Entanglement Entropy of a Random Pure State

Distribution of Bipartite Entanglement of a
Random Pure State
Satya N. Majumdar
Laboratoire de Physique Théorique et Modèles Statistiques,CNRS,
Université Paris-Sud, France
February 14, 2012
Collaborators:
C. Nadal (Oxford University, UK)
M. Vergassola (Institut Pasteur, Paris, France)
Refs: Phys. Rev. Lett. 104, 110501 (2010)
J. Stat. Phys. 142, 403 (2011) (long version)
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Plan
Plan:
• Bipartite entanglement of a random pure state
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Plan
Plan:
• Bipartite entanglement of a random pure state
• Reduced density matrix −→ random matrix theory
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Plan
Plan:
• Bipartite entanglement of a random pure state
• Reduced density matrix −→ random matrix theory
• Distribution of the Renyi entropy
Results
Coulomb gas technique for large systems
Phase transitions
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Plan
Plan:
• Bipartite entanglement of a random pure state
• Reduced density matrix −→ random matrix theory
• Distribution of the Renyi entropy
Results
Coulomb gas technique for large systems
Phase transitions
• Summary and Conclusions
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Coupled Bipartite System
A
B
N−dim.
M−dim.
| i A
|α B
Coupled Bipartite System
N M
Bipartite quantum system A × B: Hilbert space H A ⊗H B
subsystem A: dimension N (the system to study)
subsystem B: dimension M ≥ N (“environment”)
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Coupled Bipartite System
A
B
N−dim.
M−dim.
| i A
|α B
Coupled Bipartite System
N M
Bipartite quantum system A × B: Hilbert space H A ⊗H B
subsystem A: dimension N (the system to study)
subsystem B: dimension M ≥ N (“environment”)
Large system: limit N ≫ 1 and M ≫ 1 with M ≈ N.
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Pure State and Reduced Density Matrix
A
B
N−dim.
M−dim.
| i A
|α B
Coupled Bipartite System
N M
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Pure State and Reduced Density Matrix
A
B
N−dim.
M−dim.
| i A
|α B
Coupled Bipartite System
N M
Any pure state of the full system:
|ψ >= ∑ i,α
x i,α |i A > ⊗|α B >
X = [x i,α ] → (N × M) rectangular Coupling matrix
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Pure State of Bipartite System
• |ψ >= ∑ i,α
x i,α |i A > ⊗|α B >
S.N. Majumdar
Distribution of Bipartite Entanglement 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 >
S.N. Majumdar
Distribution of Bipartite Entanglement 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)
S.N. Majumdar
Distribution of Bipartite Entanglement 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)
S.N. Majumdar
Distribution of Bipartite Entanglement 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)
S.N. Majumdar
Distribution of Bipartite Entanglement 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
S.N. Majumdar
Distribution of Bipartite Entanglement 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

