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arXiv:0807.3716v1 [quant-ph] 23 Jul 2008

Entropy of entanglement and multifractal exponents for random states

Olivier Giraud, John Martin and Bertrand Georgeot

Laboratoire de Physique Th´ eorique, Universit´ e de Toulouse, UPS, CNRS, 31062 Toulouse France

(Dated: July 23, 2008)

We relate the entropy of entanglement of ensembles of random vectors to their generalized fractal

dimensions. Expanding the von Neumann entropy around its maximum we show that the first order

only depends on the participation ratio, while higher orders involve other multifractal exponents.

These results can be applied to entanglement behavior near the Anderson transition.

PACS numbers: 03.67.Mn, 03.67.Ac, 05.45.Df, 71.30.+h

Entanglement is an important characteristics of quan-

tum systems, which has been much studied in the past

few years due to its relevance to quantum information

and computation. It is a feature that is absent from clas-

sical information processing, and a crucial ingredient in

many quantum protocols. In the field of quantum com-

puting, it has been shown that a process involving pure

states with small enough entanglement can always be

simulated efficiently classically [1]. Thus a quantum al-

gorithm exponentially faster than classical ones requires

a minimal amount of entanglement (at least for pure

states). Conversely, it is possible to take advantage of

the weak entanglement in certain quantum many-body

systems to devise efficient classical algorithms to simu-

late them [2]. All these reasons make it important to

estimate the amount of entanglement present in different

types of physical systems, and relate it to other proper-

ties of the system. However, in many cases the features

specific to a system obscure its generic behavior. One

way to circumvent this problem and to extract generic

properties is to construct ensembles of systems which af-

ter averaging over random realizations can give analytic

formulas. Such an approach has proven successful, e.g. in

the quantum chaos field, where Random Matrix Theory

(RMT) can describe many properties of complex quan-

tum systems.

One of the interesting questions which have been ad-

dressed in many studies (see e.g. [3] and references

therein) is the behavior of entanglement near phase tran-

sitions. It has been shown that the entanglement of the

ground state changes close to phase transitions. For ex-

ample, in the XXZ and XY spin chain models, the en-

tanglement between a block of spins and the rest of the

system diverges logarithmically with the block size at the

transition point [4], making classical simulations harder.

However, such results cannot be applied directly to sys-

tems where the transition concerns one-particle states,

for which entanglement has to be suitably defined. A

famous example is the Anderson transition of electrons

in a disordered potential, which separates localized from

extended states, with multifractal states at the transition

point. Previous works [5] have described the lattice on

which the particle evolves as a spin chain and studied en-

tanglement in this framework. However, the lattice can

alternatively be described in terms of quantum computa-

tion with a much smaller number of two-level systems [6].

In this paper, we study entanglement of random vec-

tors which can be localized, extended or multifractal in

Hilbert space. We consider entanglement between blocks

of qubits. In the case of the Anderson transition, this

amounts to directly relate entanglement to the quantum

simulation of the system on a nr–qubit system, the num-

ber of lattice sites being 2nrrather than nr as in [5].

Entanglement of random pure states was mainly stud-

ied in the case of columns of matrices drawn from the

Circular Unitary Ensemble (CUE) [7]. However, such

vectors are extended and cannot describe systems with

various amounts of localization, from genuine localiza-

tion to multifractality. Recently it was shown in [8, 9]

that for localized random vectors, the linear entangle-

ment entropy (first order of the von Neumann entropy)

of one qubit with all the others can be related to the

localization properties. Here we develop this approach

to obtain a general description of bipartite entanglement

in terms of certain global properties for random vectors

both extended and localized. First we show that for any

bipartition, the linear entropy can be written in terms

of the participation ratio, a measure of localization. We

then show that higher-order terms also depend on higher

moments of the wavefunction. In particular, for multi-

fractal systems they are controlled by the multifractal

exponents.

