Entanglement Spectrum of Large Quantum Systems:
von Neumann entropy and Phase Transitions
Giuseppe Florio
Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”
&
Dipartimento di Fisica , Università degli Studi di Bari & INFN (Italy)
In collaboration with
P. Facchi (UniBari), F. D. Cunden (UniBari), G. Parisi (Rome, “Sapienza”),
S. Pascazio (UniBari), K.Yuasa (Tokyo, Waseda University)
“Problemi attuali di Fisica Teorica 2014: Meccanica Quantistica e Applicazioni”
(Vietri sul Mare, April 11-13, 2014)
1
Objectives and Outline
• von Neumann Entropy and bipartite entanglement
•Study of the distribution of the eigenvalues of reduced density matrices,
conditioned to the value of von Neumann Entropy
• Methods: Coulomb gas and Saddle point equations
• Result: phase transitions (sudden change of the eigenvalues distribution)
• Alternative: polarized ensembles
2
Bipartite quantum system A + B
B
Pure state |ψ�
∈H
H = HA ⊗ HB
A
dim HA = dim HB = N
dim H = N 2
Reduced density matrix of subsystem A:
von Neumann Entropy of
�A
SvN (�λ) = −trA (�A ln �A ) = −
�λ = (λ1 , . . . , λN ) ∈ ∆N −1
N
�
�A = trB |ψ��ψ|
λk ln λk
k=1
simplex of eigenvalues
λk ≥ 0,
�
λk = 1
k
A measure of the amount of bipartite entanglement between A and B
3
State of the global system
|ψ� =
N �
�
i=1

λ1
0

�A =  .
 ..
0
0
λ2
..
.
0
Schmidt
decomposition
λi |ui � ⊗ |vi �
···
···
..
.
0
0
..
.
···
λN





SvN (�λ) = −trA (�A ln �A ) = −
N
�
k=1
4
�A = trB |ψ��ψ|
Partial density matrix
λk ln λk
0 ≤ SvN ≤ ln N
Global system in a product state: no entanglement
|ψ� = |uA � ⊗ |vB �
Party A in a pure state

1
0

�A =  .
 ..

0
0
0
..
.
0
···
···
..
.
···
SvN = 0
5
0
0

.. 
.
0
Global system in a maximally bipartite entangled state
N
�
1
|ui � ⊗ |vi � Party A in a totally mixed state
|ψ� = √
N i=1


