Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

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by  Satya N. Majumdar

Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State
Satya N. Majumdar,1 Oriol Bohigas,1 and Arul Lakshminarayan 2
∗
1Laboratoire de Physique Th´eorique et Mod`eles Statistiques (UMR 8626 du CNRS),
Universit´e Paris-Sud, Bˆatiment 100, 91405 Orsay Cedex, France.
2Max-Planck-Institut fu¨r Physik komplexer Systeme,
N¨othnitzer Straβe 38, D-01187 Dresden, Germany.
Arecentconjectureregardingtheaverageoftheminimumeigenvalueofthereduceddensitymatrix
8 of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is
0 derived exactly for both the cases of a random real and a random complex state. Our results are
0
relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of
2
randommatrixtheoryandquantumchaoticsystems. Theyalsoprovidearareexactlysolvablecase

n forthedistributionoftheminimumofaset ofN strongly correlated randomvariablesforallvalues
a of N (and not just for large N).
J

4
1 Key Words: entanglement, random pure state, extreme value statistics

]
h I. INTRODUCTION
c
e
m Entangelement has been studied extensively in the recent past due to its central role in quantum information
- and possible involvement in quantum computation [1, 2]. It is desirable in many instances to create states of large
t entanglement. Measures of entanglement have been studied mostly in the context of pure bipartite states, where
a
t the von-Neumann entropy of either subsystem is one of the measures of entanglement [2]. However there exist other
s
. measures of entanglement as well, e.g. the so called concurrence for two-qubit systems [3]. The entanglement of
t
a random purequantum states is of interestas they have near maximalentanglementcontent, especially in the context
m of bipartite entanglement [4]. Apart from the issue of bipartite entanglement, statistical properties of such random
states are relevant for quantum chaotic or non-integrable systems. The applicability of random matrix theory and
-
d hence of random states to systems with well-defined chaotic classical limits was pointed out long back [5]. They are
n also of relevance to other systems with no apparent classical limit [6, 7, 8].
o
In this paper, we focus on a bipartite quantum system. More precisely, we consider a bipartite partition of a
c
[ NM-dimensionalHilbert space (NM) as (NM) = (N) (M). We canassumewithout lossofgeneralityN M.
H H HA ⊗HB ≤
As an example of such a bipartite system, may be considered a given subsystem (say a set of spins) and may
2 A B
represent the environment (e.g., a heat bath). Any quantum state ψ of the composite system can be generally
v | i
7 written as a linear combination, |ψi= Ni=1 Mα=1xi,α|iAi⊗|αBi where |iAi and |αBi denote two complete basis of
7 (N) and (M) respectivelyandthecoefficientsx ’sformthe entriesofarectangular(N M)matrixX. Mutually
6 HA HB P P i,α ×
nonexclusivepropertiesofsuchastate areentanglement,randomnessandstatisicalpurity. Sucha quantumstate ψ
0 | i
is:
.
1
1 entangled: if not expressible as a direct product of two states belonging to the two subsystems and . Only
7 in•thespecialcasewhenthecoefficientshavetheproductform,xi,α =aibα foralliandα, thestate ψA= φAB φB
| i | i⊗| i
0 can be written as a direct product of two states φA = N a iA and φB = M b αB belonging respectively
: | i i=1 i| i | i α=1 α| i
v to the two subsystems and . In this case, the composite state ψ is fully unentangled. But otherwise, it is
i generically entangled. A B P | i P
X
r random: if the coefficients xi,α are random variables drawn from an underlyting probability distribution. The
a sim•plestandthemostcommonrandomstatecorrespondstochoosingx ’sasindependentandidenticallydistributed
i,α
Gaussian variables, real or complex.
pure: if the density matrix of the composite system is simply givenby, ρ= ψ ψ with the constraintTr[ρ]=1,
• | ih |
or equivalently ψ ψ =1.
h | i
Given a random, pure and generically entangled composite state, important informations on the results of the
measurement of any observable on the subsystem can be derived from the reduced density matrix ρ = Tr [ρ],
A B
A
obtained upon tracing out the environmental degrees of freedom (i.e., those of subsystem ). It is easy to show
B
∗ Permanentaddress: DepartmentofPhysics,IndianInstituteofTechnologyMadras,Chennai,600036, India.
