Symplectic Bases and Canonical Forms

DefinitionSymplectic Basis

A symplectic basis for $(V, \omega)$ with $\dim V = 2n$ is an ordered basis $(e_1, \ldots, e_n, f_1, \ldots, f_n)$ such that:

$\omega(e_i, e_j) = 0, \quad \omega(f_i, f_j) = 0, \quad \omega(e_i, f_j) = \delta_{ij}$

In such a basis, the matrix of $\omega$ is the standard symplectic matrix $J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$ .

The existence of symplectic bases is a fundamental result in symplectic linear algebra, showing that all symplectic vector spaces of the same dimension are isomorphic. This is the linear version of Darboux's theorem for symplectic manifolds.

ExampleConstructing a Symplectic Basis

Given any non-zero vector $v_1 \in V$ , since $\omega$ is non-degenerate, there exists $w_1$ with $\omega(v_1, w_1) \neq 0$ . Normalizing, we can choose $f_1$ so that $\omega(v_1, f_1) = 1$ .

Set $e_1 = v_1$ . Then $\text{span}\{e_1, f_1\}$ is a symplectic subspace, and we can proceed inductively on the symplectic complement:

$V = \text{span}\{e_1, f_1\} \oplus (\text{span}\{e_1, f_1\})^\omega$

Remark

This construction shows that every symplectic vector space admits a direct sum decomposition into symplectic 2-planes. Each 2-plane is symplectomorphic to $(\mathbb{R}^2, dq \wedge dp)$ .

The symplectic basis allows us to express any symplectic form in canonical form, revealing the fundamental simplicity of symplectic linear algebra compared to the more complex classification problems in other geometries.

DefinitionSymplectic Normal Form for Matrices

Let $A: V \to V$ be a linear map on a symplectic vector space. The Williamson normal form theorem states that there exists a symplectic basis in which $A$ takes a canonical block-diagonal form.

For symmetric positive-definite $A$ (relevant in optics and mechanics), the form is:

$A = \bigoplus_{i=1}^n \lambda_i \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

with $\lambda_i > 0$ called the symplectic eigenvalues.

ExampleApplication to Gaussian States

In quantum mechanics, Gaussian states are characterized by their covariance matrices, which are positive-definite symmetric matrices on phase space. The symplectic eigenvalues determine the purity and entanglement properties of the quantum state.

For a two-mode system with covariance matrix $\Gamma$ , the symplectic eigenvalues $\nu_1, \nu_2$ satisfy:

$\nu_1, \nu_2 \geq 1$

with equality if and only if the state is pure (a coherent or squeezed state).

The canonical forms for symplectic matrices are richer than those for symmetric or orthogonal matrices. A symplectic matrix may have real eigenvalues, complex eigenvalues on the unit circle, or complex eigenvalues with modulus different from 1, but the eigenvalues always appear in quartets $(\lambda, \bar{\lambda}, 1/\lambda, 1/\bar{\lambda})$ .

Remark

The symplectic spectrum of a matrix $A \in \text{Sp}(2n, \mathbb{R})$ has the reciprocal symmetry: if $\lambda$ is an eigenvalue, so is $1/\lambda$ . This reflects the preservation of the symplectic volume form $\omega^n$ .

DefinitionComplex Structures Compatible with $\omega$

A complex structure $J: V \to V$ (satisfying $J^2 = -I$ ) is compatible with $\omega$ if:

$\omega(Jv, Jw) = \omega(v, w)$ for all $v, w \in V$
$g(v, w) := \omega(v, Jw)$ defines an inner product on $V$

The space of compatible complex structures is non-empty and contractible.