This symplectic group has a distinguished set of generators, which can be used to find all possible symplectic matrices. This includes the following sets
where is the set of symmetric matrices. Then, is generated by the set[1]p. 2
of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in and together, along with some power of .
Inverse matrix
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 (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
Since and we have that .
When the underlying field is real or complex, one can also show this by factoring the inequality .[2]
Block form of symplectic matrices
Suppose Ω is given in the standard form and let be a block matrix given by
where are matrices. The condition for to be symplectic is equivalent to the two following equivalent conditions[3]
symmetric, and
symmetric, and
The second condition comes from the fact that if is symplectic, then is also symplectic. When these conditions reduce to the single condition . Thus a matrix is symplectic
iff it has unit determinant.
Inverse matrix of block matrix
With in standard form, the inverse of is given by
The group has dimension . 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 independent coefficients.
A symplectic transformation is then a linear transformation which preserves , i.e.
Fixing a
basis for , can be written as a matrix and as a matrix . The condition that be a symplectic transformation is precisely the condition that M be a symplectic matrix:
Sometimes the notation is used instead of 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 but represents a very different structure. A complex structure is the coordinate representation of a linear transformation that squares to , whereas is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which is not skew-symmetric or does not square to .
where is the
metric. That and 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 such that
where the diagonal elements of D are the
eigenvalues of S.[4]
If instead M is a 2n × 2nmatrix with
complex entries, the definition is not standard throughout the literature. Many authors [6] adjust the definition above to
(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]
^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.
^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.
hdl:1808/374.
^Mackey, D. S.; Mackey, N. (2003). On the Determinant of Symplectic Matrices (Numerical Analysis Report 422). Manchester, England: Manchester Centre for Computational Mathematics.