# 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 = ∑ipiρiA ⊗ ρiB.

### Pure states

For pure states the above definition can be represented as follows. Consider two quantum systems A and B, with respective Hilbert spaces HA and HB. The Hilbert space of the composite system is the tensor product HA ⊗ HB. If the state ∣Ψ⟩ABof the composite system can be represented in the form

∣Ψ⟩AB = ∣ψA ⊗ ∣ϕB,

where ∣ψA ∈ HA and ∣ϕB ∈ HB 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.