PHYSICAL REVIEW A 67, 012313 共2003兲
Entanglement entropy of multipartite pure states
Sergei Bravyi*
Landau Institute for Theoretical Physics, Kosygina Street 2, Moscow 117940, Russia
共Received 14 May 2002; published 22 January 2003兲
Consider a system consisting of n d-dimensional quantum particles and an arbitrary pure state 兩 ⌿ 典 of the
whole system. Suppose we simultaneously perform complete von Neumann measurements on each particle.
The Shannon entropy of the outcomes’ joint probability distribution is a functional of the state 兩 ⌿ 典 and of n
measurements chosen for each particle. Denote S 关 ⌿ 兴 the minimum of this entropy over all choices of the
measurements. We show that S 关 ⌿ 兴 coincides with the entropy of entanglement for bipartite states. We compute S 关 ⌿ 兴 for some special multipartite states: the hexacode state 兩 H 典 (n⫽6, d⫽2) and the determinant states
兩 Detn 典 (d⫽n). The computation yields S 关 H 兴 ⫽4 log 2 and S 关 Detn 兴 ⫽log(n!). Counterparts of the determinant
state defined for d⬍n are also considered.
DOI: 10.1103/PhysRevA.67.012313
PACS number共s兲: 03.67.⫺a, 03.65.⫺w
I. INTRODUCTION AND MAIN RESULTS
Quantum information theory has many interesting features which have no classical analog. One of them is entanglement or quantum correlations. It has been the object of
intensive study for the past years because entanglement
makes it possible to develop effective algorithms for many
computational problems and opens new perspectives in communication and cryptography, see, for example, Ref. 关1兴, and
references therein.
However, at the present moment there is no canonical
measure of entanglement and the question how to quantify
entanglement contained in a given multipartite quantum state
remains open. Nevertheless, any functional which pretends
to be a measure of entanglement should meet certain natural
requirements, in particular it should be monotonic under local quantum operations and classical communication 关2兴.
An important example of such a functional is the entropy
of entanglement 关3兴. For a pure state 兩 ⌿ 典 of any bipartite
system the entropy of entanglement E 关 ⌿ 兴 is defined as
E 关 ⌿ 兴 ⫽⫺
兺i p i log2 p i ,
共1兲
where p i ’s denote the eigenvalues of the reduced density
matrices ␳ 1 ⫽tr2 ( 兩 ⌿ 典具 ⌿ 兩 ) and ␳ 2 ⫽tr1 ( 兩 ⌿ 典具 ⌿ 兩 ) 共they share
the same set of eigenvalues兲. This particular measure is distinguished because in the asymptotic limit 共when one takes a
large number of copies of the given shared state兲 any monotonic functional up to trivial rescaling coincides with the
entropy of entanglement, see Ref. 关4兴. Unfortunately, if a
system is divided into three or more local parts, the entropy
of entanglement is not defined.
The functional E 关 ⌿ 兴 has a very simple physical sense.
Assume that each of the two parties sharing the state 兩 ⌿ 典
performs a complete von Neumann measurement on his part
of the system. Such joint measurement is a complete measurement on the whole system. Its outcome is a random variable whose probability distribution depends upon the pair of
*Email address: serg@itp.ac.ru
1050-2947/2003/67共1兲/012313共7兲/$20.00
complete von Neumann measurements chosen by each of the
two parties. This choice is equivalent to a choice of the orthonormal basis in each party’s Hilbert space of states. One
can ask: what bases should be chosen by each party to minimize the entropy of the outcomes joint probability distribution and what the minimal value of this entropy is? If the
state 兩 ⌿ 典 is factorizable, i.e., 兩 ⌿ 典 ⫽ 兩 ⌿ 1 典 丢 兩 ⌿ 2 典 , then the
answer is trivial: the ith party should arbitrarily complement
兩 ⌿ i 典 to a complete basis and this choice yields zero entropy
because the measurement outcome will be ‘‘(⌿ 1 ,⌿ 2 )’’ with
probability 1. If 兩 ⌿ 典 is an entangled state, one can easily
show 共see Sec. II兲 that the ith party should perform his measurement in the basis diagonalizing the density matrix ␳ i 共the
Schmidt basis兲 and the minimal entropy of the outcomes
coincides with E 关 ⌿ 兴 . Thus we can interpret the entropy of
entanglement E 关 ⌿ 兴 as the minimum of the outcomes entropy over all choices of local complete von Neumann measurements.