Reduced Density Matrix of subsystem A:
A
B
N−dim.
M−dim.
| i A
|α B
Coupled Bipartite System
N M
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Reduced Density Matrix of subsystem A:
A
B
N−dim.
M−dim.
| i A
|α B
Coupled Bipartite System
N M
• Reduced Density Matrix: ˆρ A = Tr B [ˆρ] = Tr B [|ψ〉〈ψ|] with Tr[ˆρ A ] = 1
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Reduced Density Matrix of subsystem A:
A
B
N−dim.
M−dim.
| i A
|α B
Coupled Bipartite System
N M
• Reduced Density Matrix: ˆρ A = Tr B [ˆρ] = Tr B [|ψ〉〈ψ|] with Tr[ˆρ A ] = 1
Measurement → 〈ÔA〉 = Tr[Ô ˆρ A]
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Reduced Density Matrix of subsystem A:
A
B
N−dim.
M−dim.
| i A
|α B
Coupled Bipartite System
N M
• Reduced Density Matrix: ˆρ A = Tr B [ˆρ] = Tr B [|ψ〉〈ψ|] with Tr[ˆρ A ] = 1
Measurement → 〈ÔA〉 = Tr[Ô ˆρ A]
• ˆρ A = Tr B [ˆρ]
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Reduced Density Matrix of subsystem A:
A
B
N−dim.
M−dim.
| i A
|α B
Coupled Bipartite System
N M
• Reduced Density Matrix: ˆρ A = Tr B [ˆρ] = Tr B [|ψ〉〈ψ|] with Tr[ˆρ A ] = 1
Measurement → 〈ÔA〉 = Tr[Ô ˆρ A]
M∑
• ˆρ A = Tr B [ˆρ] = < α B |ˆρ|α B >
α=1
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Reduced Density Matrix of subsystem A:
A
B
N−dim.
M−dim.
| i A
|α B
Coupled Bipartite System
N M
• Reduced Density Matrix: ˆρ A = Tr B [ˆρ] = Tr B [|ψ〉〈ψ|] with Tr[ˆρ A ] = 1
Measurement → 〈ÔA〉 = Tr[Ô ˆρ A]
[
M∑
N∑ M
]
∑
• ˆρ A = Tr B [ˆρ] = < α B |ˆρ|α B > = x i,α xj,α
∗ |i A >< j A |
α=1
i,j=1
α=1
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Reduced Density Matrix of subsystem A:
A
B
N−dim.
M−dim.
| i A
|α B
Coupled Bipartite System
N M
• Reduced Density Matrix: ˆρ A = Tr B [ˆρ] = Tr B [|ψ〉〈ψ|] with Tr[ˆρ A ] = 1
Measurement → 〈ÔA〉 = Tr[Ô ˆρ A]
[
M∑
N∑ M
]
∑
• ˆρ A = Tr B [ˆρ] = < α B |ˆρ|α B > = x i,α xj,α
∗ |i A >< j A |
α=1
=
i,j=1
α=1
N∑
W i,j |i A >< j A |
i,j=1
where the N × N matrix: W = XX †
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Eigenvalues of W:
• In the diagonal representation
N∑
N∑
ˆρ A = W i,j |i A >< j A | → λ i |λ A i >< λ A i |
i,j=1
i=1
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Eigenvalues of W:
• In the diagonal representation
N∑
N∑
ˆρ A = W i,j |i A >< j A | → λ i |λ A i >< λ A i |
i,j=1
i=1
{λ 1 , λ 2 , . . . , λ N } → non-negative eigenvalues of W = XX †
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Eigenvalues of W:
• In the diagonal representation
N∑
N∑
ˆρ A = W i,j |i A >< j A | → λ i |λ A i >< λ A i |
i,j=1
i=1
{λ 1 , λ 2 , . . . , λ N } → non-negative eigenvalues of W = XX †
Tr[ˆρ A ] = 1 →
N∑
λ i = 1 ⇒ 0 ≤ λ i ≤ 1
i=1
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

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

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

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

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

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
i=1
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

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

Entanglement entropy
Subsystem A described by ˆρ A = ∑ N
i=1 λ i|λ A i 〉〈λ A i |
• Von Neuman entropy: S VN = −Tr [ˆρ A ln ˆρ A ] = − ∑ N
i=1 λ i ln λ i
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Entanglement entropy
Subsystem A described by ˆρ A = ∑ N
i=1 λ i|λ A i 〉〈λ A i |
• Von Neuman entropy: S VN = −Tr [ˆρ A ln ˆρ A ] = − ∑ N
i=1 λ i ln λ i
[
• Renyi entropy: S q = 1
1−q ln ∑N
]
i=1 λq i
and Σ q = ∑ N
i=1 λq i
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Entanglement entropy
Subsystem A described by ˆρ A = ∑ N
i=1 λ i|λ A i 〉〈λ A i |
• Von Neuman entropy: S VN = −Tr [ˆρ A ln ˆρ A ] = − ∑ N
i=1 λ i ln λ i
[
• Renyi entropy: S q = 1
1−q ln ∑N
]
i=1 λq i
and Σ q = ∑ N
i=1 λq i
• S q→1 = S VN and S q→∞ = − ln(λ max )
• Σ 2 = exp[−S q=2 ] −→ Purity
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Entanglement entropy
Subsystem A described by ˆρ A = ∑ N
i=1 λ i|λ A i 〉〈λ A i |
• Von Neuman entropy: S VN = −Tr [ˆρ A ln ˆρ A ] = − ∑ N
i=1 λ i ln λ i
[
• Renyi entropy: S q = 1
1−q ln ∑N
]
i=1 λq i
and Σ q = ∑ N
i=1 λq i
• S q→1 = S VN and S q→∞ = − ln(λ max )
• Σ 2 = exp[−S q=2 ] −→ Purity
(i) Unentangled
λ i0 = 1, λ j = 0 ∀j ≠ i 0
ˆρ A = |λ A i 0
〉〈λ A i 0
| is pure
S VN = 0 = S q is minimal
S.N. Majumdar
(ii) Maximally entangled
λ j = 1/N for all j
ˆρ A = 1 N
∑ N
i=1 |λA i 〉〈λ A i | mixed
S VN = ln N = S q is maximal
Distribution of Bipartite Entanglement of a Random Pure State

