content: import numpy as np
  import pytest
  import qutip
  
  
  def real_hermitian(n_levels):
      qobj = qutip.Qobj(0.5 - np.random.random_sample((n_levels, n_levels)))
      return qobj + qobj.dag()
  
  
  def imaginary_hermitian(n_levels):
      qobj = qutip.Qobj(1j*(0.5 - np.random.random_sample((n_levels, n_levels))))
      return qobj + qobj.dag()
  
  
  def complex_hermitian(n_levels):
      return real_hermitian(n_levels) + imaginary_hermitian(n_levels)
  
  
  def rand_bra(n_levels):
      return qutip.rand_ket(n_levels).dag()
  
  
  @pytest.mark.parametrize("hermitian_constructor", [real_hermitian,
                                                     imaginary_hermitian,
                                                     complex_hermitian])
  @pytest.mark.parametrize("n_levels", [2, 10])
  def test_transformation_to_eigenbasis_is_reversible(hermitian_constructor,
                                                      n_levels):
      """Transform n-level real-values to eigenbasis and back"""
      H1 = hermitian_constructor(n_levels)
      _, ekets = H1.eigenstates()
      Heb = H1.transform(ekets)  # In the eigenbasis (should be diagonal)
      H2 = Heb.transform(ekets, True)  # Back to original basis
      assert (H1 - H2).norm() < 1e-6
  
  
  @pytest.mark.parametrize("n_levels", [4])
  def test_ket_and_dm_transformations_equivalent(n_levels):
      """Consistency between transformations of kets and density matrices."""
      psi0 = qutip.rand_ket(n_levels)
      # Generate a random basis
      _, rand_basis = qutip.rand_dm(n_levels, density=1).eigenstates()
      rho1 = qutip.ket2dm(psi0).transform(rand_basis, True)
      rho2 = qutip.ket2dm(psi0.transform(rand_basis, True))
      assert (rho1 - rho2).norm() < 1e-6
  
  
  def test_eigenbasis_transformation_makes_diagonal_operator():
      """Check diagonalization via eigenbasis transformation."""
      cx, cy, cz = np.random.random_sample((3,))
      H = cx*qutip.sigmax() + cy*qutip.sigmay() + cz*qutip.sigmaz()
      _, ekets = H.eigenstates()
      Heb = H.transform(ekets).tidyup()  # Heb should be diagonal
      assert abs(Heb.full() - np.diag(Heb.full().diagonal())).max() < 1e-6
  