If we consider the entropy of entanglement from this
point of view, its definition can be naturally generalized to
arbitrary multipartite states. Assume that a system consists of
n d-dimensional quantum particles distributed between n remote parties and let 兩 ⌿ 典 苸(C d ) 丢 n be an arbitrary pure state
of the whole system. Denote Bi the orthonormal basis in C d
chosen by the ith party and let 兩 Bi ( j) 典 苸C d be the jth basis
vector in the basis Bi , where i苸 关 1,n 兴 , j苸 关 1,d 兴 . Now let us
define the functional S 关 ⌿ 兴 according to
S关⌿兴⫽
inf
B1 , . . . ,Bn
S 关 ⌿,B1 , . . . ,Bn 兴 ,
S 关 ⌿,B1 , . . . ,Bn 兴 ⫽⫺
兺j p 共 j 兲 log2 p 共 j 兲 ,
共2兲
p 共 j 兲 ⬅ p 共 j 1 , . . . , j n 兲 ⫽ 兩 具 ⌿ 兩 B1 共 j 1 兲 , . . . ,Bn 共 j n 兲 典 兩 2
共the sum involving multi-index j is a sum over all possible
j 1 , . . . , j n 苸 关 1,d 兴 ). It tells us to what extent the parties may
decrease the entropy of the outcomes distribution by varying
the bases, which they use for local measurements on the state
兩 ⌿ 典 . Thus we can consider this quantity as a degree of uncertainty characterizing nonlocal quantum correlations be-
67 012313-1
©2003 The American Physical Society
PHYSICAL REVIEW A 67, 012313 共2003兲
SERGEI BRAVYI
tween the parties. As was said above, S 关 ⌿ 兴 ⫽E 关 ⌿ 兴 for bipartite states so we will call S 关 ⌿ 兴 the entropy of
entanglement. Its properties immediately following from the
definition are S 关 ⌿ 兴 ⫽0 if and only if 兩 ⌿ 典 is factorizable;
S 关 ⌿ 丢 ⌽ 兴 ⫽S 关 ⌿ 兴 ⫹S 关 ⌽ 兴 共here we mean that 兩 ⌿ 典 and 兩 ⌽ 典
are the states shared by n and m parties, while 兩 ⌿ 丢 ⌽ 典 is
shared by n⫹m parties兲; S 关 ⌿ 兴 is a continuous functional of
the state 兩 ⌿ 典 . For any pure state 兩 ⌿ 典 苸(C d ) 丢 n we also have
the inequality S 关 ⌿ 兴 ⭐n log2d. It is easy to prove that the
upper bound n log2d is not achievable unless the trivial case
d⫽1 is concerned. Indeed, assume that S 关 ⌿ 兴 ⫽n log2d.
Then for any choice of the bases Bi in Eq. 共2兲 we have
p( j 1 , . . . , j n )⬅1/d n and thus the probability distribution of
the outcome j 1 conditioned on the outcomes j 2 , . . . , j n is
absolutely uniform, i.e., p( j 1 兩 j 2 , . . . , j n )⬅1/d. On the other
hand, after the parties 2,3, . . . ,n have performed their measurements, the first party becomes completely disentangled
from the others and is left in some pure state 兩 ␺ 1 典
⬅ 兩 ␺ 1 ( j 2 , . . . , j n ) 典 苸C d , which depends upon the measurement outcomes j 2 , . . . , j n . Thus the state 兩 ␺ 1 典 produces absolutely uniform distribution of the outcomes whatever basis
B1 in the Hilbert space C d is chosen. Obviously a pure state
meeting this criteria does not exist and thus the assumption
S 关 ⌿ 兴 ⫽n log2d leads to a contradiction.
Unfortunately, entropy of entanglement computation in
the multipartite case n⬎2 becomes a difficult problem.
Given some generic pure state, this problem is very likely to
be solvable only numerically 关note that the number of parameters to be optimized in the definition 共2兲 grows as O(nd 2 )].
It is relatively easy to get upper bounds on S 关 ⌿ 兴 — one just
needs to choose tentatively some basis for each party. As the
lower bound on S 关 ⌿ 兴 one can take the von Neumann entropy of the mixed state of any group of parties 共see Sec. III兲.
However, this lower bound is generally too weak. It can be
improved only if the state has some special symmetries.
In this paper, we consider two examples of such states:
the determinant states 兩 Detn 典 苸(C n ) 丢 n , see Ref. 关5兴, and the
six-qubit hexacode state 兩 H 典 , see Ref. 关6兴. The determinant
state 兩 Detn 典 is invariant under unitary transformations U 丢 n ,
where U苸SU(n) is an arbitrary local unitary operator. The
hexacode state is in some sense a ‘‘maximally uniform’’ pure
state—if we divide the six qubits into two groups of three
qubits by an arbitrary way, then each group’s mixed state
will be absolutely uniform. Due to these special properties,
the entropy of entanglement of the states 兩 Detn 典 and 兩 H 典 can
be exactly computed.