A Simple Diagram for N=3
(0,0,1)
λ 3
(1/3,1/3,1/3)
MOST
ENTANGLED
S=log(N)
λ 2
(0,1,0)
λ (1,0,0) 1
LEAST ENTANGLED
S=0
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

A Simple Diagram for N=3
(0,0,1)
λ 3
(1/3,1/3,1/3)
MOST
ENTANGLED
S=log(N)
λ 2
(0,1,0)
λ (1,0,0) 1
LEAST ENTANGLED
S=0
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Random Pure State: Haar Measure
|ψ >= ∑ x i,α |i A > ⊗|α B > =
i,α
N∑ √
λi |λ A i > ⊗|λ B i >
i=1
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Random Pure State: Haar Measure
|ψ >= ∑ i,α
x i,α |i A > ⊗|α B > =
N∑ √
λi |λ A i > ⊗|λ B i >
• random pure state −→ where X = [x i,α ] are uniformly distributed
among the sets of {x i,α } satisfying ∑ i,α |x i,α| 2 = Trρ = 1
i=1
−→ Haar measure
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Random Pure State: Haar Measure
|ψ >= ∑ i,α
x i,α |i A > ⊗|α B > =
N∑ √
λi |λ A i > ⊗|λ B i >
• random pure state −→ where X = [x i,α ] are uniformly distributed
among the sets of {x i,α } satisfying ∑ i,α |x i,α| 2 = Trρ = 1
i=1
−→ Haar measure
• Equivalently, for the N × M matrix
P(X ) ∝ δ ( Tr(XX † ) − 1 )
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Random Pure State: Haar Measure
|ψ >= ∑ i,α
x i,α |i A > ⊗|α B > =
N∑ √
λi |λ A i > ⊗|λ B i >
• random pure state −→ where X = [x i,α ] are uniformly distributed
among the sets of {x i,α } satisfying ∑ i,α |x i,α| 2 = Trρ = 1
i=1
−→ Haar measure
• Equivalently, for the N × M matrix
P(X ) ∝ δ ( Tr(XX † ) − 1 )
• In the basis |i A 〉 of H A : ˆρ A = W = XX †
−→ Distribution of the eigenvalues λ i of ˆρ A ?
−→ Distribution of the entanglement entropies S VN , S q ?
S.N. Majumdar
Distribution of Bipartite Entanglement 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
S.N. Majumdar
Distribution of Bipartite Entanglement 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 ∏
i
|λ j − λ k | β
2
i=1 i
j

