Symplectic Operations

Module name: thewalrus.symplectic

Contains some Gaussian operations and auxiliary functions.

Auxiliary functions

expand(S, modes, N)	Expands a Symplectic matrix S to act on the entire subsystem.
expand_vector(alpha, mode, N[, hbar])	Returns the phase-space displacement vector associated to a displacement.
reduced_state(mu, cov, modes)	Returns the vector of means and the covariance matrix of the specified modes.
is_symplectic(S[, rtol, atol])	Checks if matrix S is a symplectic matrix
sympmat(N)	Returns the matrix \(\Omega_n = \begin{bmatrix}0 & I_n\\ -I_n & 0\end{bmatrix}\)

Gaussian states

vacuum_state(modes[, hbar])

Returns the vacuum state.

Gates and operations

two_mode_squeezing(r, phi)	Two-mode squeezing.
interferometer(U)	Interferometer.
loss(mu, cov, T, mode[, nbar, hbar])	Loss channel acting on a Gaussian state.
mean_photon_number(mu, cov[, hbar])	Calculates the mean photon number for a given one-mode state.
beam_splitter(theta, phi)	Beam-splitter.
rotation(theta)	Rotation gate.

Code details

autonne(A, rtol=1e-05, atol=1e-08, svd_order=True)

Autonne-Takagi decomposition of a complex symmetric (not Hermitian!) matrix.

Parameters

• A (array) – square, symmetric matrix

• rtol (float) – the relative tolerance parameter between A and A.T

• atol (float) – the absolute tolerance parameter between A and A.T

• svd_order (boolean) – whether to return result by ordering the singular values of A in descending (True) or asceding (False) order.

Returns

(r, U), where r are the singular values, and U is the Autonne-Takagi unitary, such that \(A = U \diag(r) U^T\).

Return type

tuple[array, array]

beam_splitter(theta, phi)

Beam-splitter.

Parameters

• theta (float) – transmissivity parameter

• phi (float) – phase parameter

Returns

symplectic-orthogonal transformation matrix of an interferometer with angles theta and phi

Return type

array

expand(S, modes, N)

Expands a Symplectic matrix S to act on the entire subsystem.

Parameters

• S (array) – a \(2M\times 2M\) Symplectic matrix

• modes (Sequence[int]) – the list of modes S acts on

• N (int) – full size of the subsystem

Returns

the resulting \(2N\times 2N\) Symplectic matrix

Return type

array

expand_vector(alpha, mode, N, hbar=2.0)

Returns the phase-space displacement vector associated to a displacement.

Parameters

• alpha (complex) – complex displacement

• mode (int) – mode index

• N (int) – number of modes

Returns

phase-space displacement vector of size 2*N

Return type

array

interferometer(U)

Interferometer.

Parameters

U (array) – unitary matrix

Returns

symplectic transformation matrix

Return type

array

is_symplectic(S, rtol=1e-05, atol=1e-08)

Checks if matrix S is a symplectic matrix

Parameters

S (array) – a square matrix

Returns

whether the given matrix is symplectic

Return type

(bool)

loss(mu, cov, T, mode, nbar=0, hbar=2)

Loss channel acting on a Gaussian state.

Parameters

• mu (array) – means vector

• cov (array) – covariance matri

• T (float) – transmission; 1 corresponds to no loss, 0 to full loss.

• mode (int) – mode to act on

• nbar (float) – thermal mean population (default 0)

• hbar (float) – (default 2) the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\)

Returns

the means vector and covariance matrix of the resulting state

Return type

tuple[array]

mean_photon_number(mu, cov, hbar=2)

Calculates the mean photon number for a given one-mode state.

Parameters

• mu (array) – length-2 vector of means

• cov (array) – \(2\times 2\) covariance matrix

• hbar (float) – (default 2) the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\)

Returns

the photon number expectation and variance

Return type

tuple

reduced_state(mu, cov, modes)

Returns the vector of means and the covariance matrix of the specified modes.

Parameters

modes (int of Sequence[int]) – indices of the requested modes

Returns

means is an array containing the vector of means, and cov is a square array containing the covariance matrix. Both use the \(xp\)-ordering.

Return type

tuple (means, cov)

rotation(theta)

Rotation gate.

Parameters

theta (float) – rotation angle

Returns

rotation matrix by angle theta

Return type

array

squeezing(r, phi)

Squeezing. In fock space this corresponds to exp(tfrac{1}{2}r e^{i phi} (a^2 - a^{dagger 2}) ).

Parameters

• r (float) – squeezing magnitude

• phi (float) – rotation parameter

Returns

symplectic transformation matrix

Return type

array

sympmat(N)

Returns the matrix \(\Omega_n = \begin{bmatrix}0 & I_n\\ -I_n & 0\end{bmatrix}\)

Parameters

N (int) – positive integer

Returns

\(2N\times 2N\) array

Return type

array

two_mode_squeezing(r, phi)

Two-mode squeezing.

Parameters

• r (float) – squeezing magnitude

• phi (float) – rotation parameter

Returns

symplectic transformation matrix

Return type

array

vacuum_state(modes, hbar=2.0)

Returns the vacuum state.

Parameters

• modes (str) – Returns the vector of means and the covariance matrix

• hbar (float) – (default 2) the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\)

Returns

the means vector and covariance matrix of the vacuum state

Return type

list[array]

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