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.
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 = ∣ψ⟩A ⊗ ∣ϕ⟩B,
where ∣ψ⟩A ∈ HA and ∣ϕ⟩B ∈ 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.
- 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.