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

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

Random Pure State: Well Studied
• Average von Neumann entropy: 〈S VN 〉 → ln N − N
2M for M ∼ N ≫ 1
−→ Near Maximal
[Lubkin (’78), Page (’93), Foong and Kanno (’94), Sánchez-Ruiz (’95),
Sen (’96)]
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Random Pure State: Well Studied
• Average von Neumann entropy: 〈S VN 〉 → ln N − N
2M for M ∼ N ≫ 1
−→ Near Maximal
[Lubkin (’78), Page (’93), Foong and Kanno (’94), Sánchez-Ruiz (’95),
Sen (’96)]
• statistics of observables such as average density of eigenvalues,
concurrence, purity, minimum eigenvalue λ min etc.
[Lubkin (’78), Zyczkowski & Sommers (2001, 2004), Cappellini,
Sommers and Zyczkowski (2006), Giraud (2007), Znidaric (2007), S.M.,
Bohigas and Lakshminarayan (2008), Kubotini, Adachi and Toda (2008),
Chen, Liu and Zhou (2010), Vivo (2010), ...]
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Random Pure State: Well Studied
• Average von Neumann entropy: 〈S VN 〉 → ln N − N
2M for M ∼ N ≫ 1
−→ Near Maximal
[Lubkin (’78), Page (’93), Foong and Kanno (’94), Sánchez-Ruiz (’95),
Sen (’96)]
• statistics of observables such as average density of eigenvalues,
concurrence, purity, minimum eigenvalue λ min etc.
[Lubkin (’78), Zyczkowski & Sommers (2001, 2004), Cappellini,
Sommers and Zyczkowski (2006), Giraud (2007), Znidaric (2007), S.M.,
Bohigas and Lakshminarayan (2008), Kubotini, Adachi and Toda (2008),
Chen, Liu and Zhou (2010), Vivo (2010), ...]
• Laplace transform of the purity (q = 2) distribution for large N [Facchi
et. al. (2008, 2010)]
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Results: Entropy Distribution
• Scaling for large N: λ typ ∼ 1 N as ∑ N
i=1 λ i = 1
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Results: Entropy Distribution
• Scaling for large N: λ typ ∼ 1 N as ∑ N
i=1 λ i = 1
Renyi entropy:
S q = 1
1−q ln [Σ q] with Σ q = ∑ N
i=1 λq i
, q > 1
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Results: Entropy Distribution
• Scaling for large N: λ typ ∼ 1 N as ∑ N
i=1 λ i = 1
Renyi entropy:
S q = 1
1−q ln [Σ q] with Σ q = ∑ N
i=1 λq i
, q > 1
Σ q = N 1−q s for large N −→ S q = ln N − ln s
q−1
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Results: Entropy Distribution
• Scaling for large N: λ typ ∼ 1 N as ∑ N
i=1 λ i = 1
Renyi entropy:
S q = 1
1−q ln [Σ q] with Σ q = ∑ N
i=1 λq i
, q > 1
Σ q = N 1−q s for large N −→ S q = ln N − ln s
q−1
(i) Unentangled
S q = 0 is minimal
Σ q = 1 (or s → ∞)
(ii) Maximally entangled
S q = ln N is maximal
Σ q = N 1−q (or s = 1)
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Results: Entropy Distribution
• Scaling for large N: λ typ ∼ 1 N as ∑ N
i=1 λ i = 1
Renyi entropy:
S q = 1
1−q ln [Σ q] with Σ q = ∑ N
i=1 λq i
, q > 1
Σ q = N 1−q s for large N −→ S q = ln N − ln s
q−1
(i) Unentangled
S q = 0 is minimal
Σ q = 1 (or s → ∞)
(ii) Maximally entangled
S q = ln N is maximal
Σ q = N 1−q (or s = 1)
• Average entropy for large N:
〈Σ q 〉 ≈ N 1−q ¯s(q) → 〈S q 〉 ≈ ln N −
ln ¯s(q)
q−1
where ¯s(q) = Γ(q+1/2) √ πΓ(q+2)
4 q
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Results: Entropy Distribution
• Scaling for large N: λ typ ∼ 1 N as ∑ N
i=1 λ i = 1
Renyi entropy:
S q = 1
1−q ln [Σ q] with Σ q = ∑ N
i=1 λq i
, q > 1
Σ q = N 1−q s for large N −→ S q = ln N − ln s
q−1
(i) Unentangled
S q = 0 is minimal
Σ q = 1 (or s → ∞)
(ii) Maximally entangled
S q = ln N is maximal
Σ q = N 1−q (or s = 1)
• Average entropy for large N:
〈Σ q 〉 ≈ N 1−q ¯s(q) → 〈S q 〉 ≈ ln N −
ln ¯s(q)
q−1
where ¯s(q) = Γ(q+1/2) √ πΓ(q+2)
4 q
For q = 2, 〈Σ 2 〉 ≈ 2 N (purity); For q → 1, 〈S VN〉 ≈ ln N − 1 2
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Results: pdf of Σ q = ∑ i λq i
⎧
P ( Σ q = N 1−q s ) ⎪⎨
≈
⎪⎩
exp { −βN 2 Φ I (s) }
exp { −βN 2 Φ II (s) }
{
exp −βN 1+ 1 q
}
ΨIII (s)
for 1 ≤ s < s 1 (q)
for s 1 (q) < s < s 2 (q)
for s > s 2 (q)
P ( Σ q = N 1−q s )
¯s(q)
(a)
− ln [ P ( Σ q = N 1−q s )]
(b)
I II III
I II III
1 s 1 (q) s 2 (q)
max. entangled ←− unentangled
s
1s 1 (q)
¯s(q)
s 2 (q)
s
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Computation of the pdf of Σ q : Coulomb gas method
• PDF of Σ q = ∑ i λq i
∫
( ) ( )
∑ ∏
P(Σ q , N) = P(λ 1 , ..., λ N ) δ λ q i
− Σ q dλ i .
i
i
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Computation of the pdf of Σ q : Coulomb gas method
• PDF of Σ q = ∑ i λq i
∫
( ) ( )
∑ ∏
P(Σ q , N) = P(λ 1 , ..., λ N ) δ λ q i
− Σ q dλ i .
i
i
• The joint pdf of the eigenvalues can be interpreted as a Boltzmann
weight at inverse temperature β:
P(λ 1 , ..., λ N ) ∝ δ ( ∑
λ i − 1 )
i
N ∏
i=1
∝ exp {−βE [{λ i }]}
λ β 2 (M−N+1)−1
i
∏
|λ i − λ j | β
i

