Removing redundant note for doctest.

docs/intro_tutorial.rst

@@ -272,22 +272,15 @@ fidelity function between quantum states that happen to be identical.
 
     >>> from toqito.states import bell
     >>> from toqito.state_metrics import fidelity
+    >>> import numpy as np
     >>>
     >>> # Define two identical density operators.
     >>> rho = bell(0)*bell(0).conj().T
     >>> sigma = bell(0)*bell(0).conj().T
     >>> 
     >>> # Calculate the fidelity between `rho` and `sigma`
-    >>> '%.2f' % fidelity(rho, sigma)
-    '1.00'
-
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(fidelity(rho, sigma), decimals=2)
+    1.0
 
 There are a number of other metrics one can compute on two density matrices
 including the trace norm, trace distance. These and others are also available

docs/tutorials.extended_nonlocal_games.rst

@@ -316,21 +316,14 @@ This can be verified in :code:`toqito` as follows.
 
     >>> # Calculate the unentangled value of the BB84 extended nonlocal game.
     >>> from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame
+    >>> import numpy as np
     >>> 
     >>> # Define an ExtendedNonlocalGame object based on the BB84 game.
     >>> bb84 = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat)
     >>> 
     >>> # The unentangled value is cos(pi/8)**2 \approx 0.85356
-    >>> '%.2f' % bb84.unentangled_value() 
-    '0.85'
-
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(bb84.unentangled_value(), decimals=2)
+    0.85
 
 The BB84 game also exhibits strong parallel repetition. We can specify how many
 parallel repetitions for :code:`toqito` to run. The example below provides an
@@ -340,21 +333,14 @@ example of two parallel repetitions for the BB84 game.
 
     >>> # The unentangled value of BB84 under parallel repetition.
     >>> from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame
+    >>> import numpy as np
     >>> 
     >>> # Define the bb84 game for two parallel repetitions.
     >>> bb84_2_reps = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat, 2)
     >>> 
     >>> # The unentangled value for two parallel repetitions is cos(pi/8)**4 \approx 0.72855
-    >>> '%.2f' % bb84_2_reps.unentangled_value() 
-    '0.73'
-
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(bb84_2_reps.unentangled_value(), decimals=2)
+    0.73
 
 It was shown in :cite:`Johnston_2016_Extended` that the BB84 game possesses the property of strong
 parallel repetition. That is,
@@ -374,21 +360,14 @@ using :code:`toqito` as well.
 
     >>> # Calculate lower bounds on the standard quantum value of the BB84 extended nonlocal game.
     >>> from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame
+    >>> import numpy as np
     >>> 
     >>> # Define an ExtendedNonlocalGame object based on the BB84 game.
     >>> bb84_lb = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat)
     >>> 
     >>> # The standard quantum value is cos(pi/8)**2 \approx 0.85356
-    >>> '%.2f' % bb84_lb.quantum_value_lower_bound()
-    '0.85'
-
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(bb84_lb.quantum_value_lower_bound(), decimals=2)
+    0.85
 
 From :cite:`Johnston_2016_Extended`, it is known that :math:`\omega(G_{BB84}) =
 \omega^*(G_{BB84})`, however, if we did not know this beforehand, we could
@@ -409,20 +388,14 @@ Using :code:`toqito`, we can see that :math:`\omega_{ns}(G) = \cos^2(\pi/8)`.
 
     >>> # Calculate the non-signaling value of the BB84 extended nonlocal game.
     >>> from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame
+    >>> import numpy as np
     >>> 
     >>> # Define an ExtendedNonlocalGame object based on the BB84 game.
     >>> bb84 = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat)
     >>> 
     >>> # The non-signaling value is cos(pi/8)**2 \approx 0.85356
-    >>> '%.2f' % bb84.nonsignaling_value() 
-    '0.85'
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(bb84.nonsignaling_value(), decimals=2)
+    0.85
 
 So we have the relationship that
 
@@ -437,20 +410,14 @@ can observe this by the following snippet.
 
     >>> # The non-signaling value of BB84 under parallel repetition.
     >>> from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame
+    >>> import numpy as np
     >>> 
     >>> # Define the bb84 game for two parallel repetitions.
     >>> bb84_2_reps = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat, 2)
     >>> 
     >>> # The non-signaling value for two parallel repetitions is cos(pi/8)**4 \approx 0.73825
-    >>> '%.2f' % bb84_2_reps.nonsignaling_value() 
-    '0.74'
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(bb84_2_reps.nonsignaling_value(), decimals=2)
+    0.74
 
 Note that :math:`0.73825 \geq \cos(\pi/8)^4 \approx 0.72855` and therefore we
 have that
@@ -554,42 +521,29 @@ the unentangled value of :math:`G_{CHSH}`.
 
     >>> # Calculate the unentangled value of the CHSH extended nonlocal game
     >>> from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame
+    >>> import numpy as np
     >>> 
     >>> # Define an ExtendedNonlocalGame object based on the CHSH game.
     >>> chsh = ExtendedNonlocalGame(chsh_prob_mat, chsh_pred_mat)
     >>> 
     >>> # The unentangled value is 3/4 = 0.75
-    >>> '%.2f' % chsh.unentangled_value() 
-    '0.75'
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(chsh.unentangled_value(), decimals=2)
+    0.75
 
 We can also run multiple repetitions of :math:`G_{CHSH}`.
 
