Minor changes to punctuation and code rendering for documentation.

docs/intro_tutorial.rst

@@ -319,7 +319,7 @@ Taking the partial trace over the second subsystem of :math:`X` yields the follo
     X_{pt, 2} = \begin{pmatrix}
                 7 & 11 \\
                 23 & 27
-                \end{pmatrix}
+                \end{pmatrix}.
 
 By default, the partial trace function in :code:`toqito` takes the trace of the second
 subsystem.
@@ -343,7 +343,7 @@ trace over the first subsystem yields the following matrix
     X_{pt, 1} = \begin{pmatrix}
                     12 & 14 \\
                     20 & 22
-                \end{pmatrix}
+                \end{pmatrix}.
 
 .. code-block:: python
 

docs/tutorials.state_distinguishability.rst

@@ -81,7 +81,7 @@ may be obtained by solving the following semidefinite program (SDP).
         \text{maximize:} \quad & \sum_{i=0}^n p_i \langle M_i,
         \rho_i \rangle \\
         \text{subject to:} \quad & \sum_{i=0}^n M_i = \mathbb{I}_{\mathcal{X}},\\
-                                 & M_i \in \text{Pos}(\mathcal{X})
+                                 & M_i \in \text{Pos}(\mathcal{X}).
     \end{align*}
 
 This optimization problem is solved in :code:`toqito` to obtain the optimal
@@ -103,7 +103,7 @@ such that
     \rho_1 = | 1 \rangle \langle 1 | = \begin{pmatrix}
                 0 & 0 \\
                 0 & 1
-             \end{pmatrix}
+             \end{pmatrix}.
 
 
 These states are completely orthogonal to each other, and it is known that Bob
@@ -188,9 +188,9 @@ It was shown in [tCosentino13]_ and later extended in [tCR13]_ that for the foll
 .. math::
     \begin{equation}
         \begin{aligned}
-            \rho_1^{(2)} &= |\psi_0 \rangle | \psi_0 \rangle \langle \psi_0 | \langle \psi_0 |, \\
+            \rho_1^{(2)} &= |\psi_0 \rangle | \psi_0 \rangle \langle \psi_0 | \langle \psi_0 |, \quad
             \rho_2^{(2)} &= |\psi_1 \rangle | \psi_3 \rangle \langle \psi_1 | \langle \psi_3 |, \\
-            \rho_3^{(2)} &= |\psi_2 \rangle | \psi_3 \rangle \langle \psi_2 | \langle \psi_3 |, \\
+            \rho_3^{(2)} &= |\psi_2 \rangle | \psi_3 \rangle \langle \psi_2 | \langle \psi_3 |, \quad
             \rho_4^{(2)} &= |\psi_3 \rangle | \psi_3 \rangle \langle \psi_3 | \langle \psi_3 |, \\
         \end{aligned}
     \end{equation}

docs/tutorials.xor_quantum_value.rst

@@ -202,10 +202,8 @@ sets of measurements.
 
     .. math::
         \begin{equation}
-            \begin{aligned}
-                | \phi_0 \rangle &= \cos(\theta)|0 \rangle + \sin(\theta)|1 \rangle, \\
-                | \phi_1 \rangle &= -\sin(\theta)|0 \rangle + \cos(\theta)|1 \rangle,
-            \end{aligned}
+            | \phi_0 \rangle &= \cos(\theta)|0 \rangle + \sin(\theta)|1 \rangle, \quad
+            | \phi_1 \rangle &= -\sin(\theta)|0 \rangle + \cos(\theta)|1 \rangle,
         \end{equation}
 
 such that

toqito/channel_metrics/completely_bounded_spectral_norm.py

@@ -9,7 +9,7 @@
 def completely_bounded_spectral_norm(phi: np.ndarray) -> float:
     r"""Compute the completely bounded spectral norm of a quantum channel.
 
-    :cite:`Watrous_2009_Semidefinite, QETLAB_link`.
+    As defined in :cite:`Watrous_2009_Semidefinite` and :cite:`QETLAB_link`.
 
     Examples
     ========

toqito/channel_metrics/diamond_norm.py

@@ -10,7 +10,7 @@ def diamond_norm(choi_1: np.ndarray, choi_2: np.ndarray) -> float:
     The calculation uses the simplified semidefinite program of Watrous in
     :cite:`Watrous_2009_Semidefinite`.
 
-    This function has been adapted from :cite:`Rigetti_2022_Forest`
+    This function has been adapted from :cite:`Rigetti_2022_Forest`.
 
     .. note::
         This calculation becomes very slow for 4 or more qubits.

toqito/channel_props/choi_rank.py

@@ -19,25 +19,23 @@ def choi_rank(phi: np.ndarray | list[list[np.ndarray]]) -> int:
 