Bipartite entanglement of a pure state |ψ? belonging to

a Hilbert space HA⊗HBis measured through the entropy

of entanglement, which has been shown to be a unique

entanglement measure [10]. Let ρAbe the density matrix

obtained by tracing subsystem B out of ρ = |ψ??ψ|. The

entropy of entanglement of the state with respect to the

bipartition (A,B) is the von Neumann entropy of ρA,

that is S = −tr(ρAlog2ρA). It is convenient to define

the linear entropy as SL =

dimHA≤ dimHB. The scaling factor ensures that SL

varies in [0,1]. We will show that the average value of

S over a set of random states can be expressed only in

terms of averages of the moments of the wavefunction

d

d−1(1 − trρ2

A), where d =

pq=

N

?

i=1

|ψi|2q

(1)

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2

provided some natural assumptions are made. Here we

consider ensembles of random vectors of size N ≡ 2nr

with the following two properties:

the vector components are independent, uniformly dis-

tributed random variables, and ii) the joint distribution

P(x1,...,xN) of the modulus squared of the vector com-

ponents is such that all marginal distributions P(xi),

P(xi,xj) for i ?= j, P(xi,xj,xk) for i ?= j ?= k and so

on, do not depend on the indices. As a consequence,

all correlators ?|ψi1|2s1|ψi2|2s2...? of the components of

|ψ? are independent of the indices i1?= i2?= ... involved.

Random vectors realized as columns of CUE matrices are

instances of vectors having such properties.

Let us first consider the simplest case of entanglement

of one qubit with respect to the others. Then d = 2

and the linear entropy SLis simply the tangle τ, or the

square of the generalized concurrence [11]. It is given by

τ = 4detρA. If we consider a vector |ψ? of size N, the

bipartition with respect to qubit i splits the components

ψjof |ψ? into two sets, according to the value of the ith

bit of the binary decomposition of j. If |ψ(0)? and |ψ(1)?

are the two corresponding vectors, the linear entropy is

i) the phases of

τ = 4

?

?ψ(0)|ψ(0)??ψ(1)|ψ(1)? − |?ψ(0)|ψ(1)?|2?

. (2)

After averaging τ over random phases, only the diagonal

terms survive in the scalar product |?ψ(0)|ψ(1)?|2. Since

it is assumed that two-point correlators of the vector |ψ?

do not depend on indices, their average can be expressed

solely in terms of the mean moments, as ?|ψi|2|ψj|2? =

?p2

1?−?p2?

N(N−1)for i ?= j. Normalization of |ψ? implies p1= 1.

As vectors |ψ(0)? and |ψ(1)? always contain components

of |ψ? with different indices, we get

?τ? =N − 2

N − 1(1 − ?p2?). (3)

Since p2= 1/ξ where ξ is the inverse participation ratio

(IPR), Eq. (3) is exactly the Eq. (3) of Ref. [9]. Let us

now turn to the general case, and consider the entropy of

entanglement of ν qubits with nr− ν others ((ν,nr− ν)

bipartition).The vector |ψ? is now split into vectors

|ψ(j)?, 0 ≤ j ≤ 2ν− 1, depending on the values of the ν

qubits. The reduced density matrix ρAthen appears as

the Gram matrix of the |ψ(j)?, and the linear entropy is

1 −

SL=

2ν

2ν− 1

2ν−1

?

i,j=0

|?ψ(i)|ψ(j)?|2

. (4)

When averaging over random vectors, each term in

Eq. (4) with i ?= j yields 2nr−νtwo-point correlators,

while each term with i = j yields 2nr−ν(2nr−ν− 1) two-

point correlators and 2nr−νterms of the form ?|ψi|4?.

Inserting these expressions into (4) gives ?SL? = (N −

2ν)(1 − ?p2?)/(N − 1), which generalizes Eq. (3). The

first-order series expansion of the mean von Neumann

entropy around its maximum can be expressed as

?S? ≃ ν −2ν− 1

2ln2

?

1 −N − 2ν

N − 1(1 − ?p2?)

?

