Entanglement Entropy of a Random Pure State

Pure State of Bipartite System
• |ψ >= ∑ i,α
x i,α |i A > ⊗|α B >
• If x i,α = a i b α then
|ψ >= ∑ i
a i |i A > ⊗ ∑ α
b α |α B >= |Φ A > ⊗|Φ B >
−→ Fully separable (factorised)
Otherwise −→ Entangled (non-factorisable)
• Density matrix of the composite system
ˆρ = |ψ >< ψ| with Tr[ˆρ] = 1
• Pure state: ˆρ ≠ ∑ k
p k |ψ k >< ψ k | → not a Mixed state
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State