Symplectic matrix on arvau

Wikipedia / Symplectic matrix

In mathematics, a symplectic matrix matrix with real entries that satisfies the condition

denotes the transpose nonsingular, skew-symmetric matrix matrices with entries in other fieldss, such as the complex numbers, finite fields, p-adic numberss, and function fields.

is chosen to be the block matrix identity matrix has determinant.

Properties

Generators for symplectic matrices

symplectic matrices with real entries form a subgroup of the general linear group under matrix multiplication since being symplectic is a property stable under matrix multiplication. Topologically, this symplectic group is a connected noncompact real Lie group. 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 symmetric matrices is generated by the setp. 2.

Inverse matrix

Every symplectic matrix is invertible with the inverse 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.

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.

Block form of symplectic matrices

block matrix to be symplectic is equivalent to the two following equivalent conditions matrix is symplectic if and only if it has unit determinant.

Inverse matrix of block matrix

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 equipped with a nondegenerate, skew-symmetric bilinear form called the symplectic form.

Fixing a basis be a symplectic transformation is precisely the condition that M

Under a change of basis, represented by a matrix A 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 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.

given above is the block diagonal This choice differs from the previous one by a permutation of basis vectors.

for the skew-symmetric matrix. This is a particularly unfortunate choice as it leads to confusion with the notion of a complex structure.

Given a hermitian structure is the metric 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

Complex matrices

If instead M is a 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 (n=1), 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 (Bogoliubov) 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 (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.