text: # -*- coding: utf-8 -*-
  
  import platform
  
  import numpy as np
  import pytest
  
  from qutip import (
      Qobj, tensor, fock_dm, basis, destroy, qdiags, sigmax, sigmay, sigmaz,
      qeye, rand_ket, rand_super_bcsz, rand_ket_haar, rand_dm_ginibre, rand_dm,
      rand_unitary, rand_unitary_haar, to_super, to_choi, kraus_to_choi,
  )
  from qutip.qip.operations import (
      hadamard_transform, swap,
  )
  
  try:
      import cvxpy
      import cvxopt
  except ImportError:
      cvxpy, cvxopt = None, None
  
  # These ones are the metrics functions that we actually want to test.
  from qutip import (
      fidelity, tracedist, hellinger_dist, dnorm, average_gate_fidelity,
      unitarity, hilbert_dist, bures_dist,
  )
  
  
  @pytest.fixture(scope="function", params=[2, 5, 10, 15, 25, 100])
  def dimension(request):
      # There are also some cases in the file where this fixture is explicitly
      # overridden by a more local mark.  That is deliberate; this dimension is
      # intended for non-superoperators, and may cause inordinantly long tests if
      # (for example) something uses dimension=100 then makes a superoperator out
      # of it.
      return request.param
  
  
  @pytest.fixture(scope="function", params=[
      pytest.param(rand_ket_haar, id="pure"),
      pytest.param(rand_dm_ginibre, id="mixed"),
  ])
  def state(request, dimension):
      return request.param(dimension)
  
  
  # Also parametrise left, right as if they're the names of two states for tests
  # that need to take two states.
  left = right = state
  
  
  # The class names have an unusual naming convention to make them more
  # convenient to use with the `pytest -k "expr"` selection syntax.  They start
  # with the standard `Test`, but then are the name of the function they are
  # testing in the function naming convention, so it's easy to remember the
  # selector to choose a particular function.
  
  class Test_fidelity:
      def test_mixed_state_inequality(self, dimension):
          tol = 1e-7
          rho1 = rand_dm(dimension, 0.25)
          rho2 = rand_dm(dimension, 0.25)
          F = fidelity(rho1, rho2)
          assert 1 - F <= np.sqrt(1 - F*F) + tol
  
      @pytest.mark.parametrize('right_dm', [True, False], ids=['mixed', 'pure'])
      @pytest.mark.parametrize('left_dm', [True, False], ids=['mixed', 'pure'])
      def test_orthogonal(self, left_dm, right_dm, dimension):
          left = basis(dimension, 0)
          right = basis(dimension, dimension//2)
          if left_dm:
              left = left.proj()
          if right_dm:
              right = right.proj()
          assert fidelity(left, right) == pytest.approx(0, abs=1e-6)
  
      def test_invariant_under_unitary_transformation(self, dimension):
          rho1 = rand_dm(dimension, 0.25)
          rho2 = rand_dm(dimension, 0.25)
          U = rand_unitary(dimension, 0.25)
          F = fidelity(rho1, rho2)
          FU = fidelity(U*rho1*U.dag(), U*rho2*U.dag())
          assert F == pytest.approx(FU, rel=1e-5)
  
      def test_state_with_itself(self, state):
          assert fidelity(state, state) == pytest.approx(1, abs=1e-6)
  
      def test_bounded(self, left, right, dimension):
          """Test that fidelity is bounded on [0, 1]."""
          tol = 1e-7
          assert -tol <= fidelity(left, right) <= 1 + tol
  
      def test_pure_state_equivalent_to_overlap(self, dimension):
          """Check fidelity against pure-state overlap, see gh-361."""
          psi = rand_ket_haar(dimension)
          phi = rand_ket_haar(dimension)
          overlap = np.abs(psi.overlap(phi))
          assert fidelity(psi, phi) == pytest.approx(overlap, abs=1e-7)
  
      ket_0 = basis(2, 0)
      ket_1 = basis(2, 1)
      ket_p = (ket_0 + ket_1).unit()
      ket_py = (ket_0 + np.exp(0.25j*np.pi)*ket_1).unit()
      max_mixed = qeye(2).unit()
  
      @pytest.mark.parametrize(['left', 'right', 'expected'], [
          pytest.param(ket_0, ket_p, np.sqrt(0.5), id="|0>,|+>"),
          pytest.param(ket_0, ket_1, 0, id="|0>,|1>"),
          pytest.param(ket_0, max_mixed, np.sqrt(0.5), id="|0>,id/2"),
          pytest.param(ket_p, ket_py, np.sqrt(0.125 + (0.5+np.sqrt(0.125))**2),
                       id="|+>,|+'>"),
      ])
      def test_known_cases(self, left, right, expected):
          assert fidelity(left, right) == pytest.approx(expected, abs=1e-7)
  
  
  class Test_tracedist:
      def test_state_with_itself(self, state):
          assert tracedist(state, state) == pytest.approx(0, abs=1e-6)
  
