# 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,

*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⟩*A**B*of the composite system can be represented in the form

∣Ψ⟩*A**B* = ∣*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