2
(see section II for details) that for a random pure state, ρ = XX is an N N square matrix where X is the
A †
×
N M rectangularcoefficientmatrix. TheN unorderedeigenvaluesλ ,λ ,...,λ ofρ carryimportantinformations
1 2 N A
×
regarding the degree of the entanglement in the subsystem A. Given that the entries x of the coefficient matrix X
i,α
are independent Gaussian variables (real or complex), the eigenvalues λ ’s of the matrix ρ =XX are also random
i A †
variables and their joint probability density function (jpdf) is known [9, 10]
N N
P(λ1,λ2,··· ,λN)=BM,Nδ λi−1! λiβ2(M−N+1)−1 |λj −λk|β. (1)
i=1 i=1 j<k
X Y Y
Here β = 1,2 corresponding to the real and complex entries of A and B is the normalization constant that is
M,N
known explicitly [10]. Several spectral properties associated with the jpdf in Eq. (1), in particular for the complex
β =2 case, have been studied extensively in the literature, for instance see the book [11] and references therein.
Inprinciple,allinformationsaboutthespectralpropertiesofthesubsystem ,includingitsdegreeofentanglement,
A
are encoded in the jpdf (1). For example, one useful measure of entanglement is the von Neumann entropy S =
N λ ln(λ ) which is a random variable. The average entropy S (where the average is performed with the
− i=1 i i h i
measure in Eq. (1)) was computed for β = 2 by Page [12] and was found to be S ln(N) N for large
P h i ≈ − 2M
1 << N M. Noting that ln(N) is the maximal possible value of entropy of the subsystem , it follows that in
≤ A
the limit when M >> N, the average entropy, and hence the average entanglement, of a random pure state is near
maximal. Later, the same result was shown to hold for the β =1 case [13].
While the average entropy is a useful measure of entanglement, it is not the unique one. In fact, important
informations regarding the nature of entanglement of a random pure state can also be obtained (see Section II for a
detailed discussion) by studying the probability distributions of the extreme eigenvalues λ =max(λ ,λ ,...,λ )
max 1 2 N
and λ =min(λ ,λ ,...,λ ). In particular,the probability distribution of the minimum eigenvalue λ provides,
min 1 2 N min
in addition to the nature of the entanglement, an important information about the degree to which the effective
dimension of the Hilbert space of the subsystem can be reduced.
A
In fact, the average value λ (with respect to the measure in Eq. (1)) of the minimum eigenvalue was studied
min
h i
recentlybyZnidaric[14]forthecaseN =M andbasedontheexact λ forsmallvaluesofN,Znidaricconjectured
min
h i
that λ =1/N3 for all N for the complex case (β =2). The purpose of this paper is to provide exact results for
min
h i
the full probability distribution of λ for all N (for the case when N = M), both for the complex (β = 2) and the
min
real (β =1) cases. A byproduct of our general results is the proof of Znidaric’s conjecture for β =2. Our results are
summarized as follows. Let P (x)dx denote the probability that x λ x+dx, i.e., P (x) is the probability
N min N
≤ ≤
density function (pdf) of λ . We show that
min
Complex case (β =2):
•
P (x)=N(N2 1)(1 Nx)N2 2Θ(1 Nx) (2)
N −
− − −
where Θ(x) is the standard Heaviside function, Θ(x) = 1 for x > 0 and Θ(x) = 0 for x < 0. The k-th moment
µ (N)= λk is given by
k h mini
Γ(k+1)Γ(N2)
µ (N)= . (3)
k NkΓ(N2+k)
In particular, for k =1, we get µ (N)=1/N3 thus proving the recent conjecture in [14].
1
Real case (β = 1): the result for the real case turns out to be a bit more complicated. For the pdf of λ we
min
•
get
PN(x)=ANx−N/2(1 Nx)(N2+N−4)/22F1 N +2,N −1,N2+N −2, 1−Nx , 0<x 1/N (4)
− 2 2 2 − x ≤
(cid:18) (cid:19)
and P (x)=0 for x 1/N. The constant A is given by
N N
≥
NΓ(N)Γ(N2/2)
A = , (5)
N 2N 1Γ(N/2)Γ((N2+N 2)/2)
− −
and F (α,β,γ,z) is the standard Hypergeometric function defined as [15]
2 1
αβ α(α+1)β(β+1) z2 α(α+1)(α+2)β(β+1)(β+2) z3
F (α,β,γ,z)=1+ z+ + +... (6)
2 1
γ γ(γ+1) 2! γ(γ+1)(γ+2) 3!
3
The moments µ (N)= λk are also computed exactly and are given in Eq. (58). In particular, the average value
k h mini
(k =1) decays for large N as
c
µ (N) (7)
1 ≈ N3
where the prefactor c has a nontrivial value
πe
c=2 1 erfc(1/√2) =0.688641 (8)
− 2 ···
(cid:20) r (cid:21)
The paper is organized as follows. In Section II, we provide a general introduction to the random pure states of a
bipartite system and recapitulate some general facts leading to the jpdf (1). Section II and III provide the detailed
calculationsofthe distributionofthe minimum eigenvaluefor the complex andthe realcasesrespectively. Finally we
conclude inSectionIVwith asummary andopenquestions. Some details ofthe calculationsarepresentedin the two
appendices.
