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diff --git a/source/know/concept/quantum-entanglement/index.md b/source/know/concept/quantum-entanglement/index.md
index 5ae5c92..cfb6721 100644
--- a/source/know/concept/quantum-entanglement/index.md
+++ b/source/know/concept/quantum-entanglement/index.md
@@ -9,19 +9,19 @@ categories:
 layout: "concept"
 ---
 
-Consider a composite quantum system which consists of two subsystems $A$ and $B$,
-respectively with basis states $\Ket{a_n}$ and $\Ket{b_n}$.
-All accessible states of the sytem $\Ket{\Psi}$ lie in
+Consider a composite quantum system which consists of two subsystems $$A$$ and $$B$$,
+respectively with basis states $$\Ket{a_n}$$ and $$\Ket{b_n}$$.
+All accessible states of the sytem $$\Ket{\Psi}$$ lie in
 the tensor product of the subsystems'
-[Hilbert spaces](/know/concept/hilbert-space/) $\mathbb{H}_A$ and $\mathbb{H}_B$:
+[Hilbert spaces](/know/concept/hilbert-space/) $$\mathbb{H}_A$$ and $$\mathbb{H}_B$$:
 
 $$\begin{aligned}
     \Ket{\Psi} \in \mathbb{H}_A \otimes \mathbb{H}_B
 \end{aligned}$$
 
 A subset of these states can be written as the tensor product (i.e. Kronecker product in a basis)
-of a state $\Ket{\alpha}$ in $A$ and a state $\Ket{\beta}$ in $B$,
-often abbreviated as $\Ket{\alpha} \Ket{\beta}$:
+of a state $$\Ket{\alpha}$$ in $$A$$ and a state $$\Ket{\beta}$$ in $$B$$,
+often abbreviated as $$\Ket{\alpha} \Ket{\beta}$$:
 
 $$\begin{aligned}
     \Ket{\Psi}
@@ -32,20 +32,20 @@ $$\begin{aligned}
 The states that can be written in this way are called **separable**,
 and states that cannot are called **entangled**.
 Therefore, we are dealing with **quantum entanglement**
-if the state of subsystem $A$ cannot be fully described
-independently of the state of subsystem $B$, and vice versa.
+if the state of subsystem $$A$$ cannot be fully described
+independently of the state of subsystem $$B$$, and vice versa.
 
 To detect and quantify entanglement,
-we can use the [density operator](/know/concept/density-operator/) $\hat{\rho}$.
-For a pure ensemble in a given (possibly entangled) state $\Ket{\Psi}$,
-$\hat{\rho}$ is given by:
+we can use the [density operator](/know/concept/density-operator/) $$\hat{\rho}$$.
+For a pure ensemble in a given (possibly entangled) state $$\Ket{\Psi}$$,
+$$\hat{\rho}$$ is given by:
 
 $$\begin{aligned}
     \hat{\rho} = \Ket{\Psi} \Bra{\Psi}
 \end{aligned}$$
 
-From this, we would like to extract the corresponding state of subsystem $A$.
-For that purpose, we define the **reduced density operator** $\hat{\rho}_A$ of subsystem $A$ as follows:
+From this, we would like to extract the corresponding state of subsystem $$A$$.
+For that purpose, we define the **reduced density operator** $$\hat{\rho}_A$$ of subsystem $$A$$ as follows:
 
 $$\begin{aligned}
     \boxed{
@@ -55,11 +55,11 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $\Tr_B(\hat{\rho})$ is called the **partial trace** of $\hat{\rho}$,
-which basically eliminates subsystem $B$ from $\hat{\rho}$.
-For a pure composite state $\Ket{\Psi}$,
-the resulting $\hat{\rho}_A$ describes a pure state in $A$ if $\Ket{\Psi}$ is separable,
-else, if $\Ket{\Psi}$ is entangled, it describes a mixed state in $A$.
+Where $$\Tr_B(\hat{\rho})$$ is called the **partial trace** of $$\hat{\rho}$$,
+which basically eliminates subsystem $$B$$ from $$\hat{\rho}$$.
+For a pure composite state $$\Ket{\Psi}$$,
+the resulting $$\hat{\rho}_A$$ describes a pure state in $$A$$ if $$\Ket{\Psi}$$ is separable,
+else, if $$\Ket{\Psi}$$ is entangled, it describes a mixed state in $$A$$.
 In the former case we simply find:
 
