Symplectic matrix

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

[math]\displaystyle{ M^\text{T} \Omega M = \Omega, }[/math]

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(1)

where [math]\displaystyle{ M^\text{T} }[/math] denotes the transpose of [math]\displaystyle{ M }[/math] and [math]\displaystyle{ \Omega }[/math] is a fixed [math]\displaystyle{ 2n\times 2n }[/math] nonsingular, skew-symmetric matrix. This definition can be extended to [math]\displaystyle{ 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]\displaystyle{ \Omega }[/math] is chosen to be the block matrix [math]\displaystyle{ \Omega = \begin{bmatrix} 0 & I_n \\\ -I_n & 0 \\\ \end{bmatrix}, }[/math] where [math]\displaystyle{ I_n }[/math] is the [math]\displaystyle{ n\times n }[/math] identity matrix. The matrix [math]\displaystyle{ \Omega }[/math] has determinant [math]\displaystyle{ +1 }[/math] and its inverse is [math]\displaystyle{ \Omega^{-1} = \Omega^\text{T} = -\Omega }[/math].

## Contents

* 1 Properties
* 1.1 Generators for symplectic matrices
* 1.2 Inverse matrix
* 1.3 Determinantal properties
* 1.4 Block form of symplectic matrices
* 1.4.1 Inverse matrix of block matrix
* 2 Symplectic transformations
* 3 The matrix Ω
* 4 Diagonalization and decomposition
* 5 Complex matrices
* 6 Applications
* 7 See also
* 8 References

## Properties

### Generators for symplectic matrices

Every symplectic matrix has determinant [math]\displaystyle{ +1 }[/math], and the [math]\displaystyle{ 2n\times 2n }[/math] symplectic matrices with real entries form a subgroup of the general linear group [math]\displaystyle{ \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]\displaystyle{ n(2n+1) }[/math], and is denoted [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \text{Sym}(n;\mathbb{R}) }[/math] is the set of [math]\displaystyle{ n\times n }[/math] symmetric matrices. Then, [math]\displaystyle{ \mathrm{Sp}(2n;\mathbb{R}) }[/math] is generated by the set[1]p. 2 [math]\displaystyle{ \\{\Omega \\} \cup D(n) \cup N(n) }[/math] of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in [math]\displaystyle{ D(n) }[/math] and [math]\displaystyle{ N(n) }[/math] together, along with some power of [math]\displaystyle{ \Omega }[/math].

### Inverse matrix

Every symplectic matrix is invertible with the inverse matrix given by [math]\displaystyle{ 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]\displaystyle{ \mbox{Pf}(M^\text{T} \Omega M) = \det(M)\mbox{Pf}(\Omega). }[/math] Since [math]\displaystyle{ M^\text{T} \Omega M = \Omega }[/math] and [math]\displaystyle{ \mbox{Pf}(\Omega) \neq 0 }[/math] we have that [math]\displaystyle{ \det(M)=1 }[/math].

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

### Block form of symplectic matrices

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

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

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

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

When [math]\displaystyle{ n=1 }[/math] these conditions reduce to the single condition [math]\displaystyle{ \det(M)=1 }[/math]. Thus a [math]\displaystyle{ 2\times 2 }[/math] matrix is symplectic iff it has unit determinant.

#### Inverse matrix of block matrix

With [math]\displaystyle{ \Omega }[/math] in standard form, the inverse of [math]\displaystyle{ M }[/math] is given by [math]\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}. }[/math] The group has dimension [math]\displaystyle{ n(2n+1) }[/math]. This can be seen by noting that [math]\displaystyle{ ( 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]\displaystyle{ \binom{2n}{2}, }[/math] the identity [math]\displaystyle{ M^\text{T} \Omega M = \Omega }[/math] imposes [math]\displaystyle{ 2n \choose 2 }[/math] constraints on the [math]\displaystyle{ (2n)^2 }[/math] coefficients of [math]\displaystyle{ M }[/math] and leaves [math]\displaystyle{ M }[/math] with [math]\displaystyle{ 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]\displaystyle{ (V,\omega) }[/math] is a [math]\displaystyle{ 2n }[/math]-dimensional vector space [math]\displaystyle{ V }[/math] equipped with a nondegenerate, skew-symmetric bilinear form [math]\displaystyle{ \omega }[/math] called the symplectic form.