Charge Density and Effective Energy
E{λ i } = −γ
N∑
ln λ i − ∑ ln |λ i − λ j | with ∑ λ i = 1
i

Charge Density and Effective Energy
E{λ i } = −γ
N∑
ln λ i − ∑ ln |λ i − λ j | with ∑ λ i = 1
i

Charge Density and Effective Energy
E{λ i } = −γ
N∑
ln λ i − ∑ ln |λ i − λ j | with ∑
i

Saddle Point Solution
∫ ∞ ∫ ∞
E s [ρ]=− 1 (∫ ∞
dxdx ′ ρ(x)ρ(x ′ ) ln |x − x ′ | + µ 0
2 0 0
(∫ ∞
) (∫ ∞
+µ 1 dx x ρ(x) − 1 + µ 2
0
0
)
dx ρ(x) − 1
0
)
dx x q ρ(x) − s
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Saddle Point Solution
∫ ∞ ∫ ∞
E s [ρ]=− 1 (∫ ∞
dxdx ′ ρ(x)ρ(x ′ ) ln |x − x ′ | + µ 0
2 0 0
(∫ ∞
) (∫ ∞
+µ 1 dx x ρ(x) − 1 + µ 2
0
0
)
dx ρ(x) − 1
0
)
dx x q ρ(x) − s
• Saddle point method for large N:
P ( Σ q = N 1−q s, N ) ∫
∝ D [ρ] exp { −βN 2 E s [ρ] } ∝ exp { −βN 2 E s [ρ c ] }
where ρ c minimizes the energy:
δE s
δρ
∣ = 0
ρ=ρc
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Saddle Point Solution
∫ ∞ ∫ ∞
E s [ρ]=− 1 (∫ ∞
dxdx ′ ρ(x)ρ(x ′ ) ln |x − x ′ | + µ 0
2 0 0
(∫ ∞
) (∫ ∞
+µ 1 dx x ρ(x) − 1 + µ 2
0
0
)
dx ρ(x) − 1
0
)
dx x q ρ(x) − s
• Saddle point method for large N:
P ( Σ q = N 1−q s, N ) ∫
∝ D [ρ] exp { −βN 2 E s [ρ] } ∝ exp { −βN 2 E s [ρ c ] }
where ρ c minimizes the energy:
∫ ∞
0
δE s
δρ
∣ = 0 giving
ρ=ρc
dx ′ ρ c (x ′ ) ln |x − x ′ | = µ 0 + µ 1 x + µ 2 x q ≡ V (x)
V (x) → effective potential for the charges.
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Saddle Point Solution
∫ ∞ ∫ ∞
E s [ρ]=− 1 (∫ ∞
dxdx ′ ρ(x)ρ(x ′ ) ln |x − x ′ | + µ 0
2 0 0
(∫ ∞
) (∫ ∞
+µ 1 dx x ρ(x) − 1 + µ 2
0
0
)
dx ρ(x) − 1
0
)
dx x q ρ(x) − s
• Saddle point method for large N:
P ( Σ q = N 1−q s, N ) ∫
∝ D [ρ] exp { −βN 2 E s [ρ] } ∝ exp { −βN 2 E s [ρ c ] }
where ρ c minimizes the energy:
∫ ∞
0
δE s
δρ
∣ = 0 giving
ρ=ρc
dx ′ ρ c (x ′ ) ln |x − x ′ | = µ 0 + µ 1 x + µ 2 x q ≡ V (x)
V (x) → effective potential for the charges.
• Taking derivative with respect to x gives
P
∫ ∞
0
dx ′ ρ c(x ′ )
x − x ′ = µ 1 + q µ 2 x q−1 = V ′ (x)
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Steps for Computing the PDF of Σ q
(i) find the solution ρ c (x) of the integral equation
P
∫ ∞
0