 .. code-block:: python
 
     >>> # The unentangled value of CHSH under parallel repetition.
     >>> from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame
+    >>> import numpy as np
     >>> 
     >>> # Define the CHSH game for two parallel repetitions.
     >>> chsh_2_reps = ExtendedNonlocalGame(chsh_prob_mat, chsh_pred_mat, 2)
     >>> 
     >>> # The unentangled value for two parallel repetitions is (3/4)**2 \approx 0.5625
-    >>> '%.2f' % chsh_2_reps.unentangled_value() 
-    '0.56'
-
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(chsh_2_reps.unentangled_value(), decimals=2)
+    0.56
 
 Note that strong parallel repetition holds as
 
@@ -606,21 +560,14 @@ non-signaling value.
 
     >>> # Calculate the non-signaling value of the CHSH extended nonlocal game.
     >>> from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame
+    >>> import numpy as np
     >>> 
     >>> # Define an ExtendedNonlocalGame object based on the CHSH game.
     >>> chsh = ExtendedNonlocalGame(chsh_prob_mat, chsh_pred_mat)
     >>> 
     >>> # The non-signaling value is 3/4 = 0.75
-    >>> '%.2f' % chsh.nonsignaling_value() 
-    '0.75'
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
-
+    >>> np.around(chsh.nonsignaling_value(), decimals=2)
+    0.75
 
 As we know that :math:`\omega(G_{CHSH}) = \omega_{ns}(G_{CHSH}) = 3/4` and that
 
@@ -739,18 +686,11 @@ Now that we have encoded :math:`G_{MUB}`, we can calculate the unentangled value
 
 .. code-block:: python
 
+    >>> import numpy as np
     >>> g_mub = ExtendedNonlocalGame(prob_mat, pred_mat)
     >>> unent_val = g_mub.unentangled_value()
-    >>> '%.2f' % unent_val
-    '0.65'
-
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(unent_val, decimals=2)
+    0.65
 
 That is, we have that 
 
@@ -763,18 +703,11 @@ obtain.
 
 .. code-block:: python
 
+    >>> import numpy as np
     >>> g_mub = ExtendedNonlocalGame(prob_mat, pred_mat)
     >>> q_val = g_mub.quantum_value_lower_bound()
-    >>> '%.2f' % q_val
-    '0.66'
-
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(q_val, decimals=2)
+    0.66
 
 Note that as we are calculating a lower bound, it is possible that a value this
 high will not be obtained, or in other words, the algorithm can get stuck in a

docs/tutorials.nonlocal_games.rst

@@ -343,17 +343,11 @@ use :code:`toqito` to determine the lower bound on the quantum value.
 
 .. code-block:: python
 
+    >>> import numpy as np
     >>> from toqito.nonlocal_games.nonlocal_game import NonlocalGame
     >>> chsh = NonlocalGame(prob_mat, pred_mat)
-    >>> '%.2f' % chsh.quantum_value_lower_bound()
-    '0.85'
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(chsh.quantum_value_lower_bound(), decimals=2)
+    0.85
 
 In this case, we can see that the quantum value of the CHSH game is in fact
 attained as :math:`\cos^2(\pi/8) \approx 0.85355`.
@@ -419,17 +413,10 @@ value and calculate lower bounds on the quantum value of the FFL game.
     ...                     pred_mat[a_alice, b_bob, x_alice, y_bob] = 1
     >>> # Define the FFL game object.
     >>> ffl = NonlocalGame(prob_mat, pred_mat)
-    >>> '%.2f' % ffl.classical_value()
-    '0.67'
-    >>> '%.2f' % ffl.quantum_value_lower_bound()
-    '0.22'
-
-        .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(ffl.classical_value(), decimals=2)
+    0.67
+    >>> np.around(ffl.quantum_value_lower_bound(), decimals=2)
+    0.22
 
 In this case, we obtained the correct quantum value of :math:`2/3`, however,
 the lower bound technique is not guaranteed to converge to the true quantum

docs/tutorials.state_distinguishability.rst

@@ -114,6 +114,7 @@ Using :code:`toqito`, we can calculate this probability directly as follows:
 
 .. code-block:: python
 
+    >>> import numpy as np
     >>> from toqito.states import basis
     >>> from toqito.state_opt import state_distinguishability
     >>> 
@@ -132,15 +133,8 @@ Using :code:`toqito`, we can calculate this probability directly as follows:
     >>>
     >>> # Calculate the probability with which Bob can 
     >>> # distinguish the state he is provided.
-    >>> '%.2f' % state_distinguishability(states, probs)[0]
-    '1.00'
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(state_distinguishability(states, probs)[0], decimals=2)
+    1.0
 
 Specifying similar state distinguishability problems can be done so using this
 general pattern.
@@ -235,15 +229,8 @@ via PPT measurements in the following manner.
     >>>
     >>> states = [rho_1, rho_2, rho_3, rho_4]
     >>> probs = [1 / 4, 1 / 4, 1 / 4, 1 / 4]
-    >>> '%.2f' % ppt_distinguishability(vectors=states, probs=probs, dimensions=[2, 2, 2, 2], subsystems=[0, 2])[0]
-    '0.87'
-
-    .. note::
-        You do not need to use `'%.2f' %` when you use this function.
-        We use this to format our output such that `doctest` compares the calculated output to the
-        expected output upto two decimal points only. The accuracy of the solvers can calculate the
-        `float` output to a certain amount of precision such that the value deviates after a few digits
-        of accuracy.
+    >>> np.around(ppt_distinguishability(vectors=states, probs=probs, dimensions=[2, 2, 2, 2], subsystems=[0, 2])[0], decimals=2)
+    0.87
 
 Probability of distinguishing a state via separable measurements
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^