where MT denotes the transpose of M and Ω is a fixed 2n × 2n nonsingular, skew-symmetric matrix. This definition can be extended to 2n × 2n matrices with entries in other fields, such as the complex numbers.
Typically Ω is chosen to be the block matrix
Every symplectic matrix has determinant 1, and the 2n × 2n symplectic matrices with real entries form a subgroup of the special linear group SL(2n, R) under matrix multiplication. Topologically, this symplectic group is a connected noncompact real Lie group of real dimension n(2n + 1), and is denoted Sp(2n, R). 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 .
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
Since and we have that det(M) = 1.
Suppose Ω is given in the standard form and let M be a 2n×2n block matrix given by
- symmetric, and
- symmetric, and
When n = 1 these conditions reduce to the single condition det(M) = 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(2n + 1). 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(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 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
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 (up to an overall sign) is simply a consequence of the fact that the metric g is usually the identity matrix.
Diagonalisation and decomposition
- For any positive definite symmetric real symplectic matrix S there exists U in U(2n,R) such that
where the diagonal elements of D are the eigenvalues of S.
- Any real symplectic matrix S has a polar decomposition of the form:
- Any real symplectic matrix can be decomposed as a product of three matrices:
such that O and O' are both symplectic and orthogonal and D 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.
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.
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. 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). 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.
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