      @pytest.mark.parametrize('right_dm', [True, False], ids=['mixed', 'pure'])
      @pytest.mark.parametrize('left_dm', [True, False], ids=['mixed', 'pure'])
      def test_orthogonal(self, left_dm, right_dm, dimension):
          left = basis(dimension, 0)
          right = basis(dimension, dimension//2)
          if left_dm:
              left = left.proj()
          if right_dm:
              right = right.proj()
          assert tracedist(left, right) == pytest.approx(1, abs=1e-6)
  
      def test_invariant_under_unitary_transformation(self, dimension):
          rho1 = rand_dm(dimension, 0.25)
          rho2 = rand_dm(dimension, 0.25)
          U = rand_unitary(dimension, 0.25)
          D = tracedist(rho1, rho2)
          DU = tracedist(U*rho1*U.dag(), U*rho2*U.dag())
          assert D == pytest.approx(DU, rel=1e-5)
  
  
  class Test_hellinger_dist:
      @pytest.mark.parametrize('right_dm', [True, False], ids=['mixed', 'pure'])
      @pytest.mark.parametrize('left_dm', [True, False], ids=['mixed', 'pure'])
      def test_orthogonal(self, left_dm, right_dm, dimension):
          left = basis(dimension, 0)
          right = basis(dimension, dimension//2)
          if left_dm:
              left = left.proj()
          if right_dm:
              right = right.proj()
          expected = np.sqrt(2)
          assert hellinger_dist(left, right) == pytest.approx(expected, abs=1e-6)
  
      def test_state_with_itself(self, state):
          assert hellinger_dist(state, state) == pytest.approx(0, abs=1e-6)
  
      def test_known_cases_pure_states(self, dimension):
          left = rand_ket(dimension)
          right = rand_ket(dimension)
          expected = np.sqrt(2 * (1 - np.abs(left.overlap(right))**2))
          assert hellinger_dist(left, right) == pytest.approx(expected, abs=1e-7)
  
      @pytest.mark.parametrize('dimension', [2, 5, 10, 25])
      def test_monotonicity(self, dimension):
          """
          Check monotonicity w.r.t. tensor product, see. Eq. (45) in
          arXiv:1611.03449v2:
              hellinger_dist(rhoA & rhoB, sigmaA & sigmaB)
              >= hellinger_dist(rhoA, sigmaA)
          with equality iff sigmaB = rhoB where '&' is the tensor product.
          """
          tol = 1e-5
          rhoA, rhoB, sigmaA, sigmaB = [rand_dm(dimension) for _ in [None]*4]
          rho = tensor(rhoA, rhoB)
          rho_sim = tensor(rhoA, sigmaB)
          sigma = tensor(sigmaA, sigmaB)
          dist = hellinger_dist(rhoA, sigmaA)
          assert hellinger_dist(rho, sigma) + tol > dist
          assert hellinger_dist(rho_sim, sigma) == pytest.approx(dist, abs=tol)
  
  
  # TODO: resolve the Mac failures.
  @pytest.mark.skipif(
      "Darwin" in platform.system(),
      reason="average gate fidelity tests broken on macOS as of July 2019",
  )
  class Test_average_gate_fidelity:
      def test_identity(self, dimension):
          id = qeye(dimension)
          assert average_gate_fidelity(id) == pytest.approx(1, abs=1e-12)
  
      @pytest.mark.parametrize('dimension', [2, 5, 10, 20])
      def test_bounded(self, dimension):
          tol = 1e-7
          channel = rand_super_bcsz(dimension)
          assert -tol <= average_gate_fidelity(channel) <= 1 + tol
  
      @pytest.mark.parametrize('dimension', [2, 5, 10, 20])
      def test_unitaries_equal_1(self, dimension):
          """Tests that for random unitaries U, AGF(U, U) = 1."""
          tol = 1e-7
          U = rand_unitary_haar(dimension)
          SU = to_super(U)
          assert average_gate_fidelity(SU, target=U) == pytest.approx(1, abs=tol)
  
  
  class Test_hilbert_dist:
      def test_known_cases(self):
          r1 = qdiags(np.array([0.5, 0.5, 0, 0]), 0)
          r2 = qdiags(np.array([0, 0, 0.5, 0.5]), 0)
          assert hilbert_dist(r1, r2) == pytest.approx(1, abs=1e-6)
  
  
  paulis = [qeye(2), sigmax(), sigmay(), sigmaz()]
  
  
  class Test_unitarity:
      @pytest.mark.parametrize(['operator', 'expected'], [
          pytest.param(to_super(sigmax()), 1, id="sigmax"),
          pytest.param(0.25 * sum(to_super(x) for x in paulis), 0, id="paulis"),
          pytest.param(0.5 * (to_super(qeye(2)) + to_super(sigmax())), 1/3,
                       id="id+sigmax"),
      ])
      def test_known_cases(self, operator, expected):
          assert unitarity(operator) == pytest.approx(expected, abs=1e-7)
  
