0& & \begin{array}{cc}0& 1\\ -1& 0\end{array}\end{array}\right].$

This choice differs from the previous one by a permutation of basis vectors.

Sometimes the notation $J$ is used instead of $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ but represents a very different structure. A complex structure $J$ is the coordinate representation of a linear transformation that squares to $-I_n$, whereas $\mathrm{\Omega }$ is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which $J$ is not skew-symmetric or $\mathrm{\Omega }$ does not square to $-I_n$.

Given a hermitian structure on a vector space, $J$ and $\mathrm{\Omega }$ are related via

${\mathrm{\Omega }}_{ab}=-g_{ac}{J^c}_b$

where $g_{ac}$ is the metric. That $J$ and $\mathrm{\Omega }$ 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 Template:Math 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

Template:NumBlk 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 Template:Nowrap matrix with complex entries, the definition is not standard throughout the literature. Many authors adjust the definition above to Template:NumBlk 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 (Template:EquationNote) for complex matrices and call matrices satisfying (Template:EquationNote) 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 (Template:EquationNote) 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