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.
expand_passive(T, modes, N)	Returns the expanded linear optical transformation acting on specified modes, with identity acting on all other modes
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[, dtype])	Returns the matrix \(\Omega_n = \begin{bmatrix}0 & I_n\\ -I_n & 0\end{bmatrix}\)
xxpp_to_xpxp(S)	Permutes the entries of the input from xxpp ordering to xpxp ordering.
xpxp_to_xxpp(S)	Permutes the entries of the input from xpxp ordering to xxpp ordering.

Gaussian states

vacuum_state(modes[, hbar, dtype])

Returns the vacuum state.

Gates and operations

two_mode_squeezing(r, phi[, dtype])	Two-mode squeezing.
squeezing(r[, phi, dtype])	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[, dtype])	Beam-splitter.
rotation(theta[, dtype])	Rotation gate.

Code details

beam_splitter(theta, phi, dtype=<class 'numpy.float64'>)

Beam-splitter.

Parameters:

• theta (float) – transmissivity parameter

• phi (float) – phase parameter

• dtype (numpy.dtype) – datatype to represent the Symplectic matrix

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. If the input is a single mode symplectic, then extends it to act on multiple modes.

Supports scipy sparse matrices. Instances of coo_array, dia_array, bsr_array will be transformed into csr_array`.

Parameters:

• S (ndarray or spmatrix) – 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_passive(T, modes, N)

Returns the expanded linear optical transformation acting on specified modes, with identity acting on all other modes

Parameters:

• T (array) – square \(M \times M\) matrix of linear optical transformation

• modes (array) – the \(M\) modes of the transformation

• N (int) – number of modes in the new expanded transformation

Returns:

\(N \times N\) array of expanded passive transformation

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

passive_transformation(mu, cov, T, hbar=2)

Perform a covariance matrix transformation for an arbitrary linear optical channel on an \(N\) modes state mapping it to a to an \(M\) modes state.

Parameters:

• mu (array) – \(2N\)-length means vector

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

• T (array) – \(M \times N\) linear optical transformation

Keyword Arguments:

hbar (float) – the value to use for hbar

Returns:

\(2M\)-length transformed means vector array \(2M \times 2M\) tranformed covariance matrix

Return type:

array

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, dtype=<class 'numpy.float64'>)

Rotation gate.

Parameters:

• theta (float) – rotation angle

• dtype (numpy.dtype) – datatype to represent the Symplectic matrix

Returns:

rotation matrix by angle theta

Return type:

array

squeezing(r, phi=None, dtype=<class 'numpy.float64'>)

Squeezing. In fock space this corresponds to:

\[\exp(\tfrac{1}{2}r e^{i \phi} (a^2 - a^{\dagger 2}) ).\]

By passing an array of squeezing parameters and phases, it applies a tensor product of squeezing operations.

Parameters:

• r (Union[array, float]) – squeezing magnitude

• phi (Union[array, float]) – rotation parameter. If None, then the function uses zeros of the same shape as r.

• dtype (numpy.dtype) – datatype to represent the Symplectic matrix. Defaults to numpy.float64.

Returns:

symplectic transformation matrix

Return type:

array

sympmat(N, dtype=<class 'numpy.float64'>)

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

Parameters:

• N (int) – positive integer

• dtype (numpy.dtype) – datatype to represent the Symplectic matrix

Returns:

\(2N\times 2N\) array

Return type:

array

two_mode_squeezing(r, phi, dtype=<class 'numpy.float64'>)

Two-mode squeezing.

Parameters:

• r (float) – squeezing magnitude

• phi (float) – rotation parameter

• dtype (numpy.dtype) – datatype to represent the Symplectic matrix

Returns:

symplectic transformation matrix

Return type:

array

vacuum_state(modes, hbar=2.0, dtype=<class 'numpy.float64'>)

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 \([\x,\p]=i\hbar\)

• dtype (numpy.dtype) – datatype to represent the covariance matrix and vector of means

Returns:

the means vector and covariance matrix of the vacuum state

Return type:

list[array]

xpxp_to_xxpp(S)

Permutes the entries of the input from xpxp ordering to xxpp ordering.

Parameters:

S (array) – input even dimensional square matrix or vector

Returns:

permuted matrix or vector

Return type:

(array)

xxpp_to_xpxp(S)

Permutes the entries of the input from xxpp ordering to xpxp ordering.

Parameters:

S (array) – input even dimensional square matrix or array

Returns:

permuted matrix or array

Return type:

(array)

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