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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} |