Entanglement Entropy of a Random Pure State

Joint PDF of Eigenvalues
• the joint pdf P(λ 1 , λ 2 , . . . , λ N )
where {λ 1 , λ 2 , . . . , λ N } → eigenvalues of the Wishart matrix
W = XX † (X → N × M Gaussian random matrix)
with an additional constraint
N∑
λ i = 1
i=1
• Joint distribution of Wishart eigenvalues (James ’64):
[ ] (
P({λ i }) ∝ exp − β N∑ ∏
λ i λ β 2 (1+M−N)−1 ∏
N
)
∑
i
|λ j − λ k | β ×δ λ i − 1
2
i=1 i
i=1
j