dx ′ ρ c(x ′ )
x − x ′ = µ 1 + q µ 2 x q−1 = V ′ (x)
Tricomi’s solution: density ρ c (x) with finite support [L 1 , L 2 ]:
[ ∫
1
L2
√ √ ]
ρ c (x) =
π √ dy y − L1 L2 − y
√ C − P
V ′ (y) ,
x − L 1 L2 − x
π x − y
where C = ∫ L 2
L 1
dx ρ c (x) is a constant
L 1
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Steps for Computing the PDF of Σ q
(i) find the solution ρ c (x) of the integral equation
P
∫ ∞
0
dx ′ ρ c(x ′ )
x − x ′ = µ 1 + q µ 2 x q−1 = V ′ (x)
Tricomi’s solution: density ρ c (x) with finite support [L 1 , L 2 ]:
[ ∫
1
L2
√ √ ]
ρ c (x) =
π √ dy y − L1 L2 − y
√ C − P
V ′ (y) ,
x − L 1 L2 − x
π x − y
where C = ∫ L 2
L 1
dx ρ c (x) is a constant
(ii) for a given s, the unknown Lagrange multipliers µ 0 , µ 1 and µ 2 are
fixed by the three conditions:
∫ ∞
0
ρ c (x) dx = 1 ,
∫ ∞
0
L 1
x ρ c (x) dx = 1 and
∫ ∞
0
x q ρ c (x) dx = s
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Steps for Computing the PDF of Σ q
(i) find the solution ρ c (x) of the integral equation
P
∫ ∞
0
dx ′ ρ c(x ′ )
x − x ′ = µ 1 + q µ 2 x q−1 = V ′ (x)
Tricomi’s solution: density ρ c (x) with finite support [L 1 , L 2 ]:
[ ∫
1
L2
√ √ ]
ρ c (x) =
π √ dy y − L1 L2 − y
√ C − P
V ′ (y) ,
x − L 1 L2 − x
π x − y
where C = ∫ L 2
L 1
dx ρ c (x) is a constant
(ii) for a given s, the unknown Lagrange multipliers µ 0 , µ 1 and µ 2 are
fixed by the three conditions:
∫ ∞
0
ρ c (x) dx = 1 ,
∫ ∞
0
L 1
x ρ c (x) dx = 1 and
(iii) evaluate the saddle point energy E s [ρ c ]
∫ ∞
0
x q ρ c (x) dx = s
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Phase Transitions
Regime I:
1 < s < s 1
(s 1 (q = 2) = 5/4 )
Regime II:
s 1 < s < s 2
(s 2 (q = 2) = 2 + 24/3
N 1/3 )
V (x) V (x) V (x)
Regime III:
s > s 2
(weakly entangled)
I II III
0
x ∗ x 0 x 0
x
(a) (b) (c)
ρ c (λ, N) ρ c (λ, N) ρ c (λ, N)
I II III
L 1 /N L 2 /N
λ
λ
λ
0 L/N 0 ζ t
S.N. Majumdar Distribution of Bipartite Entanglement of a Random Pure State

q = 2 : Purity
• Regime I: 1 < s < 5/4 √
L2 − x √ x − L 1
ρ c (x) =
,
2π (s − 1)
P
(Σ 2 = s )
N , N ∝ e −βN2 Φ I (s) , Φ I (s) = − 1 4 ln (s − 1) − 1 8
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