     .. math::
         \begin{equation}
-            \begin{aligned}
-                \frac{1}{\sqrt{2}}
-                \begin{pmatrix}
-                    0 & i \\ -i & 0
-                \end{pmatrix}, &\quad
-                \frac{1}{\sqrt{2}}
-                \begin{pmatrix}
-                    0 & 1 \\
-                    1 & 0
-                \end{pmatrix}, \\
-                \begin{pmatrix}
-                    1 & 0 \\
-                    0 & 0
-                \end{pmatrix}, &\quad
-                \begin{pmatrix}
-                    0 & 0 \\
-                    0 & 1
-                \end{pmatrix}.
-            \end{aligned}
+            \frac{1}{\sqrt{2}}
+            \begin{pmatrix}
+                0 & i \\ -i & 0
+            \end{pmatrix}, \quad
+            \frac{1}{\sqrt{2}}
+            \begin{pmatrix}
+                0 & 1 \\
+                1 & 0
+            \end{pmatrix}, \quad
+            \begin{pmatrix}
+                1 & 0 \\
+                0 & 0
+            \end{pmatrix}, \quad
+            \begin{pmatrix}
+                0 & 0 \\
+                0 & 1
+            \end{pmatrix}.
         \end{equation}
 
     and can be generated in :code:`toqito` with the following list:

toqito/channel_props/is_quantum_channel.py

@@ -40,6 +40,7 @@ def is_quantum_channel(
         \end{pmatrix}.
 
     To check if this is a valid quantum channel or not,
+
     >>> import numpy as np
     >>> from toqito.matrices import pauli
     >>> from toqito.channel_props import is_quantum_channel
@@ -50,6 +51,7 @@ def is_quantum_channel(
     False
 
     If we instead check for the validity of depolarizing channel being a valid quantum channel,
+
     >>> from toqito.channels import depolarizing
     >>> from toqito.channel_props import is_quantum_channel
     >>> choi_depolarizing = depolarizing(dim=2, param_p=0.2)

toqito/channels/partial_trace.py

@@ -47,7 +47,7 @@ def partial_trace(
         X_{pt, 2} = \begin{pmatrix}
                     7 & 11 \\
                     23 & 27
-                 \end{pmatrix}
+                 \end{pmatrix}.
 
     By default, the partial trace function in :code:`toqito` takes the trace of the second
     subsystem.

toqito/matrix_props/has_same_dimension.py

@@ -21,13 +21,15 @@ def has_same_dimension(items: list[np.ndarray]) -> bool:
     True
 
     Check a list of matrices with the same dimension:
+
     >>> import numpy as np
     >>> from toqito.matrix_props import has_same_dimension
     >>> matrices = [np.array([[1, 0], [0, 1]]), np.array([[2, 3], [4, 5]]), np.array([[6, 7], [8, 9]])]
     >>> has_same_dimension(matrices)
     True
 
     Check a list containing items of different dimensions:
+
     >>> import numpy as np
     >>> from toqito.matrix_props import has_same_dimension
     >>> mixed = [np.array([1, 2, 3]), np.array([[1, 0], [0, 1]])]

toqito/matrix_props/is_orthonormal.py

@@ -25,6 +25,7 @@ def is_orthonormal(vectors: list[np.ndarray]) -> bool:
         \end{pmatrix}
 
     To check these are a known set of orthonormal vectors:
+
     >>> import numpy as np
     >>> from toqito.matrix_props import is_orthonormal
     >>> v_1 = np.array([1, 0, 0])

toqito/matrix_props/kp_norm.py

@@ -12,18 +12,20 @@ def kp_norm(mat: np.ndarray, k: int, p: int) -> float:
     Examples
     ========
     To compute the p-norm of a vector:
+
     >>> import numpy as np
     >>> from toqito.matrix_props import kp_norm
     >>> from toqito.states import bell
     >>> kp_norm(bell(0), 1, np.inf)
-    1
+    1.0
 
     To compute the k-largest singular values of a matrix:
+
     >>> import numpy as np
     >>> from toqito.matrix_props import kp_norm
     >>> from toqito.rand import random_unitary
     >>> kp_norm(random_unitary(5), 5, 2)
-    2.2360679774997902
+    2.23606797749979
 
     :param mat: 2D numpy ndarray
     :param k: The number of singular values to take.

toqito/state_opt/ppt_distinguishability.py

@@ -53,9 +53,9 @@ def ppt_distinguishability(
     .. math::
         \begin{equation}
             \begin{aligned}
-            \rho_1^{(2)} &= |\psi_0 \rangle | \psi_0 \rangle \langle \psi_0 | \langle \psi_0 |, \\
+            \rho_1^{(2)} &= |\psi_0 \rangle | \psi_0 \rangle \langle \psi_0 | \langle \psi_0 |, \quad
             \rho_2^{(2)} &= |\psi_1 \rangle | \psi_3 \rangle \langle \psi_1 | \langle \psi_3 |, \\
-            \rho_3^{(2)} &= |\psi_2 \rangle | \psi_3 \rangle \langle \psi_2 | \langle \psi_3 |, \\
+            \rho_3^{(2)} &= |\psi_2 \rangle | \psi_3 \rangle \langle \psi_2 | \langle \psi_3 |, \quad
             \rho_4^{(2)} &= |\psi_3 \rangle | \psi_3 \rangle \langle \psi_3 | \langle \psi_3 |, \\
             \end{aligned}
         \end{equation}

toqito/state_opt/state_distinguishability.py

@@ -35,7 +35,7 @@ def state_distinguishability(
         \begin{align*}
             \text{maximize:} \quad & \sum_{i=0}^n p_i \langle M_i, \rho_i \rangle \\
             \text{subject to:} \quad & M_0 + \ldots + M_n = \mathbb{I},\\
-                                     & M_0, \ldots, M_n \geq 0
+                                     & M_0, \ldots, M_n \geq 0.
         \end{align*}
 
     Examples