Another reason of our interest in these particular states is
that their entropy of entanglement is rather close to the upper
bound n log2d. For the hexacode state (n⫽6,d⫽2) the computation yields S 关 H 兴 ⫽4, see Sec. VI, while for the determinant state (n⫽d) we get S 关 Detn 兴 ⫽log2(n!), see Sec. IV. For
large n we can use the approximation S 关 Detn 兴 ⬇ 关 1
⫺1/ln(n)兴n log2n. It means that the determinant states asymptotically saturate the upper bound for the normalized entropy: limn→⬁ S 关 Detn 兴 /n log2n⫽1. In Sec. V, we construct
the generalized determinant states 兩 Detn,d 典 苸(C d ) 丢 n defined
for n⫽pd p with p being an arbitrary integer and show that
S 关 Detn,d 兴 ⫽log2关(dp)!兴. For fixed d and large p we can write
S 关 Detn,d 兴 /(n log2d)⬇关1⫺1/(p lnd)兴, which again asymptotically approaches unity.
We make concluding remarks and discuss some open
questions concerning the generalized entropy of entanglement in Sec. VII. The most interesting question concerns the
stability of the definition 共2兲 under the extension of each
party space of states.
II. BIPARTITE SYSTEM
It is known that up to local unitary operators any state
兩 ⌿ 典 苸C d 丢 C d of a bipartite system is specified by its
d
p i ⫽1. Being
Schmidt coefficients 兵 p i 其 i⫽1, . . . ,d , p i ⭓0, 兺 i⫽1
invariant under local unitaries S 关 ⌿ 兴 is a functional of the
Schmidt coefficients only. Thus, one suffices to compute
S 关 ⌿ 兴 only for the special states
d
兩⌿典⫽
兺 冑p i兩 i,i 典 ,
i⫽1
共3兲
where 兵 兩 i 典 苸C d 其 i⫽1, . . . ,d is the standard basis of C d . Choosing tentatively the standard basis for both parties, i.e.,
兩 B1 (i) 典 ⫽ 兩 B2 (i) 典 ⫽ 兩 i 典 , i苸 关 1,d 兴 , see Eq. 共2兲, we get
d
p i log2pi⫽E关⌿兴 which implies that
S 关 ⌿,B1 ,B2 兴 ⫽⫺ 兺 i⫽1
S 关 ⌿ 兴 ⭐E 关 ⌿ 兴 . We can also prove that E 关 ⌿ 兴 is simultaneously the lower bound for S 关 ⌿ 兴 . Indeed, let B1* , B2* be
the optimal choice of the bases 共it means that S 关 ⌿ 兴
⫽S 关 ⌿,B1* ,B2* 兴 ). Consider the density matrix ␳ 1 of the first
d
party only: ␳ 1 ⫽tr2 ( 兩 ⌿ 典具 ⌿ 兩 )⫽ 兺 i⫽1
p i 兩 i 典具 i 兩 . Denote p 1 (i)
d
*
(i)
⫽
兺
p
兩
B
(i)
兩 j 典 兩 2 the distribution of
⫽ 具 B1* (i) 兩 ␳ 1 兩 B*
典
具
j
1
j⫽1
1
the first-party outcomes in the optimal basis. Because the
entropy of the partial distribution cannot exceed the entropy
of the joint distribution, we conclude that
d
S 关 ⌿ 兴 ⭓⫺
兺
i⫽1
p 1 共 i 兲 log2 关 p 1 共 i 兲兴 .
共4兲
Using concavity of the function ⫺x log2x and the normalization 兺 dj⫽1 兩 具 B1* (i) 兩 j 典 兩 2 ⫽1 we get the next estimate,
d
⫺p 1 共 i 兲 log2 p 1 共 i 兲 ⭓⫺
兺 兩 具 B1*共 i 兲 兩 j 典 兩 2共 p j log2 p j 兲 .
j⫽1
共5兲
The summation over i can be carried out taking into account
d
兩 具 B1* (i) 兩 j 典 兩 2 ⫽1. Thus from Eqs. 共4兲
the normalization 兺 i⫽1
and 共5兲 we can infer that S 关 ⌿ 兴 ⭓E 关 ⌿ 兴 and consequently
S 关 ⌿ 兴 ⫽E 关 ⌿ 兴 .