II. A RANDOM PURE STATE OF A BIPARTITE SYSTEM
In this sectionwe recallsome generalfacts about a random pure (RP) state ofa bipartite system, its entanglement
propertiesandtheassociatedrandommatrixensemble. Asmentionedintheintroduction,letusconsideracomposite
bipartite system A B composed of two smaller subsystems A and B, whose respective Hilbert spaces (N) and
⊗ HA
(M) have dimensions N and M. The Hilbert space of the composite system (NM) = (N) (M) is thus NM-
HdimBensional. Without loss of generality we will assume that N M. Let iA Hand αBHAre⊗prHeseBnt two complete
≤ {| i} {| i}
basis states for A and B respectively. Then, any arbitrary state ψ of the composite system can be most generally
| i
written as a linear combination
N M
ψ = x iA αB (9)
i,α
| i | i⊗| i
i=1α=1
XX
where the coefficients x ’s form the entries of a rectangular (N M) matrix X =[x ].
i,α i,α
×
Now,thestate ψ isastatisticallypurestateofthecompositesystemifthedensitymatrixofthecompositesystem
| i
is given by
ρ= ψ ψ . (10)
| ih |
Note that had the composite system been in a statistically mixed state, its density matrix would have been of the
form
ρ= p ψ ψ , (11)
k k k
| ih |
k
X
where ψ ’s are the pure states of the composite system and 0 p 1 denotes the probability that the composite
k k
| i ≤ ≤
systemisinthek-thpurestate,with p =1. Inthispaper,wewillrestrictourselvestothecasewhenthecomposite
k k
system is in a pure state denoted by ψ . Then its density matrix in Eq. (10), upon using the decomposition in Eq.
P| i
(9), can be expressed as
ρ= x x iA jA αB βB , (12)
i,α ∗j,β| ih |⊗| ih |
i,α j,β
XX
where the Roman indices i and j run from 1 to N and the Greek indices α and β run from 1 to M. We also assume
that the pure state ψ is normalized to unity so that Tr[ρ] = 1. Hence the coefficients x ’s must be such that
i,α
| i
Tr[ρ]=1.
GiventhedensitymatrixofthepurecompositestateinEq. (12),onecanthencomputethereduceddensitymatrix
of, say, the subsystem by tracing over the states of the subsystem
A B
M
ρ =Tr [ρ]= αB ραB . (13)
A B
h | | i
α=1
X
4
Using the expression in Eq. (12) one gets
N M N
ρ = x x iA jA = W iA jA (14)
A i,α ∗j,α| ih | ij| ih |
i,j=1α=1 i,j=1
X X X
where W ’s are the entries of the N N square matrix W = XX . In a similar way, one can express the reduced
ij †
×
density matrix ρ =Tr [ρ] of the subsystem B in terms of the square M M dimensional matrix W =X X.
B A ′ †
×
Let λ ,λ ,...,λ denote the N eigenvalues of W =XX . Note that these eigenvalues are nonnegative, λ 0 for
1 2 N † i
≥
all i=1,2,...,N. Now the matrix W =X X has M N eigenvalues. It is easy to prove that M N of them are
′ †
≥ −
identically0 andN nonzeroeigenvaluesofW arethe sameasthoseofW. Thus,inthis diagonalrepresentation,one
†
can express ρ as
A
N
ρ = λ λA λA (15)
A i| i ih i |
i=1
X
where λA ’s are the eigenvectors of W = XX . A similar representation holds for ρ . It then follows that one can
| i i † B
represent the original composite state ψ in this diagonal representation as
| i
N
ψ = λ λA λB (16)
| i i| i i⊗| i i
i=1
Xp
where λA and λB represent the normalized eigenvectors(corresponding to nonzero eigenvalues) of W =XX and
| i i | i i †
W = X X respectively. This spectral decomposition in Eq. (16) is known as the Schimdt decomposition. The
′ †
normalization condition ψ ψ =1, or equivalently Tr[ρ]=1, imposes a constraint on the eigenvalues, N λ =1.
h | i i=1 i
Note that while each individual state λA λB in the Schimdt decomposition in Eq. (16) is unentangled, their
linear combination ψ , in general, is enta|niglie⊗d.|Tihiis simply means that the composite state ψ can nPot, in general,
| i | i
be written as a direct product ψ = φA φB of two states of the respective subsystems. The spectral properties
| i | i⊗| i
ofthe matrix W, i.e., the knowledgeofthe eigenvaluesλ ,λ ,...,λ , inassociationwith the Schimdt decomposition
1 2 N
in Eq. (16), provide useful information about how entangled a pure state is. For example, as mentioned in the
introduction, one useful measure of the entanglement is the von Neumann entropy, S = N λ ln(λ ).
