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mm.physics.bargmann_utils.bargmann_Abc_to_phasespace_cov_means

mrmustard.physics.bargmann_utils.bargmann_Abc_to_phasespace_cov_means(A, b, c, batched=False)[source]

Function to derive the covariance matrix and mean vector of a Gaussian state from its Wigner characteristic function in ABC form.

The covariance matrix and mean vector can be used to write the characteristic function of a Gaussian state :math:

Chi_G(r) = expleft( -frac{1}{2}r^T Omega^T cov Omega r + i r^TOmega^T mean right),

and the Wigner function of a Gaussian state: :math:

W_G(r) = frac{1}{sqrt{Det(cov)}} expleft( -frac{1}{2}(r - mean)^T cov^{-1} (r-mean) right).

The internal expression of our Gaussian state \(\rho\) is in Bargmann representation, one can write the characteristic function of a Gaussian state in Bargmann representation as :math:

Chi_G(alpha) = Tr(rho D) = c expleft( -frac{1}{2}alpha^T A alpha + alpha^T b right).

This function is to go from the Abc triple in characteristic phase space into the covariance and mean vector for Gaussian state.

Parameters:

  • A (ndarray[Tuple[int, int], Union[TypeVar(R, float16, float32, float64), TypeVar(C, complex64, complex128), TypeVar(Z, int16, int32, int64), TypeVar(N, uint16, uint32, uint64)]]) – The (A, b, c) triple of the state in characteristic phase space.

  • b (ndarray[Tuple[int], Union[TypeVar(R, float16, float32, float64), TypeVar(C, complex64, complex128), TypeVar(Z, int16, int32, int64), TypeVar(N, uint16, uint32, uint64)]]) – The (A, b, c) triple of the state in characteristic phase space.

  • c (Union[TypeVar(R, float16, float32, float64), TypeVar(C, complex64, complex128), TypeVar(Z, int16, int32, int64), TypeVar(N, uint16, uint32, uint64)]) – The (A, b, c) triple of the state in characteristic phase space.

Return type:

tuple[ndarray[Tuple[int, int], Union[TypeVar(R, float16, float32, float64), TypeVar(C, complex64, complex128), TypeVar(Z, int16, int32, int64), TypeVar(N, uint16, uint32, uint64)]], ndarray[Tuple[int], Union[TypeVar(R, float16, float32, float64), TypeVar(C, complex64, complex128), TypeVar(Z, int16, int32, int64), TypeVar(N, uint16, uint32, uint64)]], Union[TypeVar(R, float16, float32, float64), TypeVar(C, complex64, complex128), TypeVar(Z, int16, int32, int64), TypeVar(N, uint16, uint32, uint64)]]

Returns:

The covariance matrix, mean vector and coefficient of the state in phase space.

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