共6兲
III. CONNECTION WITH THE VON NEUMANN
ENTROPY
Consider a system consisting of n d-dimensional quantum
particles distributed between n remote parties and let 兩 ⌿ 典
苸(C d ) 丢 n be an arbitrary pure state of the whole system. Let
B1* , . . . ,Bn* be the optimal choice of bases minimizing the
entropy. Let us choose a group X傺 兵 1,2, . . . ,n 其 of k parties,
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PHYSICAL REVIEW A 67, 012313 共2003兲
ENTANGLEMENT ENTROPY OF MULTIPARTITE PURE STATES
for example X⫽ 兵 1,2, . . . ,k 其 . The chosen group of parties
shares the mixed state ␳ x ⫽tr j苸” X ( 兩 ⌿ 典具 ⌿ 兩 ). Denote
p x (i 1 , . . . ,i k )⬅p x (i) the optimal distribution of the outcomes observed by the parties from X:
p x 共 i 兲 ⫽ 具 B1* 共 i 1 兲 , . . . ,Bk* 共 i k 兲 兩 ␳ x 兩 B1* 共 i 1 兲 , . . . ,Bk* 共 i k 兲 典 .
共7兲
Let S 关 p x (i) 兴 be the entropy of the distribution 共7兲. By repeating the arguments presented in Sec. II, we can show that
⫺tr共 ␳ x log2 ␳ x 兲 ⭐S 关 p x 共 i 兲兴 ⭐S 关 ⌿ 兴 .
共8兲
Thus the von Neumann entropy of the mixed state ␳ x can
serve as the lower bound on the entropy of entanglement. Of
course the group of parties X can be chosen by an arbitrary
way.
Note that the density matrix of the parties that were not
selected to X has the same von Neumann entropy as ␳ x . It
means that one suffices to consider the groups of k⭐n/2
parties and the best lower estimate on S 关 ⌿ 兴 which we can
hope to achieve is (n/2)log2d.
As a good example consider the three-qubit GreenbergerHorne-Zeilinger 共GHZ兲 state 兩 N GHZ典 ⫽2 ⫺1/2( 兩 0,0,0 典
⫹ 兩 1,1,1 典 ), d⫽2, n⫽3. The density matrix of the first qubit
is absolutely uniform: ␳ 1 ⫽(1/2)1̂. Thus S 关 N GHZ兴 ⭓
⫺tr( ␳ 1 log2␳1)⫽1. On the other hand, we can choose the tentative bases B1 ⫽B2 ⫽B3 ⫽ 兵 兩 0 典 , 兩 1 典 其 which provide us with
the upper estimate S 关 N GHZ兴 ⭐S 关 N GHZ ,B1 ,B2 ,B2 兴 ⫽1. We
conclude that S 关 N GHZ兴 ⫽1.
IV. THE DETERMINANT STATE
Let us consider a multipartite system with d⫽n. Choose
the standard basis 兵 兩 i 典 苸C n 其 i⫽1, . . . ,n in each copy of C n and
consider the state 兩 Detn 典 苸(C n ) 丢 n defined as
兩 Detn 典 ⫽ 共 n! 兲 ⫺1/2
兺
i 1 , . . . ,i n
⑀ i 1 , . . . ,i n 兩 i 1 , . . . ,i n 典 ,
共9兲
where ⑀ i 1 , . . . ,i n is the completely antisymmetric tensor of the
rank n and the sum is over all i 1 , . . . ,i n 苸 关 1,n 兴 . This state
was used in Ref. 关5兴 to study the limitations on pairwise
entanglement in the multipartite systems. We will call the
family 兵 兩 Detn 典 其 n the determinant states. Note that 兩 Det2 典 is
the familiar singlet state 2 ⫺1/2( 兩 1,2典 ⫺ 兩 2,1典 ). The purpose of
this section is to prove the formula
S 关 Detn 兴 ⫽log2 共 n! 兲 .
兩 具 Detn 兩 ␾ 1 , . . . , ␾ n 典 兩 2 ⫽ 共 n! 兲 ⫺1
兩 具 Detn 兩 ␾ 1 , . . . , ␾ n⫺1 ,n 典 兩 2 ⫽
共11兲
兩 ␾ i 典 苸C n ,
具 ␾ i 兩 ␾ i 典 ⫽1,i苸[1,n].
共we employ a standard designation 兩 ␾ 1 , . . . , ␾ n 典 ⬅ 兩 ␾ 1 典
丢 ••• 丢 兩 ␾ n 典 ). It tells us that the projection of 兩 Detn 典 on any
factorizable state has the norm at most (n!) ⫺1 . To prove Eq.
