def factor_density_matrix(
    t: np.ndarray, axes: Sequence[int], *, validate=True, atol=1e-07
) -> tuple[np.ndarray, np.ndarray]:
    """Factors a density matrix into two independent density matrices.

    This function should only be called on density matrices that are known to
    be separable, such as immediately after a measurement or reset operation.
    It does not verify that the provided density matrix is indeed separable,
    and will return nonsense results for matrices representing entangled
    states.

    Args:
        t: The density matrix to factor.
        axes: The axes to factor out. Only the left axes should be provided.
            For example, to extract [C,A] from density matrix of shape
            [A,B,C,D,A,B,C,D], `axes` should be [2,0], and the return value
            will be two density matrices ([C,A,C,A], [B,D,B,D]).
        validate: Perform a validation that the density matrix factors cleanly.
        atol: The absolute tolerance for the validation.

    Returns:
        A tuple with the `(extracted, remainder)` density matrices, where
        `extracted` means the sub-matrix which corresponds to the axes
        requested, and with the axes in the requested order, and where
        `remainder` means the sub-matrix on the remaining axes, in the same
        order as the original density matrix.

    Raises:
        ValueError: If the tensor cannot be factored along the given aces.
    """
    extracted = partial_trace(t, axes)
    remaining_axes = [i for i in range(t.ndim // 2) if i not in axes]
    remainder = partial_trace(t, remaining_axes)
    if validate:
        t1 = density_matrix_kronecker_product(extracted, remainder)
        product_axes = list(axes) + remaining_axes
        t2 = transpose_density_matrix_to_axis_order(t1, product_axes)
        if not np.allclose(t2, t, atol=atol):
            raise ValueError('The tensor cannot be factored by the requested axes')
    return extracted, remainder
Factors a density matrix into two independent density matrices.

    This function should only be called on density matrices that are known to
    be separable, such as immediately after a measurement or reset operation.
    It does not verify that the provided density matrix is indeed separable,
    and will return nonsense results for matrices representing entangled
    states.

    Args:
        t: The density matrix to factor.
        axes: The axes to factor out. Only the left axes should be provided.
            For example, to extract [C,A] from density matrix of shape
            [A,B,C,D,A,B,C,D], `axes` should be [2,0], and the return value
            will be two density matrices ([C,A,C,A], [B,D,B,D]).
        validate: Perform a validation that the density matrix factors cleanly.
        atol: The absolute tolerance for the validation.

    Returns:
        A tuple with the `(extracted, remainder)` density matrices, where
        `extracted` means the sub-matrix which corresponds to the axes
        requested, and with the axes in the requested order, and where
        `remainder` means the sub-matrix on the remaining axes, in the same
        order as the original density matrix.

    Raises:
        ValueError: If the tensor cannot be factored along the given aces.
    
cirq-core/cirq/linalg/transformations.py