,(5)

with p2 = 1/ξ. Equation (5) shows that for any par-

tition of the system into two subsystems, the average

bipartite entanglement of random states only depends at

first order on the localization properties of the states,

through the mean participation ratio. For CUE vectors,

formula (5) reduces to the expression for the mean en-

tanglement derived earlier in [12].

this formula also applies to multifractal quantum states.

There, the asymptotic behavior of the IPR is governed

by the fractal exponent D2, where one defines generalized

fractal dimensions Dqthrough the scaling of the moments

pq∝ N−Dq(q−1). Thus the linear entropy is only sensitive

to a single fractal dimension. These results imply that

entanglement grows more slowly with the system size for

multifractal systems.

To test the relevance of Eq. (5) for describing entan-

glement in realistic settings, we consider eigenvectors of

N × N unitary matrices of the form

Ukl=eiφk

N

More interestingly,

1 − e2iπNγ

1 − e2iπ(k−l+Nγ)/N,(6)

where φk are independent random variables uniformly

distributed in [0,2π[. These random matrices display in-

termediate statistical properties [13], and possess eigen-

vectors that are multifractal [14], both features being

tuned through the value of the real parameter γ. We also

illustrate Eq. (5) with eigenstates of a many-body Hamil-

tonian with disorder and interaction H =

?

puter in presence of static disorder [15]. Here the σiare

the Pauli matrices for qubit i, energy spacing between

the two states of qubit i is given by Γirandomly and uni-

formly distributed in the interval [∆0−δ/2,∆0+δ/2], and

the Jijuniformly distributed in the interval [−J,J] repre-

sent a random static interaction. For large J and δ ≈ ∆0,

eigenstates are delocalized in the basis of register states,

but without multifractality. They display properties of

quantum chaos, with eigenvalues statistics close to the

ones of RMT [15]. For both systems (unitary matrices

and many-body Hamiltonian), components of the eigen-

vectors have been shuffled in order to reduce correlations,

but leaving the peculiarities of the distribution itself un-

altered. Figure 1 plots the first-order expansion (5) as

a function of the mean IPR for three different biparti-

tions, showing remarkable agreement with the exact ?S?,

both for multifractal (Fig. 1, left panel) and non fractal

(right) states, and even for moderately entangled states.

The agreement is better for the non-fractal system than

for the multifractal one. This can be understood from

the study of higher order terms in the entropy.

?

iΓiσz

i+

i<jJijσx

iσx

j. This system can describe a quantum com-

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3

?1/ξ?−1

160 12080400

?1/ξ?−1

?S?

60504030 20 100

5

4

3

2

1

0

FIG. 1: (Color online) Mean entropy of entanglement as a

function of the mean IPR. Left: eigenvectors of (6) with

γ = 1/3; the average is taken over 106eigenvectors. Right:

eigenvectors of the Hamiltonian H (see text) with δ = ∆0 and

J/δ = 1.5; average over N/16 central eigenstates, with a total

number of vectors ≈ 3 × 105. Triangles correspond to ν = 1,

squares to ν = 2 and circles to ν = nr/2, with nr = 4 − 10

(bipartition of the ν first qubits with the nr−ν others). Black

symbols are the theoretical predictions for ?S? at first order

(Eq. (5)) and green (grey) symbols are the computed mean

values of the exact ?S?.

Indeed, while the linear entropy does not depend on

other fractal dimensions than D2, the entropy of entan-

glement does. If we go back to the case of a (1,nr− 1)

bipartition of the system, the entropy of entanglement

can be expressed in a simple way as a function of τ as

?1 +√1 − τ

S(τ) = h

2

?