共11兲 we first note that the state 兩 Detn 典 is a SU(n) singlet, i.e.,
冏
1
具 Detn⫺1 兩 ␾ 1 , . . . , ␾ n⫺1 典 兩 2 .
n
共12兲
Here, we have denoted 兩 Detn⫺1 典 the embedding of the true
determinant state 兩 Detn⫺1 典 苸(C n⫺1 ) 丢 (n⫺1) into the space
(C n ) 丢 (n⫺1) 共the space C n⫺1 is embedded into C n by adding a
zero nth component to all vectors兲. Although in Eq. 共12兲
兩 ␾ i 典 苸C n , the right-hand side of Eq. 共12兲 obviously achieves
the maximum when all the states 兩 ␾ 1 典 , . . . , 兩 ␾ n⫺1 典 have a
zero nth component. It implies that
兩 具 Detn 兩 ␾ 1 , . . . , ␾ n 典 兩 2
sup
兩 ␾ i 典 苸C n ,
具 ␾ i 兩 ␾ i 典 ⫽1i苸[1,n]
⫽
1
n
sup
兩 具 Detn⫺1 兩 ␾ 1 , . . . , ␾ n⫺1 典 兩 2 ,
苸C n⫺1 ,
兩␾i典
具 ␾ i 兩 ␾ i 典 ⫽1,i苸[1,n⫺1]
which by induction leads to the equality 共11兲.
Now let us explain why Eq. 共11兲 implies Eq. 共10兲. Suppose that B1* , . . . ,Bn* is the optimal choice of bases, i.e.,
S 关 Detn 兴 ⫽S 关 Detn ,B1* , . . . ,Bn* 兴 .
Let
p * (i 1 , . . . ,i n )
⫽ 兩 具 B1* (i 1 ), . . . ,Bn* (i n ) 兩 Detn 典 兩 2 be the optimal distribution.
According to Eq. 共11兲 p * (i 1 , . . . ,i n )⭐(n!) ⫺1 for any outcomes i 1 , . . . ,i n and thus S 关 Detn 兴 ⭓log2(n!). On the other
hand, we can tentatively suggest all parties to perform the
measurements in the standard basis, i.e., 兩 Bi ( j) 典 ⫽ 兩 j 典 , i, j
苸 关 1,n 兴 . Then S 关 Detn ,B1 , . . . ,Bn 兴 ⫽log2(n!) which tells us
that S 关 Detn 兴 ⭐log2(n!) and thus that S 关 Detn 兴 ⫽log2(n!).
V. THE GENERALIZED DETERMINANT STATE
If the number of parties is n⫽ pd p for some integer p then
p
p
the space (C d ) 丢 n can be identified with (C d ) 丢 d and thus
p
p
the determinant state 兩 Detd p 典 苸(C d ) 丢 d has its counterpart
d 丢n
in (C ) . This simple observation allows to construct the
state 兩 Detn,d 典 苸(C d ) 丢 n , n⫽pd p such that
共10兲
This result is a simple consequence of the following property
of the determinant states:
sup
for any V苸SU(n) we have V 丢 n 兩 Detn 典 ⫽ 兩 Detn 典 . Therefore, if
the projection of 兩 Detn 典 on the state 兩 ␾ 1 , . . . , ␾ n 典 is the
highest one then the projection of 兩 Detn 典 on the state
兩 V ␾ 1 , . . . ,V ␾ n 典 is also the highest one. So while looking
for the maximum in Eq. 共11兲 we can fix one of 兩 ␾ i 典 , e.g., put
兩 ␾ n 典 ⫽ 兩 n 典 . But according to the definition 共9兲 we have
S 关 Detn,d 兴 ⫽S 关 Detd p 兴 ⫽log2 关共 d p 兲 ! 兴 ,
n⫽pd p .
共13兲
Note that 兩 Detn,d 典 and 兩 Detd p 典 formally represent one and the
same state but there are two ways in which this state is distributed between the local parties. Correspondently,
S 关 Detn,d 兴 might be greater than S 关 Detd p 兴 because in the first
case we have less freedom in the choice of bases to minimize
the entropy.
Let us explain the construction of the state 兩 Detn,d 典 and
derive Eq. 共13兲 taking as example the qubits, i.e., d⫽2. Consider any one-to-one map ␸ which maps the integers in the
interval 关 1,2p 兴 to the binary strings of the length p, e.g.,
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PHYSICAL REVIEW A 67, 012313 共2003兲
SERGEI BRAVYI
TABLE I. The normalized entropy of entanglement for the
n-qubit determinant states (n⫽p2 p ).
p
n
S 关 Detn,2兴 /n
1
2
3
4
5
10
2
8
24
64
160
10240
0.50
0.57
0.64
0.69
0.74
0.86
FIG. 1. The graph G used in the definition of the hexacode state.
