[NOTES/QCQI-03001] Entanglement

Definition 1 State of a physical system is called pure state if it is represented by a vector in Hilbert space of all states.

Definition 2 If it is given that the probability of system being in a set of states
$\{\ket{\psi_k}\}$ are $\{p_k|\sum_{p>0}p_k=1\}$. We say that such a state is a mixed state. A mixed state is described by density matrix convex linear combination, \begin{equation}\label{eq01} \rho = \sum_{k=1}^n p_k \ket{\psi_k}\bra{\psi_k} \end{equation} of projection operators $\ket{\psi_k}\bra{\psi_k}$. For a pure state a particular state $\ket{\psi}$ has $p=1$ and the corresponding density matrix is $\rho=\ket{\psi}\bra{\psi}$.

Definition 3 A system consisting to two subsystems $A,B$ is described by a vector in the tensor product of Hilbert spaces $\mathcal{H}_A\otimes \mathcal{H}_B$. In general a state of composite system has the form \begin{equation}\label{eq02} \ket{\Psi}= \sum_{k=1}^n C_K \ket{\psi_{AK}} \end{equation} A state is separable if the state vector has only one term in (\ref{eq02}) i.e. $\ket{\Psi}$ can be written as a direct product \begin{equation}\label{eq03} \ket{\Psi} = \ket{\psi_A} \otimes \ket{\psi_B} \end{equation} for some $\ket{\psi_A}$ and $\ket{\psi_B}$ in Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$.

Definition 4 We say that the system is in an entangled state if the state vector cannot be written as product (\ref{eq03}). i.e. the state vector is not separable.

Bell States:} For a two level system, Bell states are given by \begin{eqnarray*} \ket{\Psi_\pm} &=& {1\over\sqrt{2}}(\ket{0,0}\pm\ket{1,1})\\ \ket{\Phi_\pm} &=& {1\over\sqrt{2}}(\ket{0,1}\pm\ket{1,0}) \end{eqnarray*} These states are entangled states and play an important role in quantum information theory.

Definition 5  We will now define a mixed entangled state: If the density operator $\rho$ of a composite system can be written as $$ \rho = \sum p_i \rho^i_A\otimes \rho^i_B $$ on Hilbert space $\mathcal{H}_A\otimes\mathcal{H}_B$, we say that the mixed state is separable; otherwise the system is in non-separable, or mixed entangled state. }