q = 2 : Purity
• Regime I: 1 < s < 5/4 √
L2 − x √ x − L 1
ρ c (x) =
,
2π (s − 1)
P
(Σ 2 = s )
N , N ∝ e −βN2 Φ I (s) , Φ I (s) = − 1 4 ln (s − 1) − 1 8
• Regime II: 5/4 < s < 2 + 24/3
N 1/3
ρ c (x) = 1 √
L − x
(
(A + B x) with L = 2 3 − √ )
9 − 4s
π x
P
(Σ 2 = s )
N , N ∝ e −βN2 Φ II (s) , Φ II (s) = − 1 ( ) L
2 ln + 6 4 L 2 − 5 L + 7 8
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

q = 2 : Purity
• Regime I: 1 < s < 5/4 √
L2 − x √ x − L 1
ρ c (x) =
,
2π (s − 1)
P
(Σ 2 = s )
N , N ∝ e −βN2 Φ I (s) , Φ I (s) = − 1 4 ln (s − 1) − 1 8
• Regime II: 5/4 < s < 2 + 24/3
N 1/3
ρ c (x) = 1 √
L − x
(
(A + B x) with L = 2 3 − √ )
9 − 4s
π x
P
(Σ 2 = s )
N , N ∝ e −βN2 Φ II (s) , Φ II (s) = − 1 ( ) L
2 ln + 6 4 L 2 − 5 L + 7 8
• Regime III: s > 2 + 24/3 Continuous density with support [0, ζ] with
N 1/3
ζ ≈ 4 N and separated λ max ≫ ζ: λ max = t ≈ √ √
s−2
N
P
(Σ 2 = s )
N , N ≈ e −βN 3 2 Ψ III (s)
, Ψ III (s) =
√ s − 2
2
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Numerical simulations
Monte Carlo Simulations (non-standard Metropolis algorithm):
pdf of Σ 2 (purity) for N = 1000 (second phase transition)
− ln P (Σ 2 = s N )
0.008
βN 2
0.006
0.004
Φ III = Ψ III √
N
0.002
Φ II
0.000
2.0 2.1 2.2 2.3 2.4 2.5
¯s(2) s 2 (2)
S.N. Majumdar Distribution of Bipartite Entanglement of a Random Pure State
s

Numerical Simulations (2)
Jump of the maximal eigenvalue (rightmost charge) at s = s 2 for q = 2
and different values of N
N t = N λ max
for Σ 2 = s N
20
15
N = 1000
N = 500
10
5
N = 50
0
2.0 2.1 2.2 2.3 2.4 2.5
¯s s 2
(N = 1000)
s2
(N = 500)
S.N. Majumdar
s
Distribution of Bipartite Entanglement of a Random Pure State

The limit q → 1 — von Neumann Entropy
S VN = S q→1 = − ∑ N
i=1 λ i ln(λ i ) = ln(N) − z
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

The limit q → 1 — von Neumann Entropy
00 11
00 11
00 11
z
01 01
01
01
S VN = S q→1 = − ∑ N
i=1 λ i ln(λ i ) = ln(N) − z
⎧
exp { −βN 2 φ I (z) } for 0 ≤ z < z 1 = 2 3 − ln ( 3
2)
= 0.26.
⎪⎨
(S VN = ln(N) − z) ≈ exp { −βN 2 φ II (z) } for 0.26.. < z < z 2 = 1/2
{
}
⎪⎩ exp −β N2
ln(N) φ III (z) for z > z 2 = 1/2
P(S VN
= ln(N)−z)
z=0
II
I
maximally entangled
z 1
=0.26..
III
2
= 0.5
z
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

The limit q → ∞ — Maximal eigenvalue
• S q→∞ = − ln (λ max )
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

The limit q → ∞ — Maximal eigenvalue
• S q→∞ = − ln (λ max )
(
P λ max = t )
≈
N
⎧
⎪⎨
⎪⎩
exp { −βN 2 χ I (t) } for 1 ≤ t < t 1 = 4 3
exp { −βN 2 χ II (t) } for 4 3 < t < t 2 = 4
exp {−βN χ III (t)} for t > t 2 = 4
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