− i=1 i i
In addition, the two extreme eigevalues, the largest λ = max(λ ,λ ,...,λ ) and the smallest λ =
max 1 2 N min
P
min(λ ,λ ,...,λ ) also provide useful information about the entanglement. Note that due to the constraint
1 2 N
N λ = 1 and the fact that all eigenvalues are nonnegative, it follows that 1/N λ 1 and 0
i=1 i ≤ max ≤ ≤
λ 1/N. Consider, for instance, the following limiting situations. Suppose that the largest eigenvalue
min
λP =≤ max(λ ,λ ,...,λ ) takes its maximum allowed value 1. Then due to the constraint N λ = 1 and
max 1 2 N i=1 i
the fact that λ 0 for all i, it follows that all the rest (N 1) eigenvalues must be identically 0. In that case, it
i
≥ − P
followsfromEq. (16) that ψ is fully unentangled. Onthe other hand,if λ =1/N (i.e., it takesits lowestallowed
max
| i
value), it follows that all the eigenvalues must have the same value, λ = 1/N for all i, again due to the constraint
i
N λ = 1. In this case, one can show that the pure state ψ is maximally entangled, as this state maximizes the
i=1 i | i
von Neumann entropy S =ln(N).
P
In this paper, we will focus on the smallest eigenvalue 0 λ 1/N. As in the case of the largest eigenvalue
min
≤ ≤
above, let us consider the two limiting situations. When λ takes its maximal allowedvalue λ =1/N, it follows
min min
again from the constraint N λ = 1 that all the eigenvalues must have the same value λ = 1/N. This will thus
i=1 i i
makethestate ψ maximally entangled. Intheoppositecase,whenλ =0takesitssmallestallowedvalue,whileit
min
| i P
doesnotprovideanyinformationonthe entanglementofthe state ψ , oneseesfromtheSchmidtdecompositionthat
| i
the dimensionofthe effective Hilbert spaceof the subsystem Agets reducedfromN to N 1. Indeed, if λ is very
min
−
closetozero,onecaneffectivelyignorethetermcontainingλ inEq. (16)andthusachieveareducedHilbertspace,
min
a process called ‘dimensional reduction’ that is often used in the compression of large data structures in computer
vision [16, 17, 18]. Thus the knowledge of λ and in particular its proximity to its upper and lower limits provide
min
informationson boththe entanglementphenomenonas wellasonthe efficiency of the dimensionalreductionprocess.
So far, our discussion is valid for an arbitrary pure state in Eq. (9) with any fixed coefficient matrix X = [x ].
i,α
Now, such a pure state will be called a random pure state if the coefficients x ’s are random variables, drawn
i,α
from an underlying probability distribution. In particular, we will consider the case when the elements of X are
independent and identically distributed random variables, real or complex, drawn from a Gaussian distribution:
Prob[X] exp βTr(X X) , where the Dyson index β = 1,2 corresponds respectively to the real and complex
∝ −2 †
h i
5
X matrices. The product W = XX is called the random Wishart matrix [19]. The joint distribution of the N
†
nonnegative eigenvalues of W is known [20]
N
PW(λ1,λ2,...,λN)∝ e−β2PNi=1λi λiβ2(1+M−N)−1 |λj −λk|β. (17)
i=1 j<k
Y Y
Note however,that in case of a random pure state ψ in Eq. (9), the eigenvalues of the matrix W =XX are not
†
| i
quite the same as that of the Wishart matrix, due to the additional constraint that Tr[ρ] = Tr[W] = 1. Thus, the
eigenvaluesofW thatappearinthe SchimdtdecompositioninEq. (16), aredistributedaccordingtothe Wishartlaw
inEq. (17), butinadditionhavetosatisfythe constraint N λ =1. Thisconstraintcanbe explicitly incorporated
i=1 i
bymultiplyingadeltafunctionδ( N λ 1)totheWishartmeasureinEq. (17). Withthisadditionaldeltafunction
i=1 i− P
multiplying the Wishart measure, the exponential term in Eq. (17) just becomes a constant and can be absorbed
P
into the overallnormalizationconstantandonearrivesatthe jpdf ofthe eigenvaluesofW mentionedinEq (1) inthe
introduction.
