Symplectic matrix


In mathematics, a symplectic matrix is a matrix M with real entries that satisfies the condition
where MT denotes the transpose of M and Ω is a fixed nonsingular, skew-symmetric matrix. This definition can be extended to matrices with entries in other fields, such as the complex numbers.
Typically Ω is chosen to be the block matrix
where In is the identity matrix. The matrix Ω has determinant +1 and has an inverse given by Ω−1 = ΩT = −Ω.
Every symplectic matrix has determinant 1, and the symplectic matrices with real entries form a subgroup of the special linear group SL under matrix multiplication. Topologically, this symplectic group is a connected noncompact real Lie group of real dimension, and is denoted Sp. The symplectic group can be defined as the set of linear transformations that preserve the symplectic form of a real symplectic vector space.
Examples of symplectic matrices include the identity matrix and the matrix.

Properties

Every symplectic matrix is invertible with the inverse matrix given by
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 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
Since and we have that det = 1.
When the underlying field is real or complex, one can also show this by factoring the inequality.
Suppose Ω is given in the standard form and let M be a 2n×2n block matrix given by
where A, B, C, D are n×n matrices. The condition for M to be symplectic is equivalent to the two following equivalent conditions
When n = 1 these conditions reduce to the single condition det = 1. Thus a 2×2 matrix is symplectic iff it has unit determinant.
With Ω in standard form, the inverse of M is given by
The group has dimension n. This can be seen by noting that is anti-symmetric. Since the space of anti-symmetric matrices has dimension, the identity imposes constraints on the coefficients of and leaves with n 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 is a 2n-dimensional vector space V equipped with a nondegenerate, skew-symmetric bilinear form ω called the symplectic form.
A symplectic transformation is then a linear transformation L : V → V which preserves ω, i.e.
Fixing a basis for V, ω can be written as a matrix Ω and L as a matrix M. The condition that L be a symplectic transformation is precisely the condition that M be a symplectic matrix:
Under a change of basis, represented by a matrix A, we have
One can always bring Ω 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 Ω. As explained in the previous section, Ω 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 Ω given above is the block diagonal form
This choice differs from the previous one by a permutation of basis vectors.
Sometimes the notation J is used instead of Ω 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 Ω but represents a very different structure. A complex structure J is the coordinate representation of a linear transformation that squares to −1, whereas Ω is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which J is not skew-symmetric or Ω does not square to −1.
Given a hermitian structure on a vector space, J and Ω are related via
where is the metric. That J and Ω usually have the same coordinate expression is simply a consequence of the fact that the metric g is usually the identity matrix.

Diagonalisation and decomposition

such that and are both symplectic and orthogonal and 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.

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, 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 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 intermitted by a layer of active non-linear squeezing transformations. 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.