, (7)

where h(x) = −xlog2x − (1 − x)log2(1 − x). The series

expansion of S(τ) up to order m in (1 − τ) reads

1

ln2

Sm(τ) = 1 −

m

?

n=1

(1 − τ)n

2n(2n − 1). (8)

The tangle τ corresponds, up to a linear transformation,

to S1(τ).Let us now calculate the average of higher

orders in this expansion. The second-order expansion of

S(τ) involves calculating the mean value of

|?ψ(0)|ψ(1)?|4=

N/2

?

i,j,k,l=1

u∗

iuju∗

kulviv∗

jvkv∗

l,(9)

where the star denotes complex conjugation and ui, viare

the components of |ψ(0)?, |ψ(1)? respectively. Under the

assumption of random phases, only terms whose phases

cancel survive in (9). Since the phases of all components

of |ψ? are independent, cancelation of the phase can only

occur if the sets {i,k} and {j,l} are equal. Thus

N/2

?

i,k=1

i?=k

?|?ψ(0)|ψ(1)?|4? = 2

?|uiukvivk|2? +

N/2

?

i=1

?|uivi|4?.

(10)

The correlators in Eq. (10) can be expressed as a func-

tion of the moments as follows. Using standard notations

[16], we will denote by λ ⊢ n a partition λ = (λ1,λ2,...)

of n, with λ1 ≥ λ2 ≥ .... For any partition λ ⊢ n, we

define pλ= pλ1pλ2..., where pλiare given by (1). The

monomial symmetric polynomials are defined as mλ =

??|ψ1|2λ1|ψ2|2λ2...? (the sum runs over all Pλpermuta-

tions of the λi), and we set cλ= mλ/Pλ. The pλand mλ

are related by the simple linear relation pλ=?

where Lλµis an invertible integer lower-triangular matrix

([16], p.103). Upon our assumption ii), any correlator of

the form ?|ψi1|2s1|ψi2|2s2...? is equal to a cλ for some

partition λ of n, and thus can be expressed as a func-

tion of the moments. For instance the two correlators in

Eq. (10) are respectively equal to c1111and c22. Treating

similarly all terms involved in τ2gives

µLλµmµ,

?τ2? = N(N − 2)(N2− 6N + 16)c1111

+ 4N(N − 2)(N − 4)c211+ 4N(N − 2)c22.

This term involves the calculation of three correlators.

Using the relation between the cλ and pλ and the fact

that the vectors are normalized to one we get

(11)

c22 =

?p2

2? − ?p4?

N(N − 1), c211=?p2? − ?p2

1 − 6?p2? + 8?p3? + 3?p2

N(N − 1)(N − 2)(N − 3)

The calculation of the general term ?τn? can be per-

formed along the same lines. Expanding (2) we get

2? − 2?p3? + 2?p4?

N(N − 1)(N − 2)

2? − 6?p4?

.

,

c1111 =(12)

τn= 4n

n

?

k=0

?n

k

?

(−1)n−k

(13)

×

N/2

?

i=1

|ui|2

??

k

N/2

?

i=1

|vi|2

k

?

?t

i,j

u∗

iviujv∗

j

n−k

.

The expansion of

the form (ui1...uit)∗uj1...ujt. Only terms where the

phases coming from the u∗

ikcompensate those coming

from the ujksurvive when averaging over random phases.

Thus we keep only terms where {j1,...,jt} is a permuta-

tion of {i1,...,it}. If PKis the number of permutations

of a set K, the average of (13) over random vectors reads

i,ju∗

iviujv∗

j

contains products of

?τn? =

?

4n

n

?

k=0

?n

k

?

(−1)n−k

?

q1,...,qk

p1,...,pk

k?

j=1

|upj|2|vqj|2

×

?

i1,...,in−k

P{i1,...,in−k}|ui1vi1|2...|uin−kvin−k|2?

.(14)

Terms with the same correlator can be grouped together.

Each correlator in (14) is some cλ∪λ′, with λ,λ′parti-

tions of n. For λ ⊢ n and µ ⊢ k we define the coeffi-

cient Aλµ =

µ!

k!

?(n−k)!

(s−k)!, where the sum runs over all

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4

nr

98765432

nr

?S?−?Sm?

?S?