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

[math]\displaystyle{ \omega(Lu, Lv) = \omega(u, v). }[/math]

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

[math]\displaystyle{ M^\text{T} \Omega M = \Omega. }[/math]

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

[math]\displaystyle{ \Omega \mapsto A^\text{T} \Omega A }[/math]
[math]\displaystyle{ M \mapsto A^{-1} M A. }[/math]

One can always bring [math]\displaystyle{ \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]\displaystyle{ \Omega }[/math]. As explained in the previous section, [math]\displaystyle{ \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]\displaystyle{ \Omega }[/math] given above is the block diagonal form

[math]\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}. }[/math]

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

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

Given a hermitian structure on a vector space, [math]\displaystyle{ J }[/math] and [math]\displaystyle{ \Omega }[/math] are related via

[math]\displaystyle{ \Omega_{ab} = -g_{ac}{J^c}_b }[/math]

where [math]\displaystyle{ g_{ac} }[/math] is the metric. That [math]\displaystyle{ J }[/math] and [math]\displaystyle{ \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

* For any positive definite symmetric real symplectic matrix S there exists U in [math]\displaystyle{ \mathrm{U}(2n,\mathbb{R}) = \mathrm{O}(2n) }[/math] such that

[math]\displaystyle{ 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}), }[/math]
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]

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

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

[math]\displaystyle{ S = O\begin{pmatrix}D & 0 \\\ 0 & D^{-1}\end{pmatrix}O', }[/math]

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(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.

## Complex matrices

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

[math]\displaystyle{ M^* \Omega M = \Omega\,. }[/math]

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(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.

## 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.[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]

## See also

* Symplectic vector space
* Symplectic group
* Symplectic representation
* Orthogonal matrix
* Unitary matrix
* Hamiltonian mechanics
* Linear complex structure

## References

1. ↑ Habermann, Katharina, 1966- (2006). Introduction to symplectic Dirac operators. Springer. ISBN 978-3-540-33421-7. OCLC 262692314. http://worldcat.org/oclc/262692314.
2. ↑ Rim, Donsub (2017). "An elementary proof that symplectic matrices have determinant one". Adv. Dyn. Syst. Appl. 12 (1): 15–20. doi:10.37622/ADSA/12.1.2017.15-20.
3. ↑ de Gosson, Maurice. "Introduction to Symplectic Mechanics: Lectures I-II-III". https://www.ime.usp.br/~piccione/Downloads/LecturesIME.pdf.
4. ↑ 4.0 4.1 de Gosson, Maurice A. (2011) (in en). Symplectic Methods in Harmonic Analysis and in Mathematical Physics - Springer. doi:10.1007/978-3-7643-9992-4. ISBN 978-3-7643-9991-7.
5. ↑ Ferraro, Alessandro; Olivares, Stefano; Paris, Matteo G. A. (31 March 2005). "Gaussian states in continuous variable quantum information". Sec. 1.3, p. 4. arXiv:quant-ph/0503237.
6. ↑ Xu, H. G. (July 15, 2003). "An SVD-like matrix decomposition and its applications". Linear Algebra and Its Applications 368: 1–24. doi:10.1016/S0024-3795(03)00370-7.
7. ↑ Mackey, D. S.; Mackey, N. (2003). On the Determinant of Symplectic Matrices. Numerical Analysis Report. 422. Manchester, England: Manchester Centre for Computational Mathematics.
8. ↑ Weedbrook, Christian; Pirandola, Stefano; García-Patrón, Raúl; Cerf, Nicolas J.; Ralph, Timothy C.; Shapiro, Jeffrey H.; Lloyd, Seth (2012). "Gaussian quantum information". Reviews of Modern Physics 84 (2): 621–669. doi:10.1103/RevModPhys.84.621. Bibcode: 2012RvMP...84..621W.
9. ↑ Braunstein, Samuel L. (2005). "Squeezing as an irreducible resource". Physical Review A 71 (5): 055801. doi:10.1103/PhysRevA.71.055801. Bibcode: 2005PhRvA..71e5801B.
10. ↑ Chakhmakhchyan, Levon; Cerf, Nicolas (2018). "Simulating arbitrary Gaussian circuits with linear optics". Physical Review A 98 (6): 062314. doi:10.1103/PhysRevA.98.062314. Bibcode: 2018PhRvA..98f2314C.

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