Merge branch 'master' into fix/partial-trace-validation

Author: aman-coder03

docs/examples/quantum_states/classification.py

@@ -184,7 +184,7 @@ def povm_residual(states: list[np.ndarray], povm: dict[tuple[int, int], np.ndarr
 #   \end{equation}
 # 
 # We start by finding a basis for :math:`S := \text{range}(M)`, which can be done by picking a linearly independent set
-# of :math:`r = 3` columns of :math:`M$` :math:`S = \operatorname{span}\{(2,1,1,-1), (1,2,0,1), (1,0,2,-1)\}`. Then
+# of :math:`r = 3` columns of :math:`M`: :math:`S = \operatorname{span}\{(2,1,1,-1), (1,2,0,1), (1,0,2,-1)\}`. Then
 # :math:`R_0 = \{S\}` and we proceed recursively:
 #
 # .. math::
@@ -250,7 +250,7 @@ def povm_residual(states: list[np.ndarray], povm: dict[tuple[int, int], np.ndarr
 print(sum(result["factors"]))
 
 # %%
-# To similarly determine whether or not :math:`M`` is :math:`2`-incoherent, we proceed further with the recursive
+# To similarly determine whether or not :math:`M` is :math:`2`-incoherent, we proceed further with the recursive
 # construction by computing
 #
 # .. math::

docs/examples/quantum_states/state_distinguishability.py

@@ -80,12 +80,14 @@
 # may be obtained by solving the following semidefinite program (SDP).
 #
 # .. math::
-#    \begin{align*}
-#        \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}).
-#    \end{align*}
+#    \begin{equation}
+#        \begin{aligned}
+#            \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}).
+#        \end{aligned}
+#    \end{equation}
 #
 # This optimization problem is solved in :code:`|toqito⟩` to obtain the optimal
 # probability with which Bob can distinguish state :math:`\rho_i`.

docs/examples/quantum_states/state_exclusion.py

@@ -14,7 +14,7 @@
 # tutorial on this topic.
 #
 # Further information beyond the scope of this tutorial can be found in the text
-# `:footcite:Pusey_2012_On` as well as the course :footcite:`Bandyopadhyay_2014_Conclusive`.
+# :footcite:`Pusey_2012_On` as well as the course :footcite:`Bandyopadhyay_2014_Conclusive`.
 #
 #
 # The state exclusion problem
@@ -207,7 +207,7 @@
 print(f"Optimal SDP value: {np.around(opt_value, decimals=2)}")
 
 # %%
-# The SDP not only gives us the optimal value, which is $0$ in this case, indicating that the states are
+# The SDP not only gives us the optimal value, which is :math:`0` in this case, indicating that the states are
 # antidistinguishable, but we also get a set of optimal measurement operators. These should look familiar to the
 # measurements we explicitly constructed earlier.
 #
@@ -225,7 +225,7 @@
 # for which
 #
 # .. math::
-#    X = \sum_{j=0}^{m-1} c_j |psi_j\rangle \langle \psi_j|.
+#    X = \sum_{j=0}^{m-1} c_j |\psi_j\rangle \langle \psi_j|.
 #
 # It turns out that antidistinguishability is equivalent to :math:`k`-incoherence in the :math:`k = n - 1` case.
 # Reproducing one of the results from :footcite:`Johnston_2025_Tight`, we have the following theorem.