From d0613f8dbf23b090732bac715b07bb8d8898ac59 Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Wed, 9 Aug 2023 12:18:30 +0400
Subject: [PATCH 01/10] `purity` doc

---
 src/qibo/quantum_info/metrics.py | 5 ++++-
 1 file changed, 4 insertions(+), 1 deletion(-)

diff --git a/src/qibo/quantum_info/metrics.py b/src/qibo/quantum_info/metrics.py
index a15c0769d3..e82c20de54 100644
--- a/src/qibo/quantum_info/metrics.py
+++ b/src/qibo/quantum_info/metrics.py
@@ -8,7 +8,10 @@
 
 
 def purity(state):
-    """Purity of a quantum state :math:`\\rho`, which is given by :math:`\\text{tr}(\\rho^{2})`.
+    """Purity of a quantum state :math:`\\rho`, which is given by
+
+    .. math::
+        \\text{purity}(\\rho) = \\text{tr}(\\rho^{2}) \\, .
 
     Args:
         state (ndarray): statevector or density matrix.

From e55692a9a7011b8393930464dda0ce7f8a75e7dd Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Wed, 9 Aug 2023 12:21:22 +0400
Subject: [PATCH 02/10] `entanglement_entropy` doc

---
 src/qibo/quantum_info/metrics.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/qibo/quantum_info/metrics.py b/src/qibo/quantum_info/metrics.py
index e82c20de54..46769eb1f3 100644
--- a/src/qibo/quantum_info/metrics.py
+++ b/src/qibo/quantum_info/metrics.py
@@ -250,8 +250,8 @@ def entanglement_entropy(
     check_hermitian: bool = False,
     backend=None,
 ):
-    """Calculates the entanglement entropy :math:`S` of ``state`` :math:`\\rho`,
-    which is given by
+    """Calculates the entanglement entropy :math:`S` of bipartition :math:`A`
+    of``state`` :math:`\\rho`. This is given by
 
     .. math::
         S(\\rho_{A}) = -\\text{tr}(\\rho_{A} \\, \\log(\\rho_{A})) \\, ,

From fb5fc836b112cca50091eb254c6f0d60e3cae9e7 Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Wed, 9 Aug 2023 12:24:00 +0400
Subject: [PATCH 03/10] `bures_distance` doc

---
 src/qibo/quantum_info/metrics.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/qibo/quantum_info/metrics.py b/src/qibo/quantum_info/metrics.py
index 46769eb1f3..2f707a8f7c 100644
--- a/src/qibo/quantum_info/metrics.py
+++ b/src/qibo/quantum_info/metrics.py
@@ -588,7 +588,7 @@ def bures_distance(state, target, check_hermitian: bool = False, backend=None):
     and a ``target`` state :math:`\\sigma`. This is given by
 
     .. math::
-        D_{B}(\\rho, \\, \\sigma) = \\sqrt{2 \\, (1 - \\sqrt{F(\\rho, \\, \\sigma)})}
+        D_{B}(\\rho, \\, \\sigma) = \\sqrt{2 \\, \\left(1 - \\sqrt{F(\\rho, \\, \\sigma)}\\right)}
 
     where :math:`F(\\rho, \\sigma)` is the :func:`qibo.quantum_info.fidelity`
     between `state` and `target`.

From e4d02815bc828060792bdde8039d02560c5beef7 Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Wed, 9 Aug 2023 12:25:00 +0400
Subject: [PATCH 04/10] `entanglement_fidelity` api doc

---
 doc/source/api-reference/qibo.rst | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/doc/source/api-reference/qibo.rst b/doc/source/api-reference/qibo.rst
index 739252b6e6..574c4568c1 100644
--- a/doc/source/api-reference/qibo.rst
+++ b/doc/source/api-reference/qibo.rst
@@ -1351,7 +1351,7 @@ Bures distance
 Entanglement fidelity
 """""""""""""""""""""
 
-.. autofunction::qibo.quantum_info.entanglement_fidelity
+.. autofunction:: qibo.quantum_info.entanglement_fidelity
 
 
 Process fidelity

From b08c7fa334c12a7728fbaa46c7dc3463709e1d2e Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Wed, 9 Aug 2023 12:36:50 +0400
Subject: [PATCH 05/10] `diamond_norm` doc

