Symplectic matrix

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 ${\displaystyle (V,\omega )}$ is a ${\displaystyle 2n}$-dimensional vector space ${\displaystyle V}$ equipped with a nondegenerate, skew-symmetric bilinear form ${\displaystyle \omega }$ called the symplectic form.

A symplectic transformation is then a linear transformation ${\displaystyle L:V\to V}$ which preserves ${\displaystyle \omega }$, i.e.

${\displaystyle \omega (Lu,Lv)=\omega (u,v).}$

Fixing a basis for ${\displaystyle V}$, ${\displaystyle \omega }$ can be written as a matrix ${\displaystyle \Omega }$ and ${\displaystyle L}$ as a matrix ${\displaystyle M}$. The condition that ${\displaystyle L}$ be a symplectic transformation is precisely the condition that M be a symplectic matrix:

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

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

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

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

One can always bring ${\displaystyle \Omega }$ to either the standard form given in the introduction or the block diagonal form described below by a suitable choice of A.

Symplectic matrices are defined relative to a fixed nonsingular, skew-symmetric matrix ${\displaystyle \Omega }$. As explained in the previous section, ${\displaystyle \Omega }$ 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 ${\displaystyle \Omega }$ given above is the block diagonal form

${\displaystyle \Omega ={\begin{bmatrix}{\begin{matrix}0&1\\-1&0\end{matrix}}&&0\\&\ddots &\\0&&{\begin{matrix}0&1\\-1&0\end{matrix}}\end{bmatrix}}.}$

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

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

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

${\displaystyle \Omega _{ab}=-g_{ac}{J^{c}}_{b}}$

where ${\displaystyle g_{ac}}$ is the metric. That ${\displaystyle J}$ and ${\displaystyle \Omega }$ 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.

• For any positive definite symmetric real symplectic matrix S there exists U in U(2n,R) such that

${\displaystyle S=U^{\text{T}}DU\quad {\text{for}}\quad D=\operatorname {diag} (\lambda _{1},\ldots ,\lambda _{n},\lambda _{1}^{-1},\ldots ,\lambda _{n}^{-1}),}$
where the diagonal elements of D are the eigenvalues of S.[4]

• Any real symplectic matrix S has a polar decomposition of the form:[4]

${\displaystyle S=UR\quad {\text{for}}\quad U\in \operatorname {U} (2n,\mathbb {R} ){\text{ and }}R\in \operatorname {Sp} (2n,\mathbb {R} )\cap \operatorname {Sym} _{+}(2n,\mathbb {R} ).}$

• Any real symplectic matrix can be decomposed as a product of three matrices:

${\displaystyle S=O{\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}O',}$

(2)

such that O and O' are both symplectic and orthogonal and D is positive-definite and diagonal.[5] This decomposition is closely related to the singular value decomposition of a matrix and is known as an 'Euler' or 'Bloch-Messiah' decomposition.

If instead M is a 2n × 2n matrix with complex entries, the definition is not standard throughout the literature. Many authors [6] adjust the definition above to

${\displaystyle M^{*}\Omega M=\Omega \,.}$

(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 [7] retain the definition (1) for complex matrices and call matrices satisfying (3) conjugate symplectic.

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.[8] 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).[9] 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.[10]

• symplectic vector space
• symplectic group
• symplectic representation
• orthogonal matrix
• unitary matrix
• Hamiltonian mechanics
• Linear complex structure