Entanglement Entropy of a Random Pure State
Entanglement Entropy of a Random Pure State
Entanglement Entropy of a Random Pure State
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Eigenvalues <strong>of</strong> W:<br />
• In the diagonal representation<br />
N∑<br />
N∑<br />
ˆρ A = W i,j |i A >< j A | → λ i |λ A i >< λ A i |<br />
i,j=1<br />
i=1<br />
{λ 1 , λ 2 , . . . , λ N } → non-negative eigenvalues <strong>of</strong> W = XX †<br />
Tr[ˆρ A ] = 1 →<br />
N∑<br />
λ i = 1 ⇒ 0 ≤ λ i ≤ 1<br />
i=1<br />
M∑<br />
• Similarly ˆρ B = Tr A [ˆρ] = W α,β ′ |α B >< β B |<br />
α,β=1<br />
W ′ = X † X → (M × M) matrix with M eigenvalues (recall N ≤ M)<br />
S.N. Majumdar<br />
Distribution <strong>of</strong> Bipartite <strong>Entanglement</strong> <strong>of</strong> a <strong>Random</strong> <strong>Pure</strong> <strong>State</strong>