Given the jpdf (1), we are interested here in the distribution of the minimum eigenvalue λ . Let Q (x) =
min N,M
Prob[λ x] be the cumulative distribution of λ . The pdf of λ is simply obtained by taking the derivative,
min min min
≥
P (x) = dQ (x)/dx. Since the event λ x necessarily implies that all the eigenvalues λ x (for all
N,M N,M min i
− ≥ ≥
i = 1,2,...,N), it follows, upon using the explicit jpdf (1), that Q (x) is precisely given by the multiple integral
N,M
(with N M)
≤
N N
QN,M(x)=BM,N Zx∞···Zx∞δ i=1λi−1!j<k|λj −λk|β i=1λiβ2(M−N+1)−1dλi. (18)
X Y Y
The real technical challenge is to evaluate this multiple integral. In the next two sections, we show how to compute
this integral exactly respectively for β = 2 and β = 1, for all M = N, i.e., when the Hilbert spaces of the two
subsystems have equal dimensions. In this case, i.e., when M = N, we will denote, for simplicity of notations,
Q (x) = Q (x) for the cumulative distribution of the minimum eigenvalue and the corresponding density by
N,N N
P (x)=P (x)= dQ (x)/dx.
N,N N N
−
III. A COMPLEX RANDOM VECTOR
Thissectionisdevotedtofindingexactlythedistributionoftheminimumeigenvalueλ ortheminimumSchmidt
min
coefficient for random complex states. Let
Q (x)=Prob[λ x]=Prob[λ x,λ x,...,λ x]. (19)
N min 1 2 N
≥ ≥ ≥ ≥
Therefore
N N
Q (x)=B ∞ ∞δ λ 1 (λ λ )2 dλ (20)
N N,N i j k i
Zx ···Zx i=1 − !j<k − i=1
X Y Y
An evaluation of this multiple integral proceeds by introducing an auxiliary one defined by
N N
I(x,t)= ∞ ∞δ λ t (λ λ )2 dλ , (21)
i j k i
Zx ···Zx i=1 − !j<k − i=1
X Y Y
so that Q (x)=B I(x,1). Consider the following Laplace transform of I(x,t):
N N,N
N
∞I(x,t)e−stdt= ∞ ∞e−sPNi=1λi (λj λk)2 dλi. (22)
··· −
Z0 Zx Zx j<k i=1
Y Y
A linear shift and scaling z =s(λ x) results in
i i
−
∞I(x,t)e−stdt= e−sNsN2x ∞··· ∞e−PNi=1zi (zj −zk)2 N dzi. (23)
Z0 Z0 Z0 j<k i=1
Y Y
6
Thus the dependence on s and x just factors out of the integral. The integral happens to be one of the Selberg
integrals which can be evaluated explicitly [21] and this gives
∞I(x,t)e stdt= e−sNx N−1Γ(j+2)Γ(j+1). (24)
− sN2
Z0 j=0
Y
An inverse Laplace transform yields
I(x,t)= Nj=−01Γ(j+2)Γ(j+1) (t Nx)N2−1 Θ(t Nx). (25)
Γ(N2) − −
Q
Using the known normalization constant [10]
Γ(N2)
B = (26)
N,N N−1Γ(N j)Γ(N j+1)
j=0 − −
we finally arrive at Q
QN(x)=Prob[λmin x]=BN,NI(x,1)=(1 Nx)N2−1Θ(1 Nx). (27)
≥ − −
Subsequently, the pdf is given by
P (x)= dQN(x) =N(N2 1)(1 Nx)N2 2Θ(1 Nx). (28)
N −
− dx − − −
A plot of this pdf can be found in Fig. 1 for N =4. Thus P (x) in x [0,1/N] has the limiting behavior
N
∈
P (x) N(N2 1) as x 0
N
→ − →
= N(N2 1)(1 Nx)N2 2 as x 1/N (29)
−
− − →
Notethatintheregimewherex<<1/N,thepdfinEq. (28)becomesexponential,P (x) N(N2 1)exp[ N(N2
N
≈ − − −
1)x]. Let us also note that the distribution of the smallest eigenvalue in Eq. (27) is identical to that of the smallest
intensity component of a complex random state derived recently [22], provided one replaces N2 (in the exponent in
Eq. (27)) by N.
