Symplectic matrix

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

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

(1)

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

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

Every symplectic matrix has determinant ${\displaystyle +1}$, and the ${\displaystyle 2n\times 2n}$ symplectic matrices with real entries form a subgroup of the general linear group ${\displaystyle \mathrm {GL} (2n;\mathbb {R} )}$ 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 ${\displaystyle n(2n+1)}$, and is denoted ${\displaystyle \mathrm {Sp} (2n;\mathbb {R} )}$. 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 $${\displaystyle {\begin{aligned}D(n)=&\left\{{\begin{pmatrix}A&0\\0&(A^{T})^{-1}\end{pmatrix}}:A\in {\text{GL}}(n;\mathbb {R} )\right\}\\N(n)=&\left\{{\begin{pmatrix}I_{n}&B\\0&I_{n}\end{pmatrix}}:B\in {\text{Sym}}(n;\mathbb {R} )\right\}\end{aligned}}}$$ where ${\displaystyle {\text{Sym}}(n;\mathbb {R} )}$ is the set of ${\displaystyle n\times n}$ symmetric matrices. Then, ${\displaystyle \mathrm {Sp} (2n;\mathbb {R} )}$ is generated by the set^{[1]p. 2} $${\displaystyle \{\Omega \}\cup D(n)\cup N(n)}$$ of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in ${\displaystyle D(n)}$ and ${\displaystyle N(n)}$ together, along with some power of ${\displaystyle \Omega }$.

Every symplectic matrix is invertible with the inverse matrix given by $${\displaystyle M^{-1}=\Omega ^{-1}M^{\text{T}}\Omega .}$$ 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.

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 $${\displaystyle {\mbox{Pf}}(M^{\text{T}}\Omega M)=\det(M){\mbox{Pf}}(\Omega ).}$$ Since ${\displaystyle M^{\text{T}}\Omega M=\Omega }$ and ${\displaystyle {\mbox{Pf}}(\Omega )\neq 0}$ we have that ${\displaystyle \det(M)=1}$.

When the underlying field is real or complex, one can also show this by factoring the inequality ${\displaystyle \det(M^{\text{T}}M+I)\geq 1}$.^[2]

Suppose Ω is given in the standard form and let ${\displaystyle M}$ be a ${\displaystyle 2n\times 2n}$ block matrix given by $${\displaystyle M={\begin{pmatrix}A&B\\C&D\end{pmatrix}}}$$

where ${\displaystyle A,B,C,D}$ are ${\displaystyle n\times n}$ matrices. The condition for ${\displaystyle M}$ to be symplectic is equivalent to the two following equivalent conditions^[3]

${\displaystyle A^{\text{T}}C,B^{\text{T}}D}$ symmetric, and ${\displaystyle A^{\text{T}}D-C^{\text{T}}B=I}$

${\displaystyle AB^{\text{T}},CD^{\text{T}}}$ symmetric, and ${\displaystyle AD^{\text{T}}-BC^{\text{T}}=I}$

The second condition comes from the fact that if ${\displaystyle M}$ is symplectic, then ${\displaystyle M^{T}}$ is also symplectic. When ${\displaystyle n=1}$ these conditions reduce to the single condition ${\displaystyle \det(M)=1}$. Thus a ${\displaystyle 2\times 2}$ matrix is symplectic iff it has unit determinant.

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

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 ${\displaystyle \mathrm {U} (2n,\mathbb {R} )=\mathrm {O} (2n)}$ 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]

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

• 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
• Williamson theorem