      @pytest.mark.parametrize('n_qubits', [1, 2, 3, 4, 5])
      def test_bounded(self, n_qubits):
          tol = 1e-7
          operator = rand_super_bcsz(2**n_qubits)
          assert -tol <= unitarity(operator) <= 1 + tol
  
  
  class TestComparisons:
      """Test some known inequalities between two different metrics."""
  
      def test_inequality_tracedist_to_fidelity(self, left, right):
          tol = 1e-7
          assert 1 - fidelity(left, right) <= tracedist(left, right) + tol
  
      def test_inequality_hellinger_dist_to_bures_dist(self, left, right):
          tol = 1e-7
          hellinger = hellinger_dist(left, right)
          bures = bures_dist(left, right)
          assert bures <= hellinger + tol
  
  
  def overrotation(x):
      return to_super((1j * np.pi * x * sigmax() / 2).expm())
  
  
  def had_mixture(x):
      id_chan = to_choi(qeye(2))
      S_eye = to_super(id_chan)
      S_H = to_super(hadamard_transform())
      return (1 - x) * S_eye + x * S_H
  
  
  def swap_map(x):
      base = (1j * x * swap()).expm()
      dims = [[[2], [2]], [[2], [2]]]
      return Qobj(base, dims=dims, type='super', superrep='super')
  
  
  def adc_choi(x):
      kraus = [
          np.sqrt(1 - x) * qeye(2),
          np.sqrt(x) * destroy(2),
          np.sqrt(x) * fock_dm(2, 0),
      ]
      return kraus_to_choi(kraus)
  
  
  # dnorm tests have always been slightly flaky; in some cases, cvxpy will fail
  # to solve the problem, and this can cause an entire test-suite failure.  As
  # long as we are using random tests (perhaps not ideal), this will happen
  # occasionally.  This isn't entirely a bug, it's just a reality of using a
  # one-size-fits-all solver; we've historically assumed users who come up
  # against this sort of thing will be accepting of the fact that dnorm
  # calculation is nontrivial, and isn't always entirely feasible.
  #
  # To deal with it, we allow each test to be rerun twice, using
  # pytest-rerunfailures.  This should forbid pathological cases where the test
  # is failing every time, but not penalise one-off failures.  As far as we know,
  # the failing tests always involve a random step, so triggering a re-run will
  # have them choose new variables as well.
  #
  # The warning filter is to account for cvxpy < 1.1.10 which uses np.complex,
  # which is deprecated as of numpy 1.20.
  #
  # Skip dnorm tests if we don't have cvxpy or cvxopt available, since dnorm
  # depends on them.
  @pytest.mark.skipif(cvxpy is None or cvxopt is None,
                      reason="Skipping dnorm tests because dnorm requires cvxpy"
                      " and cvxopt which are not installed.")
  @pytest.mark.skipif("Windows" in platform.system(),
                      reason="Skipping dnorm tests -- cvxpy and cvxopt can be"
                      " installed on Windows but fail to correctly solve"
                      " the given optimization problems.")
  @pytest.mark.flaky(reruns=2)
  @pytest.mark.filterwarnings(
      "ignore:`np.complex` is a deprecated alias:DeprecationWarning:cvxpy"
  )
  class Test_dnorm:
      @pytest.fixture(params=[2, 3])
      def dimension(self, request):
          return request.param
  
      @pytest.fixture(params=[True, False], ids=['sparse', 'dense'])
      def sparse(self, request):
          return request.param
  
      @pytest.mark.parametrize("variable", [0.1, 0.5, 0.9])
      def test_sparse_against_dense_adc(self, variable):
          """
          Test sparse versus dense dnorm calculation for a sample of
          amplitude-damping channels.
          """
          # Choi matrix for identity channel on 1 qubit
          A = kraus_to_choi([qeye(2)])
          B = adc_choi(variable)
          dense = dnorm(A, B, force_solve=True, sparse=False)
          sparse = dnorm(A, B, force_solve=True, sparse=True)
          assert dense == pytest.approx(sparse, abs=1e-7)
  