Moments of λ : From the explicit expression of the pdf in Eq. (28) one can easily compute all the moments of
min
λ . For the k-th moment we get
min
µ (N)= λk = ∞xkP (x)dx= Γ(k+1)Γ(N2). (30)
k h mini N NkΓ(N2+k)
Z0
In particular, for k =1, we obtain for all N
1
µ (N)= λ = , (31)
1 h mini N3
thus proving the recent conjecture by Znidaric [14] based on evaluations for small N. Putting k = 2 in Eq. (30), we
get the second moment µ = 2 . Thus the variance is given by
2 N4(N2+1)
1 N2 1
σ2 =µ (N) [µ (N)]2 = − . (32)
2 − 1 N6 N2+1
(cid:18) (cid:19)
IV. A REAL RANDOM VECTOR
While complex random vectors are “generic”, real vectors are important as well. For instance in the case when
the system has a time-reversal symmetry or any anti-unitary symmetry the eigenfunctions can be in general chosen
to be real and the relevant ensembles are the “orthogonal” ones (such as the Gaussian orthogonal ensemble and
the circular orthogonal ensemble), wherein general orthogonal transformations leave the ensemble invariant [21, 23].
The entanglement properties of real and complex random states may, in general, differ. For instance for so called
7
“single-particle” states or one-magnon states, real states have lower entanglement measured in terms of two-spin
entanglement content than the case of the complex states [24]. In general, much less is known for random real states
thanthecomplexones,althoughforinstanceseveralmany-bodyHamiltonians(sayofspins)havenaturaltime-reversal
symmetry. In this section the distribution of the minimum eigenvalue of the real case is calculated exactly.
The jpdf of the eigenvalues λ in this case (we again restrict ourselves to the case M =N) is
i
N N
1
P (λ , ,λ )=C δ λ 1 λ λ , (33)
N 1 N N,N i j k
··· i=1 − !j<k| − |i=1√λi
X Y Y
where C is the normalization constant and is known to be [10]
N,N
2N N−1 j+1 j+3
CN−,1N = πN/2Γ(N2/2) Γ 2 Γ 2 . (34)
j=0 (cid:18) (cid:19) (cid:18) (cid:19)
Y
The cumulative distribution of the smallest eigenvalue, Q (x)=Prob[λ x], is given by
N min
≥
N N
∞ ∞ 1
Q (x)=C δ λ 1 λ λ dλ . (35)
N N,N i j k i
Zx ···Zx i=1 − !j<k| − |i=1√λi
X Y Y
To evaluate this multiple integral, we proceed, as in the previous section, by defining an auxiliary integral J(x,t)
as
N N
∞ ∞ 1
J(x,t)= δ λ t λ λ dλ , (36)
i j k i
Zx ···Zx i=1 − !j<k| − |i=1√λi
X Y Y
so that Q (x)=C J(x,1).
N N,N
Taking the Laplace transform of Eq. (36) leads to
N
Z0∞J(x,t)e−stdt= (2s)1N2/2 Z2s∞x···Z2s∞xe−12PNi=1yij<k|yj −yk|i=1√1yi dyi, (37)
Y Y
where the scaled variable y = 2sλ . We next use a result due to Edelman [25] for the Wishart orthogonal ensemble
i i
whose jpdf is given by
N
1
PNW(y1,··· ,yN)=aN,Ne−21PNi=1yi √yi |yj −yk| (38)
i=1 j<k
Y Y
where the normalization constant a is
N,N
C
N,N
a = . (39)
N,N 2N2/2Γ(N2/2)
For such an ensemble Edelman [25] showed that the distribution of the smallest eigenvalue QW(z)=Prob[y z]
min
≥
is given explicitly by
QW(z)= ∞ ∞PW(y , ,y ) N dy dy = NΓ(N) ∞ e−Ny/2 U N −1, 1,y dy. (40)
··· 1 ··· N 1··· N 2N 1/2Γ(N/2) √y 2 −2 2
Zz Zz i=1 − Zz (cid:18) (cid:19)
Y
whereU(a,b;z)isthe confluenthypergeometricfunction[26]ofthesecondkindthatsatisfiesthedifferentialequation
d2U dU
z +(b z) aU =0 (41)
dz2 − dz −
with the boundary conditions
Γ(1 b)
U(a,b,0)= − , U(a,b,z )=0. (42)
Γ(1+a b) →∞
−
8
Working back we therefore obtain
Z0∞J(x,t)e−stdt= (2s)1N2/2 (cid:20)2N−1/N2ΓΓ(N(N/2))aN,N(cid:21)Z2s∞x e−√Nyy/2 U(cid:18)N2−1,−12,y2(cid:19) dy. (43)
To make further progress, it turns out to be easier to work with the probability density function rather than the