␸ 共 1 兲 ⫽ 共 0,0, . . . ,0,0 兲 ,
certain maximal self-dual linear subspace of GF(4) 6 , see
Ref. 关6兴, p. 30 for details. In this section, we present the
alternative and more explicit definition of this state only
briefly discussing its connection with quantum codes. We
also prove the equality
␸ 共 2 兲 ⫽ 共 0,0, . . . ,0,1 兲 ,
S 关 H 兴 ⫽4,
•••
共14兲
␸ 共 2 p ⫺1 兲 ⫽ 共 1,1, . . . ,1,0 兲 ,
␸ 共 2 p 兲 ⫽ 共 1,1, . . . ,1,1 兲 .
Define the state 兩 Detn,2典 苸(C 2 ) 丢 n , n⫽p2 p as
兩 Detn,2典 ⬃
兺
i 1 , . . . ,i 2 p
⑀ i 1 , . . . ,i 2 p兩 ␸ 共 i 1 兲 , . . . , ␸ 共 i 2 p 兲 典 , 共15兲
mentioned in Sec. I.
Consider the graph G⫽(V,E) shown on Fig. 1 with the
set of vertices V⫽ 兵 1,2,3,4,5,6其 and the set of edges E
⫽ 兵 (12),(13),(14), . . . ,(56) 其 .
We associate a qubit with the each vertex i苸V. Let A i j be
6⫻6 adjacency matrix of G, i.e., A i j ⫽1 if (i j)苸E and A i j
⫽0 if (i j)苸
” E, A i j ⫽A ji . The diagonal elements A ii are assumed to be zeros.
The six-qubit hexacode state 兩 H 典 苸(C 2 ) 丢 6 is defined as
follows:
where the sum is over all i 1 , . . . ,i 2 p 苸 关 1,2p 兴 and we omit
the normalizing factor 关 (2 p )! 兴 ⫺1/2. It is written here in the
standard qubit basis 兵 兩 0 典 , 兩 1 典 其 . While computing S 关 Detn,2兴
we minimize over all the choices of p2 p one-qubit bases, see
Eq. 共2兲. But any such choice is also a special choice of 2 p
p
bases in the 2 p copies of C 2 . Therefore, we can say that
S 关 Detn,2兴 ⭓S 关 Det2 p 兴 ⫽log2 关共 2 p 兲 ! 兴 .
共16兲
On the other hand, if we tentatively measure each of p2 p
qubits in the basis 兵 兩 0 典 , 兩 1 典 其 the entropy of the outcomes’
distribution will be exactly log2关(2p)!兴, see Eq. 共15兲. Therefore, S 关 Detn,2兴 ⭐log2关(2p)!兴 and thus the equality 共13兲 is
proven 共the proof for arbitrary d copies, the proof for d
⫽2).
Let us consider the determinant states 兩 Detn,2典 corresponding to n-qubit systems. In Table I, we present the normalized entropy of entanglement S 关 Detn,2兴 /n and the number of qubits n for p⫽1, . . . ,5 and p⫽10. We see that the
normalized entropy of these states is sufficiently far from
unity for small values of n. Thus for small n’s, pure states
with the entropy close to the upper bound should be constructed using some other ideas. In the following section, we
will describe the six-qubit hexacode state 兩 H 典 such that
S 关 H 兴 ⫽4. Note that its normalized entropy is 2/3 which exceeds the normalized entropy of the determinant states for
p⭐3.
VI. THE HEXACODE STATE
The hexacode state was originally described in the context
of quantum error correcting codes. It was associated with
共17兲
兩 H 典 ⫽ 共 2 6 兲 ⫺1/2
兺
x苸B 6
a共 x 兲⫽
共 ⫺1 兲 a(x) 兩 x 典 ,
共18兲
A i jx ix j ,
兺
i⬍ j
where B 6 denotes the set of all binary strings of the length 6
and 兩 x 典 ⫽ 兩 x 1 , . . . ,x 6 典 .
Note that 兩 H 典 can also be defined in terms of stabilizer
operators. Denote ␴ ␣i the Pauli matrix ␴ ␣ acting on the ith
qubit. To each vertex of the graph we can assign the operator
X i ⫽ ␴ xi
兿
j:(i j)苸E
␴ zj ,
i苸V.