1 0 ··· 0


0
1
·
·
·
0
1 

�A =
 .. ..
..
.. 
N . .
.
.
0 0 ··· 1
SvN = ln N
6
Random states
What is a random state |ψ� ?
Uniform measure on the projective Hilbert space:
States distributed according to the Haar measure of the
unitary group U(H)
|ψ� = V |ψ0 �
V ∈ U (H) = U (HA ⊗ HB )
Induced measure on the reduced
density matrices of subsystem A
dµ(�A )
�A = trB |ψ��ψ|
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|ψ�
V
|ψ0 �
Average von Neumann Entropy
SvN
1
= ln N −
2
Page, PRL 71, 1291 (1993)
Small fluctuations for large N. The typical spectral
distribution follows a Marchenko-Pastur Law
∝
�
4−N λ
Nλ
Concentration phenomenon
Hayden, Leung, Winter,
Commun. Math. Phys. 265, 95 (2006)
0
4�N
8
Concentration of Measure
•
The uniform measure on the N-dimensional sphere concentrates
very strongly about any equator as N gets large (any polar cap has
volume exponentially small in N).
•
Levy’s Lemma
Any slowly varying function on the sphere takes values close to the
average except for an exponentially small set.
f (φ) = ⟨f ⟩
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Induced Measure
�A = U ΛU †
Λ = diag(λ1 , . . . , λN )
dµ(�A ) = dµH (U ) × dν(Λ)
Haar measure
dµH (U ),
Simplex of Eigenvalues
tr�A = trΛ =
0 ≤ λi ≤ 1
�
U ∈ U(N )
∆N −1
λi = 1
i
Zyczkowski, Sommers, J Phys A 34, 7111 (2001)
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Joint distribution of Eigenvalues
ρA = U ΛU †
Λ = diag(λ1 , . . . , λN )
dµ(�A ) = dµH (U ) × dν(Λ)
dν(Λ) = pN (Λ)dλ1 . . . dλN
pN (�λ) = CN
�
1≤j<k≤N
(λj − λk )
2
normalization constant
Eigenvalue repulsion!
Lloyd, Pagels, Ann Phys 188, 186 (1988)
11
What is the typical distribution of the eigenvalues in a system with a certain
amount of bipartite entanglement?
Isoentropic manifolds (given value of the von Neumann entropy)
Sinolecka, Zyckowski, Kus, Acta Phys. Pol. B2002
Constrained maximization problem:
H
given u ∈ [0, ln N ] find �λ such that
pN (�λ) = max pN with
(3)
SvN
SvN (�λ) = ln N − u
(2)
SvN
(1)
SvN
Previous work on “Purity” and “Rényi entropy”
Facchi et al. PRL2008, DePasquale et al. PRA2010,
Nadal et al. PRL2010
12
Gas of eigenvalues in the interval [0,1]
��
�
��
�
�
2
�
V (λ, ξ, β) = − 2
ln |λj − λk | + ξ
λk − 1 + β
λk ln N λk − u
N
j<k
k
k
repulsive 2D Coulomb
ξ, β
Lagrange multipliers
external potential
Unconstrained minimization problem
Hard wall
from positivity
Hard wall
from unit trace
0
1
λi
Dyson, J. Math. Phys 3, 157 (1962)
13
Equivalent picture from statistical mechanics
�
−βN 2 h(�
λ)
ZN =
e
pN (�λ) dN λ
∆N −1
h(�λ) = ln N − SvN (�λ)
β
“Inverse temperature”
N → +∞
Thermodynamic limit
Minimization of V (�
λ, ξ, β)
Maximization of the integrand
β=0
β large
“Energy density”
random states (distributed according to the Haar
measure on the unitary group U (H) )
highly entangled states
14
Saddle point equations
1
2 ��
β(ln N λk + 1) + 2
+ξ =0
N j λj − λk
∂V
∂V
∂V
=
=
=0
∂λi
∂ξ
∂β
�
λi = 1
i
Trace condition
large N, λ = N λj
�
λ σ(λ) dλ = 1
β(ln λ + 1) + 2P
�
σ(λ) =
�
1
N
�
j
λj ln(N λj ) = u
Fixed entropy
�
empirical
δ(λ
−
N
λ
)
j
j
distribution
λ ln λ σ(λ) dλ = u
σ(λ� )
λ� −λ
dλ� + ξ = 0
σ(λ) can be found using a theorem by Tricomi
15
Tricomi, Integral Equations,
(CUP, 1957)
Results for the Entanglement spectrum
SvN = ln N − u
Deformed Wigner’s
semicircle law
u < uc � 0.26
towards max entanglement
∝
Σ�Λ�
1.0
u�
(λ+ − λ)(λ − λ− )
�a�
20
u�
1
10
u�
0.4
uc = u(βc )
1
5
βc = 3/2
2 2
uc �ln �
3 3
0.2
0.5
1.0
1.5
2.0
16
(λ+ , λ− )
Extremes of the
support
1
0.8
0.6
�
2.5
Λ
Results for the Entanglement spectrum
SvN = ln N − u
Deformed MarchenkoPastur law
u > uc � 0.26
Σ�Λ�
1.0
0.8
�b�
2 2
uc �ln �
3 3
0.6
0.4
u�
0.2
1
2
3
17
u = 1/2
β=0
1
3
u�
1
2
4
Λ
Results for the Entanglement spectrum
SvN = ln N − u
towards separable states
u > 1/2
�λ � (1, 0, . . . , 0)
Σ�Λ�
1.0
λ1 = µ = O(1)
0.8
0.6
u � Μ ln N �
0.4
�c�
1
β<0
2
0.2
4�1�Μ�
18
NΜ
Λ
4
s
Logarithm of the volume of the
isoentropic manifolds
�1�2
�2
�4
maximally entangled
s ∝ ln pN
separable
�6
Discontinuity of the derivative at u = 1/2
�8
0 uc 1�2
1
2
3
ln N
is of order O (1/ ln N )
u
N = 50 in the figure
FIG.
4: (Color online) Logarithm of the volume of the isoend 3 s�du3
−2
tropic
ln pN vs u = ln N −SvN , for N = 50.
200 manifolds s = N
See Eq. (30). The discontinuity of the derivative at u = 1/2
is O(1/ ln N ).
150
Jump in the fourth derivative at
u = uc � 0.26
d 3 s�du3
100
200
50
150
0
100
0.2
uc
0.3
0.4
0.5
19
u
Limit N → +∞ in the figure
An alternative procedure: polarized ensembles
dim HA = N ≤ dim HB = M
H = HA ⊗ HB
dim H = N M
πAB (|ψ�) = trA (�2A ) =
N
�
λ2k
Purity (linear entropy)
k=1
1
≤ πAB ≤ 1
N
separable state
maximally entangled state
20
Superposition of two pure quantum state
�
|ψ� = �1AB +
|φ0 � ∈ H
�
�
1 − �2 UAB |φ0 �
UAB ∈ U (H)
reference state
random unitary
� ∈ [0, 1]
|φ� = UAB |φ0 �
random state
New (polarized) ensemble of states!
�
π
¯AB = E[πAB | |φ0 �, �] = � π0 + 1 − �
4
π0 = πAB (|φ0 �)
πunb
21
N +M
=
NM
4
�
πunb
Two cases
|φ0 � = |φsep � = |φ0 �A ⊗ |φ0 �B
separable state
M +N
4 MN − M − N
E [πAB | |φsep �, �] =
+�
MN
MN
|φ0 � = |φent �
maximally entangled state with
M +N
�4
E [πAB | |φent �, �] =
−
MN
M
22
π0 = 1
1
TrB |φent ��φent | = 1A
N
1
π0 =
N
Polarized ensembles of random pure states
9
Π AB
N�30 M�30
1.0
0.8
0.6
0.4 Maximally Entangled
Φ ent �
0.2
0.0
1
0.5
Separable
Φ sep �
0
0.5
1
Ε4
N�8 M�8
Π AB
A strategy
1.0for generating random pure states with fixed value of the purity
0.8
�
�
4
4
1) choose0.6
� such that πAB = � π0 + 1 − � πunb
Separable
0.4 Maximally Entangled
�
1
Φ ent �
π0 = Φ1sep
or�π0 =
2 |φ�
2) generate
|ψ�
=
�|φ
�
+
1
−
�
0.2
0
N
|φ0 � = |φsep � or |φent �
0.0
1
0.5
0
0.5
1
23
Conclusions
• Typical distribution of the eigenvalues given the value of the von Neumann
Entropy
• Phase transitions (sudden change of the eigenvalues distribution)
• Bipartite entanglement from polarized ensembles
Facchi, GF, Parisi, Pascazio, Yuasa, Physical Review A 87, 052324 (2013)
Cunden, Facchi, GF, Journal of Physics A: Mathematical and Theoretical 46, 315306 (2013)
Thank you for your attention
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