---
 src/qibo/quantum_info/metrics.py | 12 ++++++++++--
 1 file changed, 10 insertions(+), 2 deletions(-)

diff --git a/src/qibo/quantum_info/metrics.py b/src/qibo/quantum_info/metrics.py
index 2f707a8f7c..d7fa393688 100644
--- a/src/qibo/quantum_info/metrics.py
+++ b/src/qibo/quantum_info/metrics.py
@@ -859,8 +859,16 @@ def gate_error(channel, target=None, check_unitary: bool = False, backend=None):
 
 def diamond_norm(channel, target=None, **kwargs):
     """Calculates the diamond norm :math:`\\|\\mathcal{E}\\|_{\\diamond}` of
-    ``channel`` :math:`\\mathcal{E}`. If a ``target`` channel :math:`\\Lambda`
-    is specified, then it calculates :math:`\\| \\mathcal{E} - \\Lambda\\|_{\\diamond}`.
+    ``channel`` :math:`\\mathcal{E}`, which is given by
+
+    .. math::
+        \\|\\mathcal{E}\\|_{\\diamond} = \\max_{\\rho} \\, \\| \\left(\\mathcal{E} \\bigotimes I_{d^{2}}\\right)(\\rho) \\| \\, ,
+
+    where :math:`I_{d^{2}}` is the :math:`d^{2} \times d^{2}` Identity operator,
+    :math:`d = 2^{n}`, and :math:`n` is the number of qubits.
+
+    If a ``target`` channel :math:`\\Lambda` is specified,
+    then it calculates :math:`\\| \\mathcal{E} - \\Lambda\\|_{\\diamond}`.
 
     Example:
 

From 597a80248da701bfa1dda1041a3f28e5b53c523b Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Wed, 9 Aug 2023 12:45:27 +0400
Subject: [PATCH 06/10] `random_density_matrix` doc

---
 src/qibo/quantum_info/random_ensembles.py | 13 ++++++++++++-
 1 file changed, 12 insertions(+), 1 deletion(-)

diff --git a/src/qibo/quantum_info/random_ensembles.py b/src/qibo/quantum_info/random_ensembles.py
index 64411e8ac6..72a0af9acc 100644
--- a/src/qibo/quantum_info/random_ensembles.py
+++ b/src/qibo/quantum_info/random_ensembles.py
@@ -374,7 +374,18 @@ def random_density_matrix(
     seed=None,
     backend=None,
 ):
-    """Creates a random density matrix :math:`\\rho`.
+    """Creates a random density matrix :math:`\\rho`. If ``pure=True``,
+
+    .. math::
+        \\rho = \\ketbra{\\psi} \\, ,
+
+    where :math:`\\ket{\\psi}` is a :func:`qibo.quantum_info.random_statevector`.
+    If ``pure=False``, then
+
+    .. math::
+        \\rho = \\sum_{k} \\, p_{k} \\, \\ketbra{\\psi_{k}} \\, .
+
+    is a mixed state.
 
     Args:
         dims (int): dimension of the matrix.

From e2c210c283e67bd402f8904a4d959baa79b55dfc Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Wed, 9 Aug 2023 13:57:18 +0000
Subject: [PATCH 07/10] Update src/qibo/quantum_info/random_ensembles.py

Co-authored-by: vodovozovaliza <55856544+vodovozovaliza@users.noreply.github.com>
---
 src/qibo/quantum_info/random_ensembles.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/qibo/quantum_info/random_ensembles.py b/src/qibo/quantum_info/random_ensembles.py
index 72a0af9acc..45d1f10ff4 100644
--- a/src/qibo/quantum_info/random_ensembles.py
+++ b/src/qibo/quantum_info/random_ensembles.py
@@ -377,7 +377,7 @@ def random_density_matrix(
     """Creates a random density matrix :math:`\\rho`. If ``pure=True``,
 
     .. math::
-        \\rho = \\ketbra{\\psi} \\, ,
+        \\rho = \\ketbra{\\psi}{\\psi} \\, ,
 
     where :math:`\\ket{\\psi}` is a :func:`qibo.quantum_info.random_statevector`.
     If ``pure=False``, then

From 79c88aa5fe42d7671f7f5177b5f426611710ae1d Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Wed, 9 Aug 2023 13:57:26 +0000
Subject: [PATCH 08/10] Update src/qibo/quantum_info/random_ensembles.py

Co-authored-by: vodovozovaliza <55856544+vodovozovaliza@users.noreply.github.com>
---
 src/qibo/quantum_info/random_ensembles.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/qibo/quantum_info/random_ensembles.py b/src/qibo/quantum_info/random_ensembles.py
index 45d1f10ff4..d86503f3cf 100644
--- a/src/qibo/quantum_info/random_ensembles.py
+++ b/src/qibo/quantum_info/random_ensembles.py
@@ -383,7 +383,7 @@ def random_density_matrix(
     If ``pure=False``, then
 
     .. math::
-        \\rho = \\sum_{k} \\, p_{k} \\, \\ketbra{\\psi_{k}} \\, .
+        \\rho = \\sum_{k} \\, p_{k} \\, \\ketbra{\\psi_{k}}{\\psi_{k}} \\, .
 
     is a mixed state.
 