 $$\begin{aligned}
@@ -70,9 +70,9 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-We call $\Ket{\Psi}$ **maximally entangled**
+We call $$\Ket{\Psi}$$ **maximally entangled**
 if its reduced density operators are **maximally mixed**,
-where $N$ is the dimension of $\mathbb{H}_A$ and $\hat{I}$ is the identity matrix:
+where $$N$$ is the dimension of $$\mathbb{H}_A$$ and $$\hat{I}$$ is the identity matrix:
 
 $$\begin{aligned}
     \hat{\rho}_A
@@ -80,10 +80,10 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Suppose that we are given an entangled pure state
-$\Ket{\Psi} \neq \Ket{\alpha} \otimes \Ket{\beta}$.
-Then the partial traces $\hat{\rho}_A$ and $\hat{\rho}_B$
-of $\hat{\rho} = \Ket{\Psi} \Bra{\Psi}$ are mixed states with the same probabilities $p_n$
-(assuming $\mathbb{H}_A$ and $\mathbb{H}_B$ have the same dimensions,
+$$\Ket{\Psi} \neq \Ket{\alpha} \otimes \Ket{\beta}$$.
+Then the partial traces $$\hat{\rho}_A$$ and $$\hat{\rho}_B$$
+of $$\hat{\rho} = \Ket{\Psi} \Bra{\Psi}$$ are mixed states with the same probabilities $$p_n$$
+(assuming $$\mathbb{H}_A$$ and $$\mathbb{H}_B$$ have the same dimensions,
 which is usually the case):
 
 $$\begin{aligned}
@@ -97,9 +97,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 There exists an orthonormal choice
-of the subsystem basis states $\Ket{a_n}$ and $\Ket{b_n}$,
-such that $\Ket{\Psi}$ can be written as follows,
-where $p_n$ are the probabilities in the reduced density operators:
+of the subsystem basis states $$\Ket{a_n}$$ and $$\Ket{b_n}$$,
+such that $$\Ket{\Psi}$$ can be written as follows,
+where $$p_n$$ are the probabilities in the reduced density operators:
 
 $$\begin{aligned}
     \Ket{\Psi}
@@ -108,18 +108,18 @@ $$\begin{aligned}
 
 This is the **Schmidt decomposition**,
 and the **Schmidt number** is the number of nonzero terms in the summation,
-which can be used to determine if the state $\Ket{\Psi}$
+which can be used to determine if the state $$\Ket{\Psi}$$
 is entangled (greater than one) or separable (equal to one).
 
 By looking at the Schmidt decomposition, we can notice that,
-if $\hat{O}_A$ and $\hat{O}_B$ are the subsystem observables
-with basis eigenstates $\Ket{a_n}$ and $\Ket{b_n}$,
+if $$\hat{O}_A$$ and $$\hat{O}_B$$ are the subsystem observables
+with basis eigenstates $$\Ket{a_n}$$ and $$\Ket{b_n}$$,
 then measurement results of these operators
-will be perfectly correlated across $A$ and $B$.
+will be perfectly correlated across $$A$$ and $$B$$.
 This is a general property of entangled systems,
 but beware: correlation does not imply entanglement!
 
-But what if the composite system is in a mixed state $\hat{\rho}$?
+But what if the composite system is in a mixed state $$\hat{\rho}$$?
 The state is separable if and only if:
 
 $$\begin{aligned}
@@ -129,13 +129,13 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $p_m$ are probabilities,
-and $\hat{\rho}_A$ and $\hat{\rho}_B$ can be any subsystem states.
+Where $$p_m$$ are probabilities,
+and $$\hat{\rho}_A$$ and $$\hat{\rho}_B$$ can be any subsystem states.
 In reality, it is very hard to determine, using this criterium,
-whether an arbitrary given $\hat{\rho}$ is separable or not.
+whether an arbitrary given $$\hat{\rho}$$ is separable or not.
 
 As a final side note, the expectation value
-of an obervable $\hat{O}_A$ acting only on $A$ is given by:
+of an obervable $$\hat{O}_A$$ acting only on $$A$$ is given by:
 
 $$\begin{aligned}
     \expval{\hat{O}_A}