qml.liealg.check_cartan_decomp

check_cartan_decomp(k, m, verbose=True)

Helper function to check the validity of a Cartan decomposition \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{m}.\)

Check whether of not the following properties are fulfilled.

\[\begin{split}[\mathfrak{k}, \mathfrak{k}] \subseteq \mathfrak{k} & \text{ (subalgebra)}\\ [\mathfrak{k}, \mathfrak{m}] \subseteq \mathfrak{m} & \text{ (reductive property)}\\ [\mathfrak{m}, \mathfrak{m}] \subseteq \mathfrak{k} & \text{ (symmetric property)}\end{split}\]

Warning

This function is expensive to compute

Parameters:

• k (List[Union[PauliSentence, TensorLike]]) – List of operators of the vertical subspace.

• m (List[Union[PauliSentence, TensorLike]]) – List of operators of the horizontal subspace.

• verbose – Whether failures to meet one of the criteria should be printed.

Returns:

Whether or not all of the Cartan commutation relations are fulfilled.

Return type:

bool

See also

cartan_decomp()

Example

We first construct a Lie algebra.

>>> from pennylane import X, Z
>>> from pennylane.liealg import concurrence_involution, even_odd_involution, cartan_decomp
>>> generators = [X(0) @ X(1), Z(0), Z(1)]
>>> g = qml.lie_closure(generators)
>>> g
[X(0) @ X(1),
 Z(0),
 Z(1),
 -1.0 * (Y(0) @ X(1)),
 -1.0 * (X(0) @ Y(1)),
 -1.0 * (Y(0) @ Y(1))]

We compute the Cartan decomposition with respect to the concurrence_involution().

>>> k, m = cartan_decomp(g, concurrence_involution)
>>> k, m
([-1.0 * (Y(0) @ X(1)), -1.0 * (X(0) @ Y(1))],
 [X(0) @ X(1), Z(0), Z(1), -1.0 * (Y(0) @ Y(1))])

We can check the validity of the decomposition using check_cartan_decomp.

>>> from pennylane.liealg import check_cartan_decomp
>>> check_cartan_decomp(k, m)
True

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