cumulative distribution Q (x),
N
dQ (x) dJ(x,1)
N
P (x)= = C . (44)
N N,N
− dx − dx
Taking the derivative of Eq. (43) with respect to x leads to
− ∞ dJd(xx,t)e−stdt=bN√1xs(eN−2Ns1x)/2U N2−1,−12,sx , (45)
Z0 − (cid:18) (cid:19)
where
NΓ(N)
b = . (46)
N Γ(N/2)a 2(N2+2N 2)/2
N,N −
The task then is to find the Laplace inverse:
dJ(x,t) b e Nsx N 1 1
− dx = √NxL−s1 s(N−2 1)/2U 2− ,−2,sx . (47)
(cid:20) − (cid:18) (cid:19)(cid:21)
First, an application of the convolution theorem leads to
L−s1 s(eN−2N1sx)/2 = N12 1 (t−Nx)(N2−3)/2Θ(t−Nx). (48)
(cid:20) − (cid:21) Γ −
2
(cid:18) (cid:19)
Second, using an integral representation of the hypergeometric function U(a,b,z) [26] namely
U(a,b,z)= 1 ∞e ztta 1(1+t)b a 1dt (49)
− − − −
Γ(a)
Z0
one obtains the following inverse:
N 1 1 x3/2
L−s1 U 2− ,−2,sx = Γ(N 1)t(N−3)/2(x+t)−(N+2)/2. (50)
(cid:20) (cid:18) (cid:19)(cid:21) 2−
Using the two inverses in Eqs. (48) and (50) and the convolution theorem, we get upon simplifying
Ls−1 s(eN−2N1sx)/2U N2−1,−12,sx = Γ N2 x13/Γ2 N 1 t−Nxt′N2−2 (x+t′)−N2+2 (t−t′−Nx)N22−3 dt′ (51)
(cid:20) − (cid:18) (cid:19)(cid:21) 2− 2− Z0
x(cid:0)−(N−1(cid:1))/2(cid:0) (cid:1) N2+N−4 N +2 N 1 N2+N 2 t Nx
= Γ N2+N 2 (t−Nx) 2 2F1 2 , 2− , 2 − ,− −x .
2 − (cid:18) (cid:19)
Here F (a,b,c,z) is the standard hyperge(cid:0)ometric fu(cid:1)nction [26], and the integral can be found in [15]. Using this
2 1
along with Eqs. (44,47) and substituting t=1, we finally get the p.d.f. of the minimum eigenvalue λ as
min
PN(x)=ANx−N/2(1 Nx)(N2+N−4)/22F1 N +2,N −1,N2+N −2, 1−Nx , 0<x 1/N (52)
− 2 2 2 − x ≤
(cid:18) (cid:19)
and P (x)=0 for x 1/N. The constant A is given by
N N
≥
NΓ(N)Γ(N2/2)
A = . (53)
N 2N 1Γ(N/2)Γ((N2+N 2)/2)
− −
9
This solves exactly for the distribution of the minimum eigenvalue of the reduced density matrices of bipartite
randomrealstates when the dimensions ofthe subspacesare equal. In the simplest possible case of realstates of two
qubits, N =2, the distribution is simply
1 2x
P (x)= − , 0<x 1/2; P (x)=0, x 1/2. (54)
2 2
x(1 x) ≤ ≥
−
p
This follows from Eq. (52) as F (2,1/2,2,x)=1/√1 x. Alternatively it almost immediately follows from the jpdf
2 1
−
in Eq. (33) as there are only two eigenvalues that sum to unity in this case, and the distribution of the one which is
less than one-half is precisely P (x). In Fig. 1, we plot the pdf P (x) of λ for N = 4, both for the complex case
2 N min
given in Eq. (28) and the real case given in Eq. (52)
In appendix-A, we work out the limiting behavior of P (x) as x 0 and x 1/N. For general N, one finds
N
→ →
√πΓ(N)Γ(N2/2)
P (x) x 1/2 as x 0
N ≈ 2N 1Γ2(N/2)Γ((N 1)/2) − →
(cid:20) − − (cid:21)
A N N/2(1 Nx)(N2+N 4)/2 as x 1/N (55)
N − −
≈ − →
Comparing this limiting behavior in the real case in Eq. (55) with that of the complex case in Eq. (29) one finds
that while in the former P (x) diverges as x 1/2 as x 0, in the latter it approaches a constant. In the other limit
N −
x 1/N, both the densities approach zero as a power→law (1 Nx)ν, but with different exponents ν = N2 2 (for
→ − −
the complex case) and ν =(N2+N 4)/2 for the real case.
−
100
complex case
real case
80
60
)
x
(
4
P
40
20
0
0 0.05 0.1
x
FIG. 1: The p.d.f PN(x) of the minimum eigenvalue λmin vs. x for N =4, for the complex and the real cases (Eqs. (28) and
(52) respecively). In the complex case, the density approaches a constant as x → 0, whereas for the real case, it diverges as
x−1/2 as x→0.