The limit q → ∞ — Maximal eigenvalue
• S q→∞ = − ln (λ max )
(
P λ max = t )
≈
N
⎧
⎪⎨
⎪⎩
exp { −βN 2 χ I (t) } for 1 ≤ t < t 1 = 4 3
exp { −βN 2 χ II (t) } for 4 3 < t < t 2 = 4
exp {−βN χ III (t)} for t > t 2 = 4
• Around its mean, 〈λ max 〉 = 4/N, the typical fluctuations
(∼ O(N −5/3 )) are described by the Tracy-Widom distribution
λ max = 4 N + 24/3 N −5/3 Y β
Y β → random variable distributed via Tracy-Widom law
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

The limit q → ∞ — Maximal eigenvalue
• S q→∞ = − ln (λ max )
(
P λ max = t )
≈
N
⎧
⎪⎨
⎪⎩
exp { −βN 2 χ I (t) } for 1 ≤ t < t 1 = 4 3
exp { −βN 2 χ II (t) } for 4 3 < t < t 2 = 4
exp {−βN χ III (t)} for t > t 2 = 4
• Around its mean, 〈λ max 〉 = 4/N, the typical fluctuations
(∼ O(N −5/3 )) are described by the Tracy-Widom distribution
λ max = 4 N + 24/3 N −5/3 Y β
Y β → random variable distributed via Tracy-Widom law
• For finite N and M, see P. Vivo (2011)
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Summary and Conclusions
• Exact distribution of the Renyi entanglement entropy S q for a
random pure state in a large bipartite quantum system (M ∼ N ≫ 1)
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Summary and Conclusions
• Exact distribution of the Renyi entanglement entropy S q for a
random pure state in a large bipartite quantum system (M ∼ N ≫ 1)
• Coulomb gas method → saddle point solution
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Summary and Conclusions
• Exact distribution of the Renyi entanglement entropy S q for a
random pure state in a large bipartite quantum system (M ∼ N ≫ 1)
• Coulomb gas method → saddle point solution
• 2 critical points → direct consequence of 2 phase transitions in the
associated Coulomb gas
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Summary and Conclusions
• Exact distribution of the Renyi entanglement entropy S q for a
random pure state in a large bipartite quantum system (M ∼ N ≫ 1)
• Coulomb gas method → saddle point solution
• 2 critical points → direct consequence of 2 phase transitions in the
associated Coulomb gas
• at the second critical point: λ max suddenly jumps to a value
∼ N −(1−1/q) much larger than the other λ i ∼ 1/N
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Summary and Conclusions
• Exact distribution of the Renyi entanglement entropy S q for a
random pure state in a large bipartite quantum system (M ∼ N ≫ 1)
• Coulomb gas method → saddle point solution
• 2 critical points → direct consequence of 2 phase transitions in the
associated Coulomb gas
• at the second critical point: λ max suddenly jumps to a value
∼ N −(1−1/q) much larger than the other λ i ∼ 1/N
• While the average entropy of a random pure state is indeed close to its
maximal value ln N, the probability of occurrence of the maximally
entangled state (all λ i = 1/N) is actually very small.
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

Summary and Conclusions
• Exact distribution of the Renyi entanglement entropy S q for a
random pure state in a large bipartite quantum system (M ∼ N ≫ 1)
• Coulomb gas method → saddle point solution
• 2 critical points → direct consequence of 2 phase transitions in the
associated Coulomb gas
• at the second critical point: λ max suddenly jumps to a value
∼ N −(1−1/q) much larger than the other λ i ∼ 1/N
• While the average entropy of a random pure state is indeed close to its
maximal value ln N, the probability of occurrence of the maximally
entangled state (all λ i = 1/N) is actually very small.
Work in progress (C. Nadal, S.M with C. Pineda and T. Seligman)
Distribution of entropy for N finite but M ≫ N (large environment)?
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State

References
On the large N distribution of the Renyi entropy:
• C. Nadal, S.M. and M. Vergassola, Phys. Rev. Lett. 104, 110501
(2010)
• long version: J. Stat. Phys. 142, 403 (2011)
On the distribution of the minumum Schmidt number λ min :
• S.M., O. Bohigas and A. Lakshminarayan, J. Stat. Phys. 131, 33
(2008)
• see also S.M. in “ Handbook of Random Matrix Theory” (ed. by G.
Akemann, J. Baik and P. Di Francesco and forwarded by F.J. Dyson)
(Oxford Univ. Press, 2011) (arXiv:1005.4515)
S.N. Majumdar
Distribution of Bipartite Entanglement of a Random Pure State