Symplectic matrix

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

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

Typically <math>\Omega</math> is chosen to be the block matrix <math display="block"> \Omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}, </math> where <math>I_n</math> is the <math>n\times n</math> identity matrix. The matrix <math>\Omega</math> has determinant <math>+1</math> and its inverse is <math>\Omega^{-1} = \Omega^\text{T} = -\Omega</math>.

Properties

Generators for symplectic matrices

Every symplectic matrix has determinant <math>+1</math>, and the <math>2n\times 2n</math> symplectic matrices with real entries form a subgroup of the general linear group <math>\mathrm{GL}(2n;\mathbb{R})</math> 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 <math>n(2n+1)</math>, and is denoted <math>\mathrm{Sp}(2n;\mathbb{R})</math>. 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 <math display="block">\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}</math> where <math>\text{Sym}(n;\mathbb{R})</math> is the set of <math>n\times n</math> symmetric matrices. Then, <math>\mathrm{Sp}(2n;\mathbb{R})</math> is generated by the set<ref>Template:Cite book</ref>^{p. 2} <math display="block">\{\Omega \} \cup D(n) \cup N(n)</math> of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in <math>D(n)</math> and <math>N(n)</math> together, along with some power of <math>\Omega</math>.

Inverse matrix

Every symplectic matrix is invertible with the inverse matrix given by <math display="block"> M^{-1} = \Omega^{-1} M^\text{T} \Omega. </math> 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 <math display="block">\mbox{Pf}(M^\text{T} \Omega M) = \det(M)\mbox{Pf}(\Omega).</math> Since <math>M^\text{T} \Omega M = \Omega</math> and <math>\mbox{Pf}(\Omega) \neq 0</math> we have that <math>\det(M) = 1</math>.

When the underlying field is real or complex, one can also show this by factoring the inequality <math>\det(M^\text{T} M + I) \ge 1</math>.<ref>Template:Cite journal</ref>

Block form of symplectic matrices

Suppose Ω is given in the standard form and let <math>M</math> be a <math>2n\times 2n</math> block matrix given by <math display="block">M = \begin{pmatrix}A & B \\ C & D\end{pmatrix}</math> where <math>A,B,C,D</math> are <math>n\times n</math> matrices. The condition for <math>M</math> to be symplectic is equivalent to the two following equivalent conditions<ref>{{#invoke:citation/CS1|citation |CitationClass=web

}}</ref>

<math>A^\text{T}C,B^\text{T}D</math> symmetric, and <math>A^\text{T} D - C^\text{T} B = I</math>

<math>AB^\text{T},CD^\text{T}</math> symmetric, and <math>AD^\text{T} - BC^\text{T} = I</math>

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

Inverse matrix of block matrix

With <math>\Omega</math> in standard form, the inverse of <math>M</math> is given by <math display="block"> M^{-1} = \Omega^{-1} M^\text{T} \Omega=\begin{pmatrix}D^\text{T} & -B^\text{T} \\-C^\text{T} & A^\text{T}\end{pmatrix}.</math> The group has dimension <math>n(2n+1)</math>. This can be seen by noting that <math>( M^\text{T} \Omega M)^\text{T} = -M^\text{T} \Omega M</math> is anti-symmetric. Since the space of anti-symmetric matrices has dimension <math>\binom{2n}{2},</math> the identity <math> M^\text{T} \Omega M = \Omega</math> imposes <math>2n \choose 2</math> constraints on the <math>(2n)^2</math> coefficients of <math>M</math> and leaves <math>M</math> with <math>n(2n+1)</math> 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 <math>(V,\omega)</math> is a <math>2n</math>-dimensional vector space <math>V</math> equipped with a nondegenerate, skew-symmetric bilinear form <math>\omega</math> called the symplectic form.

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

Under a change of basis, represented by a matrix A, we have <math display="block">\Omega \mapsto A^\text{T} \Omega A</math> <math display="block">M \mapsto A^{-1} M A.</math> One can always bring <math>\Omega</math> 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 <math>\Omega</math>. As explained in the previous section, <math>\Omega</math> 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 <math>\Omega</math> given above is the block diagonal form <math display="block">\Omega = \begin{bmatrix} \begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\

& \ddots & \\

0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix} \end{bmatrix}.</math> This choice differs from the previous one by a permutation of basis vectors.

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

Given a hermitian structure on a vector space, <math>J</math> and <math>\Omega</math> are related via <math display="block">\Omega_{ab} = -g_{ac}{J^c}_b</math> where <math>g_{ac}</math> is the metric. That <math>J</math> and <math>\Omega</math> 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

Template:Bullet list\begin{pmatrix} I_n & i I_n \\ I_n & -i I_n \end{pmatrix} \right], </math>
|Template:EquationRef}} with <math> V \in \mathrm{U}(n,\mathbb{C})</math>. 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 <ref>Template:Cite journal</ref> 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 <ref>Template:Cite report</ref> 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.<ref>Template:Cite journal</ref> 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).<ref>Template:Cite journal</ref> 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.<ref>Template:Cite journal</ref>

See also

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