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Separable and entangled states

An entangled state is defined as a state that is not separable. A separable state can be written as a probability distribution over uncorrelated states, product states,

$\rho = \sum_i p_i \rho_i^A \otimes \rho_i^B$ .

Pure states

For pure states the above definition can be represented as follows. Consider two quantum systems $A$ and $B$ , with respective Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$ . The Hilbert space of the composite system is the tensor product $\mathcal{H}_A\otimes\mathcal{H}_B$ . If the state $|\Psi\rangle_{AB}$ of the composite system can be represented in the form

$|\Psi\rangle_{AB}=|\psi\rangle_A \otimes |\phi\rangle_B$ ,

where $|\psi\rangle_A\in\mathcal{H}_A$ and $|\phi\rangle_B\in\mathcal{H}_B$ are the states of the systems $A$ and $B$ respectively, then this state is called a separable state. If a state is not separable, it is known as an entangled state.