Entanglement Entropy of a Random Pure State

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Entanglement N∑ √ • Schmidt decomposition: |ψ >= λi |λ A i > ⊗|λ B i > N∑ • Reduced density matrix: ˆρ A = Tr B [|ψ >< ψ|] = λ i |λ A i >< λ A i | i=1 • {λ 1 , λ 2 , . . . , λ N } → eigenvalues of W = XX † with N∑ 0 ≤ λ i ≤ 1 and λ i = 1 i=1 i=1 S.N. Majumdar Distribution of Bipartite Entanglement of a Random Pure State
Entanglement • Schmidt decomposition: |ψ >= N∑ √ λi |λ A i > ⊗|λ B i > i=1 • Reduced density matrix: ˆρ A = Tr B [|ψ >< ψ|] = • {λ 1 , λ 2 , . . . , λ N } → eigenvalues of W = XX † with N∑ 0 ≤ λ i ≤ 1 and λ i = 1 (i) Unentangled λ i0 = 1, λ j = 0 ∀j ≠ i 0 |ψ〉 = |λ A i 0 〉 ⊗ |λ B i 0 〉 is separable ˆρ A = |λ A i 0 〉〈λ A i 0 | is pure i=1 N∑ λ i |λ A i >< λ A i | i=1 (ii) Maximally entangled λ j = 1/N for all j (all eigenvalues equal) |ψ〉 is a superposition of all product states ˆρ A = 1 N ∑ N i=1 |λA i 〉〈λ A i | = 1 N 1 A is completely mixed S.N. Majumdar Distribution of Bipartite Entanglement of a Random Pure State
Page 1 and 2: Distribution of Bipartite Entanglem
Page 3 and 4: Plan Plan: • Bipartite entangleme
Page 5 and 6: Plan Plan: • Bipartite entangleme
Page 7 and 8: Coupled Bipartite System A B N−di
Page 9 and 10: Pure State and Reduced Density Matr
Page 11 and 12: Pure State of Bipartite System •
Page 17 and 18: Reduced Density Matrix of subsystem
Page 25 and 26: Eigenvalues of W: • In the diagon
Page 31: Entanglement N∑ √ • Schmidt d
Page 35 and 36: Entanglement entropy Subsystem A de
Page 37 and 38: Entanglement entropy Subsystem A de
Page 39 and 40: A Simple Diagram for N=3 (0,0,1) λ
Page 41 and 42: Random Pure State: Haar Measure |ψ
Page 43 and 44: Random Pure State: Haar Measure |ψ
Page 45 and 46: Joint PDF of Eigenvalues • the jo
Page 47 and 48: Joint PDF of Eigenvalues • the jo
Page 49 and 50: Random Pure State: Well Studied •
Page 51 and 52: Results: Entropy Distribution • S
Page 57 and 58: Results: pdf of Σ q = ∑ i λq i
Page 59 and 60: Computation of the pdf of Σ q : Co
Page 61 and 62: Charge Density and Effective Energy
Page 63 and 64: Saddle Point Solution ∫ ∞ ∫
Page 65 and 66: Saddle Point Solution ∫ ∞ ∫
Page 67 and 68: Steps for Computing the PDF of Σ q
Page 69 and 70: Steps for Computing the PDF of Σ q
Page 71 and 72: q = 2 : Purity • Regime I: 1 < s
Page 73 and 74: q = 2 : Purity • Regime I: 1 < s
Page 75 and 76: Numerical Simulations (2) Jump of t
Page 77 and 78: The limit q → 1 — von Neumann E
Page 79 and 80: The limit q → ∞ — Maximal eig
Page 81 and 82: The limit q → ∞ — Maximal eig
Page 83 and 84:
Summary and Conclusions • Exact d
Page 85 and 86:
Page 87 and 88:
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Entanglement Entropy of a Random Pure State

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