共19兲
They commute with each other and stabilize the state 兩 H 典 ,
i.e., X i 兩 H 典 ⫽ 兩 H 典 . The six operators X i generate the group S
of all stabilizers, 兩 S 兩 ⫽2 6 . Each stabilizer from S is a tensor
product of several Pauli matrices 共probably with ‘‘-’’ sign兲.
As was shown in Ref. 关6兴, any nontrivial stabilizer from S is
a tensor product of at least four Pauli matrices so that 兩 H 典 is
an additive quantum code coding 0 qubits into 6 qubits with
the minimal stabilizer weight 4. Of course this quantum code
cannot be used to protect quantum information from the errors. It is just a symplectic state with some special properties.
Note, however, that if some symplectic state has the minimal
stabilizer weight d and if any 关 (d⫺1)/2兴 or less qubits were
decohered then the syndrom measurement allows to determine the positions of the decohered qubits.
The proof of the equality 共17兲 consists of three steps. The
first step is to prove that S 关 H 兴 ⭐4. On the second step, we
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ENTANGLEMENT ENTROPY OF MULTIPARTITE PURE STATES
establish a remarkable symmetry of the state 兩 H 典 which is
used in the third step to prove that S 关 H 兴 ⭓4.
共1兲 Let us choose the tentative bases Bi ⫽ 兵 兩 0 典 , 兩 1 典 其 for the
qubits i⫽2,3,4,5 and B1 ⫽B6 ⫽ 兵 兩 ⫹ 典 , 兩 ⫺ 典 其 , where 兩 ⫾ 典
⫽2 ⫺1/2( 兩 0 典 ⫾ 兩 1 典 ). Simple calculations show that the distribution
of
the
outcomes
p(x)⬅ p(x 1 , . . . ,x 6 )
⫽ 兩 具 B1 (x 1 ), . . . ,B6 (x 6 ) 兩 H 典 兩 2 measured in these bases is the
following:
p共 x 兲⫽
再
再
x 1 ⫽x 2 ⫹x 3 ⫹x 4 共 mod 2 兲 ,
1
if
16
x 6 ⫽x 3 ⫹x 4 ⫹x 5 共 mod 2 兲 ,
0
共20兲
otherwise.
Thus S 关 H,B1 , . . . ,B6 兴 ⫽4 and consequently S 关 H 兴 ⭐4.
共2兲 Suppose that the vertices of G are colored by black
and white colors such that there are three black and three
white vertices. Denote B and W the subsets of black and
white vertices, B艛W⫽V. Consider the three-qubit density
matrix ␳ w describing the state of the white qubits only: ␳ w
⫽trB ( 兩 H 典具 H 兩 ). We claim that for any partition V⫽B艛W,
␳ w ⫽2 ⫺3 1̂ where 1̂ is the unital matrix. In other words, the
mixed state of any triple of the qubits is absolutely uniform.
To verify this property fix some partition and renumber the
vertices of the graph to make W⫽ 兵 1,2,3 其 and B⫽ 兵 4,5,6 其 .
The adjacency matrix A can be split into four 3⫻3 blocks:
A⫽
冉
A ww
A wb
A bw
A bb
冊
共21兲
,
which are the adjacency matrices between white and white,
white and black, black and white, black and black vertices
关 A wb ⫽(A bw ) T 兴 . The density matrix ␳ w depends only upon
A ww and A bw . Simple calculations yield
具 x 兩 ␳ w 兩 y 典 ⫽ 共 ⫺1 兲 1/2(x
⫻2 ⫺6
T A ww x⫹y T A ww y)
兺
z苸B
3
共 ⫺1 兲 z
T A bw (x⫹y)
,
共22兲
where x⬅(x 1 ,x 2 ,x 3 ), y⬅(y 1 ,y 2 ,y 3 ), z⬅(z 1 ,z 2 ,z 3 ) and we
treat A ww , A bw as 3⫻3 matrices over binary field acting on
binary vectors. Observe that the sum over z is zero if
A bw (x⫹y)⫽(0,0,0) and is equal to 2 3 if A bw (x⫹y)
⫽(0,0,0). One can explicitly verify that the matrix A bw is
nondegenerate over the binary field 关7兴 for any partition V
⫽B艛W 共note that due to the symmetry of the graph G there
are only three nonequivalent partitions, for example, B
⫽ 兵 1,2,3 其 , B⫽ 兵 1,2,6 其 , B⫽ 兵 1,2,4 其 ). It means that A bw (x
⫹y)⫽(0,0,0) only if x⫽y. Therefore 具 x 兩 ␳ w 兩 y 典 ⫽0 if x⫽y
and 具 x 兩 ␳ w 兩 x 典 ⫽2 ⫺3 , i.e., ␳ w ⫽2 ⫺3 1̂, see also Ref. 关8兴.