      @pytest.mark.repeat(3)
      def test_sparse_against_dense_random(self, dimension):
          """
          Test sparse versus dense dnorm calculation for random superoperators.
          """
          A = rand_super_bcsz(dimension)
          dense_run_result = dnorm(A, force_solve=True, sparse=False)
          sparse_run_result = dnorm(A, force_solve=True, sparse=True)
          assert dense_run_result == pytest.approx(sparse_run_result, abs=1e-7)
  
      def test_bounded(self, dimension, sparse):
          """dnorm(A - B) in [0, 2] for random superops A, B."""
          tol = 1e-7
          A, B = rand_super_bcsz(dimension), rand_super_bcsz(dimension)
          assert -tol <= dnorm(A, B, sparse=sparse) <= 2 + tol
  
      def test_qubit_simple_known_cases(self, sparse):
          """Check agreement for known qubit channels."""
          id_chan = to_choi(qeye(2))
          X_chan = to_choi(sigmax())
          depol = to_choi(Qobj(
              qeye(4), dims=[[[2], [2]], [[2], [2]]], superrep='chi',
          ))
          # We need to restrict the number of iterations for things on the
          # boundary, such as perfectly distinguishable channels.
          assert (
              dnorm(id_chan, X_chan, sparse=sparse)
              == pytest.approx(2, abs=1e-7)
          )
          assert (
              dnorm(id_chan, depol, sparse=sparse)
              == pytest.approx(1.5, abs=1e-7)
          )
          # Finally, we add a known case from Johnston's QETLAB documentation,
          #   || Phi - I ||_♢,
          # where Phi(X) = UXU⁺ and U = [[1, 1], [-1, 1]] / sqrt(2).
          U = np.sqrt(0.5) * Qobj([[1, 1], [-1, 1]])
          assert (
              dnorm(U, qeye(2), sparse=sparse)
              == pytest.approx(np.sqrt(2), abs=1e-7)
          )
  
      @pytest.mark.parametrize(["variable", "expected", "generator"], [
          [1.0e-3, 3.141591e-03, overrotation],
          [3.1e-3, 9.738899e-03, overrotation],
          [1.0e-2, 3.141463e-02, overrotation],
          [3.1e-2, 9.735089e-02, overrotation],
          [1.0e-1, 3.128689e-01, overrotation],
          [3.1e-1, 9.358596e-01, overrotation],
          [1.0e-3, 2.000000e-03, had_mixture],
          [3.1e-3, 6.200000e-03, had_mixture],
          [1.0e-2, 2.000000e-02, had_mixture],
          [3.1e-2, 6.200000e-02, had_mixture],
          [1.0e-1, 2.000000e-01, had_mixture],
          [3.1e-1, 6.200000e-01, had_mixture],
          [1.0e-3, 2.000000e-03, swap_map],
          [3.1e-3, 6.199997e-03, swap_map],
          [1.0e-2, 1.999992e-02, swap_map],
          [3.1e-2, 6.199752e-02, swap_map],
          [1.0e-1, 1.999162e-01, swap_map],
          [3.1e-1, 6.173918e-01, swap_map],
      ])
      def test_qubit_known_cases(self, variable, expected, generator, sparse):
          """
          Test cases based on comparisons to pre-existing dnorm implementations.
          In particular, the targets for the following test cases were generated
          using QuantumUtils for MATLAB (https://goo.gl/oWXhO9).
          """
          id_chan = to_choi(qeye(2))
          channel = generator(variable)
          assert (
              dnorm(channel, id_chan, sparse=sparse)
              == pytest.approx(expected, abs=1e-7)
          )
  
      def test_qubit_scalar(self, dimension):
          """dnorm(a * A) == a * dnorm(A) for scalar a, qobj A."""
          a = np.random.random()
          A = rand_super_bcsz(dimension)
          B = rand_super_bcsz(dimension)
          assert dnorm(a*A, a*B) == pytest.approx(a*dnorm(A, B), abs=1e-7)
  
      def test_qubit_triangle(self, dimension):
          """Check that dnorm(A + B) <= dnorm(A) + dnorm(B)."""
          A = rand_super_bcsz(dimension)
          B = rand_super_bcsz(dimension)
          assert dnorm(A + B) <= dnorm(A) + dnorm(B) + 1e-7
  
      @pytest.mark.repeat(3)
      @pytest.mark.parametrize("generator", [
          pytest.param(rand_super_bcsz, id="super"),
          pytest.param(rand_unitary_haar, id="unitary"),
      ])
      def test_force_solve(self, dimension, generator):
          """
          Metrics: checks that special cases for dnorm agree with SDP solutions.
          """
          A, B = generator(dimension), generator(dimension)
          assert (
              dnorm(A, B, force_solve=False)
              == pytest.approx(dnorm(A, B, force_solve=True), abs=1e-5)
          )
  
      @pytest.mark.repeat(3)
      def test_cptp(self, dimension, sparse):
          """Check that the diamond norm is one for CPTP maps."""
          A = rand_super_bcsz(dimension)
          assert A.iscptp
          assert dnorm(A, sparse=sparse) == pytest.approx(1, abs=1e-7)
  