spheres.symplectic

Functions for converting between Gaussian Hamiltonians, complex symplectic matrices, and real symplectic matrices.

Functions

complex_real_symplectic(S, s)	Converts a complex symplectic transformation into a real symplectic transformation.
complex_real_symplectic2(S, s)	Converts a complex symplectic matrix/vector to a real symplectic matrix/vector.
gaussian_complex_symplectic(H, h[, expm, theta])	Converts a Gaussian transformation (in the form of a Hermitian matrix and a displacement vector) into a complex symplectic transformation (in the form of a complex symplectic matrix and displacement vector).
is_complex_symplectic(S)	Test if an matrix is complex symplectic.
is_real_symplectic(R)	Test if an matrix is real symplectic.
make_gaussian_operator(A[, B, h])
omega_c(n)	\(2n \times 2n\) complex symplectic form: \(\Omega_{c} = \begin{pmatrix}I_{n} & 0 \\ 0 & -I_{n} \end{pmatrix}\).
omega_r(n)	\(2n \times 2n\) real symplectic form: \(\Omega_{c} = \begin{pmatrix}0 & I_{n} \\ -I_{n} & 0 \end{pmatrix}\).
operator_real_symplectic(O[, expm, theta])	Converts a first quantized operator into a real symplectic matrix via: make_gaussian_operator(), gaussian_complex_symplectic(), \(complex_real_symplectic\).
random_gaussian_operator(n)	Returns random Gaussian transformation in the form of a Hermitian matrix and a displacement vector.
symplectic_xyz()	Returns Pauli matrices expressed as real symplectic transformations.
upgrade_single_mode_operator(O, i, n_modes)	Upgrades a single mode real symplectic operator O to act on the i’th of n modes (where the latter are represented in terms of their first and second moments.)
upgrade_two_mode_operator(O, i, j, n_modes)	Upgrades a two mode real symplectic matrix to act on subsystems i and j of n modes, (where the modes are represented in terms of their first and second moments).