1098765432

100

10−1

10−2

10−3

10−4

10−5

10−6

FIG. 2: (Color online) Relative difference of the entropy of

entanglement (7) and its successive approximations Sm (m =

1,2) with respect to the number of qubits for eigenvectors of

(6) for (left) γ = 1/3 and (right) γ = 1/7. The average is

taken over 107eigenvectors, yielding an accuracy ? 10−6on

the computed mean values. Green triangles correspond to the

first-order expansion S1, blue squares and red circles to the

second-order expansion S2. The difference between the latter

two is that for blue squares ?p2

been replaced by ?p2?2yielding a less accurate approximation.

Dashed line is a linear fit yielding 1 − ?S1?/?S? ∼ N−0.84for

γ = 1/3 and N−1.58for γ = 1/7.

2? appearing in Eq. (12) has

vectors s = (s1,...,sN/2) which are permutations of λ,

and k = (µ1,...,µN/2).We have used the notations

a = (a1,a2,...) and a! = a1!a2!.... Finally we get

?τn? = 4n?

λ,λ′⊢n

?

n

?

k=0

?n

k

?

(−1)k?

µ⊢k

PµAλµAλ′µ

?

cλ∪λ′,

(15)

which provides an expression for the nth order for ?S?

as a function of the ?pλ?. In the special case of CUE

random vectors, by resumming the whole series we re-

cover after some algebra the well-known result

?S(τ)? =

can be derived for a general (ν,nr−ν) bipartition. In this

case, the entropy S = −tr(ρAlog2ρA) can be expanded

around the maximally mixed state ρ0= 1/2ν, as

[17]

1

ln2

?N−1

k=N/2+1

1

k. Note that similar expressions

S = ν +

1

ln2

∞

?

n=1

(−2ν)n

n(n + 1)tr((ρA− ρ0)n+1). (16)

After averaging over random vectors, one can check that

the traces in (16) can be written as ?trρk

with a(k)

λ

some integer combinatorial coefficient. The en-

tropy can thus be written as a linear combination of cλ

with rational coefficients that can be expressed in terms

of the a(k)

λ.

In Fig. 2 we illustrate the accuracy of higher-order

terms in the series expansion of S for multifractal random

vectors by comparing the first and second-order expan-

sion for eigenvectors of the matrices (6). As expected, the

second-order expansion is much more accurate than the

first order one and gives a much better estimate of the

mean entropy of entanglement already for small system

A? =?

λ⊢ka(k)

λcλ,

sizes. For large N, the dominant term in S2 is ∝ ?p2

Numerically we obtained ?p2

?p2

with the slopes of the linear fit of log2(1−?S1?/?S?) (see

Fig. 2). If one replaces ?p2

?p2?2(squares in Fig. 2), the second-order expansion is

now governed only by three multifractal dimensions D2,

D3, D4. Although it becomes less and less accurate with

the system size because of the increase of the variance of

p2, it remains a very good improvement over the first or-

der in the case of moderate multifractality (Fig. 2, right).

2?.

2? ∼ N−0.81for γ = 1/3, and

2? ∼ N−1.53for γ = 1/7, which is indeed consistent

2? appearing in Eq. (12) by

Our results show that the entanglement of random vec-

tors directly depends on whether they are localized, mul-

tifractal or extended. The numerical simulations for dif-

ferent physical examples show that our theory describes

well individual systems whose correlations are averaged

out.Previous results [9] have shown that Anderson-

localized states have entanglement going to zero for large

system size. The present work shows that multifractal

states, such as those appearing at the Anderson transi-

tion, approach the maximal value of entanglement in a

way controlled by the multifractal exponents. Although

extended and multifractal states are both close to maxi-

mal entanglement, the way multifractal states approach

the maximal value for large system size is slower.

The authors thank CalMiP in Toulouse for access to

their supercomputers. This work was supported by the

Agence Nationale de la Recherche (ANR project INFOS-

YSQQ, contract number ANR-05-JCJC-0072) and the

European program EC IST FP6-015708 EuroSQIP.

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