Symplectic matrix

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 $$\Omega$$ 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 $$ \Omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}, $$ where $$I_n$$ is the $$n\times n$$ identity matrix. The matrix $$\Omega$$ has determinant $$+1$$ and its inverse is $$\Omega^{-1} = \Omega^\text{T} = -\Omega$$.

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 $$\mathrm{GL}(2n;\mathbb{R})$$ 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 $$\mathrm{Sp}(2n;\mathbb{R})$$. 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 $$\begin{align} D(n) =& \left\{ \begin{pmatrix} A & 0 \\ 0 & (A^T)^{-1} \end{pmatrix} : A \in \text{GL}(n;\mathbb{R}) \right\} \\ N(n) =& \left\{ \begin{pmatrix} I_n & B \\ 0 & I_n \end{pmatrix} : B \in \text{Sym}(n;\mathbb{R}) \right\} \end{align}$$ where $$\text{Sym}(n;\mathbb{R})$$ is the set of $$n\times n$$ symmetric matrices. Then, $$\mathrm{Sp}(2n;\mathbb{R})$$ 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 $$ \mbox{Pf}(M^\text{T} \Omega M) = \det(M)\mbox{Pf}(\Omega).$$ Since $$M^\text{T} \Omega M = \Omega$$ and $$\mbox{Pf}(\Omega) \neq 0$$ 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 $$M = \begin{pmatrix}A & B \\ C & D\end{pmatrix}$$ 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 $A^\text{T} D - C^\text{T} B = I$""$AB^\text{T},CD^\text{T}$ symmetric, and $AD^\text{T} - BC^\text{T} = I$"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 $$ M^{-1} = \Omega^{-1} M^\text{T} \Omega=\begin{pmatrix}D^\text{T} & -B^\text{T} \\-C^\text{T} & A^\text{T}\end{pmatrix}.$$ 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 $$\binom{2n}{2},$$ the identity $$ M^\text{T} \Omega M = \Omega$$ imposes $$2n \choose 2$$ 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
 * $$\Omega \mapsto A^\text{T} \Omega A$$
 * $$M \mapsto A^{-1} M A.$$

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 = \begin{bmatrix}

\begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\ & \ddots & \\ 0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix} \end{bmatrix}.$$ 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 real symplectic matrix $S$ there exists $U$ in $$\mathrm{U}(2n,\mathbb{R}) = \mathrm{O}(2n)$$ such that
 * $$S = U^\text{T} D U \quad \text{for} \quad D = \operatorname{diag}(\lambda_1,\ldots,\lambda_n,\lambda_1^{-1},\ldots,\lambda_n^{-1}),$$ where the diagonal elements of $D$ are the eigenvalues of $S$.


 * Any real symplectic matrix $S$ has a polar decomposition of the form:
 * $$S = UR \quad$$ for $$\quad U \in \operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{U}(2n,\mathbb{R})$$ and $$R \in \operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{Sym}_+(2n,\mathbb{R}).$$

such that $O$ and $O'$ 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.
 * Any real symplectic matrix can be decomposed as a product of three 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

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 ($$) for complex matrices and call matrices satisfying ($$) 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 ($$) 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.