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,

*ρ* = ∑_{i}*p*_{i}*ρ*_{i}^{A} ⊗ *ρ*_{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 H_{A} and H_{B}. The Hilbert space of the composite system is the tensor product H_{A} ⊗ H_{B}. If the state ∣Ψ⟩_{AB}of the composite system can be represented in the form

∣Ψ⟩_{AB} = ∣*ψ*⟩_{A} ⊗ ∣*ϕ*⟩_{B}

,

where ∣*ψ*⟩_{A} ∈ H_{A} and ∣*ϕ*⟩_{B} ∈ 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*.

### Ralated papers

- R. F. Werner,
*Quantum states with Einstein-Rosen-Podolsky correlations admitting a hidden-variable model*, Phys. Rev. A 40, pp. 4277-4281, 1989. - D. Bruss,
*Characterizing Entanglement*, Math. Phys. 43, pp. 4237, 2002.

Category:Entanglement Category:Handbook of Quantum Information