From 4d1ac15a6c6f9cd2ba2fb02416b1ebd4752c095f Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Wed, 9 Aug 2023 13:57:35 +0000
Subject: [PATCH 09/10] Update src/qibo/quantum_info/metrics.py

Co-authored-by: vodovozovaliza <55856544+vodovozovaliza@users.noreply.github.com>
---
 src/qibo/quantum_info/metrics.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/qibo/quantum_info/metrics.py b/src/qibo/quantum_info/metrics.py
index d7fa393688..1010d6914f 100644
--- a/src/qibo/quantum_info/metrics.py
+++ b/src/qibo/quantum_info/metrics.py
@@ -864,7 +864,7 @@ def diamond_norm(channel, target=None, **kwargs):
     .. math::
         \\|\\mathcal{E}\\|_{\\diamond} = \\max_{\\rho} \\, \\| \\left(\\mathcal{E} \\bigotimes I_{d^{2}}\\right)(\\rho) \\| \\, ,
 
-    where :math:`I_{d^{2}}` is the :math:`d^{2} \times d^{2}` Identity operator,
+    where :math:`I_{d^{2}}` is the :math:`d^{2} \\times d^{2}` Identity operator,
     :math:`d = 2^{n}`, and :math:`n` is the number of qubits.
 
     If a ``target`` channel :math:`\\Lambda` is specified,

From e03e773911070ad98d0085acee1c849426ea3521 Mon Sep 17 00:00:00 2001
From: Renato Mello <renato.msf@gmail.com>
Date: Wed, 9 Aug 2023 18:01:44 +0400
Subject: [PATCH 10/10] Liza's suggestions

---
 src/qibo/quantum_info/metrics.py                       | 5 +++--
 src/qibo/quantum_info/random_ensembles.py              | 2 +-
 src/qibo/quantum_info/superoperator_transformations.py | 2 +-
 src/qibo/quantum_info/utils.py                         | 2 +-
 4 files changed, 6 insertions(+), 5 deletions(-)

diff --git a/src/qibo/quantum_info/metrics.py b/src/qibo/quantum_info/metrics.py
index 1010d6914f..60a5ada5e7 100644
--- a/src/qibo/quantum_info/metrics.py
+++ b/src/qibo/quantum_info/metrics.py
@@ -619,9 +619,10 @@ def entanglement_fidelity(
     :math:`\\rho`, which is given by
 
     .. math::
-        \\begin{align*}
+        \\begin{align}
           F_{\\mathcal{E}} &= \\text{fidelity}(\\rho_{f}, \\rho) \\nonumber \\\\
-          &= \\text{tr}(\\rho_{f} \\, \\rho)
+                           &= \\text{tr}(\\rho_{f} \\, \\rho) \\nonumber
+        \\end{align}
 
     where
 
diff --git a/src/qibo/quantum_info/random_ensembles.py b/src/qibo/quantum_info/random_ensembles.py
index d86503f3cf..7ffeb4ea07 100644
--- a/src/qibo/quantum_info/random_ensembles.py
+++ b/src/qibo/quantum_info/random_ensembles.py
@@ -148,7 +148,7 @@ def random_hermitian(
 
 
 def random_unitary(dims: int, measure: Optional[str] = None, seed=None, backend=None):
-    """Returns a random Unitary operator :math:`U`,, i.e.
+    """Returns a random Unitary operator :math:`U`, i.e.
     a random operator such that :math:`U^{-1} = U^{\\dagger}`.
 
     Args:
diff --git a/src/qibo/quantum_info/superoperator_transformations.py b/src/qibo/quantum_info/superoperator_transformations.py
index fb59753cf2..1ca8d416c4 100644
--- a/src/qibo/quantum_info/superoperator_transformations.py
+++ b/src/qibo/quantum_info/superoperator_transformations.py
@@ -1671,7 +1671,7 @@ def stinespring_to_choi(
     to its Choi representation :math:`\\Lambda`.
 
     .. math::
-        \\Lambda = \\text{kraus_to_choi}(\\text{stinespring_to_kraus}(U_{0}}))
+        \\Lambda = \\text{kraus_to_choi}(\\text{stinespring_to_kraus}(U_{0}))
 
     Args:
         stinespring (ndarray): quantum channel in the Stinespring representation.
diff --git a/src/qibo/quantum_info/utils.py b/src/qibo/quantum_info/utils.py
index 6ab993fba1..35a61a99d0 100644
--- a/src/qibo/quantum_info/utils.py
+++ b/src/qibo/quantum_info/utils.py
@@ -43,7 +43,7 @@ def hamming_weight(bitstring, return_indexes: bool = False):
 
 def hadamard_transform(array, implementation: str = "fast", backend=None):
     """Calculates the (fast) Hadamard Transform :math:`\\text{HT}` of a
-    :math:`2^{n}`-dimensional vector or :math:`2^{n} \times 2^{n}` matrix :math:`A`,
+    :math:`2^{n}`-dimensional vector or :math:`2^{n} \\times 2^{n}` matrix :math:`A`,
     where :math:`n` is the number of qubits in the system. If :math:`A` is a vector, then
 
     .. math::