Moments of λ : One can use the explicit result for the p.d.f. P (x) of λ in Eq. (52) to calculate its k-th
min N min
10
moment
µk(N)= λmink = AN 1/Nxk−N/2(1 Nx)(N2+N−4)/22F1 N +2,N −1,N2+N −2, 1−Nx dx
h i − 2 2 2 − x
Z0 (cid:18) (cid:19)
= AN ∞y(N2+N−4)/2(N +y)−(N2+2k)/22F1 N +2,N −1,N2+N −2, y (56)
2 2 2 −
Z0 (cid:18) (cid:19)
where we made a change of variable y = N +1/x in the first line. We next use the following known integral [15]
−
∞xγ−1(x+z)−σ2F1(α,β,γ, x)dx= Γ(γ)Γ(α−γ+σ)Γ(β−γ+σ)2F1(α γ+σ,β γ+σ,α+β γ+σ,1 z)
− Γ(σ)Γ(α+β γ+σ) − − − −
Z0 −
(57)
in Eq. (56) and also the value of A from Eq. (53) to arrive at an explicit expressionfor the k-th moment (valid for
N
all N),
Γ(N +1)Γ(N2/2)Γ(k+2)Γ(k+1/2) N +3
µ (N)= F k+2,k+1/2,k+ ,1 N . (58)
k Γ(N/2)Γ(k+N2/2)Γ(k+(N +3)/2)2N 1 2 1 2 −
− (cid:18) (cid:19)
One can verify that µ (N)=1, thus ensuring the correct normalization. For the average value of λ we use k =1
0 min
and get
√πΓ(N) 3 N +5
µ (N)= λ = F 3, , ,1 N . (59)
1 h mini NΓ(N/2)Γ((N +5)/2)2N 1 2 1 2 2 −
− (cid:18) (cid:19)
Thus the expressionfor λ for arbitraryN in the real case is considerably more complicated than its counterpart
min
h i
in Eq. (31) for the complex case. One finds, from Eq. (59), that µ (N) decreases with increasing N, e.g., µ (1)=1,
1 1
µ (2)=(4 π)/8,µ (3)=(2 √3)/9etc. Inappendix-B,we showthat asymptoticallyfor largeN, µ (N) decaysas
1 1 1
− −
c πe
µ (N) ; where c=2 1 erfc(1/√2) =0.688641 (60)
1 ≈ N3 − 2 ···
(cid:20) r (cid:21)
where erfc(x) = √2π x∞e−u2du is the complementary error function. The large N result in Eq. (60) for the real
case should be compared to that of the complex case where µ (N) = 1/N3. One sees that the average value of the
R 1
minimum eigenvalue in the former case is less by a constant factor c = 0.688641 compared to the later case. In
···
Fig. 2, we plot both the exact formula for µ (N) in Eq. (59) and the asymptotic form in Eq. (60) against N.
1
V. CONCLUSION
In this paper we havecomputed the exactprobabilitydistribution ofthe minimum eigenvalueλ of anentangled
min
random (both real and complex) pure state of a bipartite system composed of two subsystems whose respective
Hilbert spaces have equal dimensions M =N. We have also computed exactly all the moments of λ for all N. As
min
a byproduct, we provethat λ =1/N3 for allN for complex matrices,a resultrecently conjectured[14]. The pdf
min
h i
of the minimum eigenvalue in the real case differs significantly from its complex counterpart.
Apartfromprovidingimportantinformationsonthenatureoftheentanglementofarandompurestateaswellason
the degree to which the dimension of the Hilbert space of a subsystem can be reduced, our result for the distribution
of the minimum eigenvalue has some relevance in the general context of extreme value statistics. This subject has
been around for a long time [27], but has seen a recent resurgence due to its many applications in diverse areas
such as engineering, economics and physical sciences [28]. If the underlying random variables are independent and
identically distributedthentherearethreepossiblelimitinguniversaldistributionsfortheextremeevents,theFr´echet,
the Gumbel and the Weibull distributions. However, much less is known when the underlying random variables are
stronglycorrelated. Insuchcases,thelimitingdistribution(forlargeN)ofthemaximumisknownexactlyonlyinvery
fewcases. Forexample,thelimitingdistributionofthelargesteigenvalueofaN N Gaussianunitaryrandommatrix
×
(GUE) is given by the celebrated Tracy-Widom law [29], which has found many recent applications [30]. Similarly,
the Tracy-Widom law also describes the limiting distribution of the largest eigenvalue of Wishart matrices [31, 32],
for random matrices with certain non-Gaussian entries [33] and the scaled height of a (1+1)-dimensional growth
models [30, 34]. The probabilities of large deviations of λ , outside the regime of the validity of the Tracy-Widom
max
law, have also been computed recently both for Gaussian [35] and Wishart matrices [18, 31]. Other examples for