共3兲 Denote B*
i the optimal basis for the ith qubit such that
S 关 H 兴 ⫽S 关 H,B1* , . . . ,B6* 兴 .
Let
p * (x 1 , . . . ,x 6 )
2
(x
)
兩
H
兩
be
the
optimal
probability
⫽ 兩 具 B1* (x 1 ), . . . ,B*
典
6
6
distribution. Consider any partition V⫽B艛W. The probability distribution of the three outcomes 兵 x i :i苸W 其 measured at
the white vertices only is
兺
x i :i苸B
p * 共 x 1 , . . . ,x 6 兲 ⫽ 具 丢 i苸W B*
i 共 x i 兲 兩 ␳ w 兩 丢 i苸W B*
i 共 x i 兲典
1
⫽ ,
8
共23兲
regardless of configuration 兵 x i :i苸W 其 . In other words, the
optimal distribution has a nice property: the partial distribution of any triple of the bits is absolutely uniform, see Ref.
关9兴. Call such property of a distribution as 3-uniformity. An
example of a 3-uniform distribution is the absolutely uniform
distribution. There are also 3-uniform distributions which are
not absolutely uniform, e.g., the distribution 共20兲. Denote P 36
the set of all 3-uniform distributions of six bits. In the Appendix, we show that
inf S 关 p 共 x 兲兴 ⫽4,
共24兲
3
p(x)苸 P 6
where S 关 p(x) 兴 is the entropy of the probability distribution
p(x). We know that p * (x 1 , . . . ,x 6 )苸 P 36 and thus we have
S 关 H 兴 ⫽S 关 p * (x 1 , . . . ,x 6 ) 兴 ⭓4. It completes the proof of Eq.
共17兲.
The definition like Eq. 共18兲 can be used to assign a state
兩 G 典 苸(C 2 ) 丢 兩 V 兩 to any unoriented graph G⫽(V,E). Reasoning as above one can show that if 兩 V 兩 ⫽2m and the adjacency
matrix A bw is nondegenerate over the binary field for any
partition V⫽B艛W, 兩 B 兩 ⫽ 兩 W 兩 ⫽m, then in the state 兩 G 典 the
density matrix of any m qubits is absolutely uniform. Such
state 兩 G 典 can be called a maximally uniform pure state because any subset of the qubits which is not forbidden by the
Schmidt constraint to have the absolutely uniform density
matrix does have the absolutely uniform density matrix. This
property of a quantum state is interesting by itself. Surprisingly for m⭐15 共i.e., for 兩 V 兩 ⭐30), appropriate graphs exist
only if m⫽3 共e.g., the graph shown on Fig. 1兲 and if m⫽1
关e.g., V⫽ 兵 1,2其 and E⫽ 兵 (12) 其 ]. It follows from the bounds
on additive quantum codes. Indeed, consider a state 兩 G 典 assigned to such graph. Like as the hexacode state we can
specify 兩 G 典 by the stabilizer operators 共19兲 which generate
the whole group S of stabilizers of the order 2 2m . Any stabilizer X苸S is a tensor product of several Pauli matrices
共possibly with a sign ‘‘-’’兲 and X 兩 G 典 ⫽ 兩 G 典 . But for any operator Y acting on m or less qubits we have 具 G 兩 Y 兩 G 典
⫽2 ⫺m tr(Y ) because the density matrix of any m qubits is
proportional to 1̂. Thus any stabilizer X苸S acts on m⫹1
qubits at least. It means that 兩 G 典 is an additive quantum code
coding 0 qubits into 2m qubits with the minimal stabilizer
weight m⫹1 or greater. The results of Ref. 关6兴 imply that for
m⭐15 such codes exist only for m⫽1,3.
The tensor powers of the hexacode state 兩 H 丢 k 典 have the
highest 共to our knowledge兲 entropy of entanglement for sufficiently small k. For example, the state 兩 H 丢 4 典 has the entropy of entanglement greater than the determinant state
兩 Det24,2典 if one takes 24 qubits, see Table I. However, we do
not expect that 兩 H 典 has the maximal entropy of entanglement
among all six-qubit states.
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SERGEI BRAVYI
VII. CONCLUSIONS
Although the generalization of the entropy of entanglement to a multipartite case suggested in the present work
looks rather natural, one faces a lot of difficulties while trying to compute the entropy of entanglement for a generic
state. A progress can be achieved only if the state has some
special properties or some symmetry.
For fixed n and d the maximum S * (n,d) of the entropy of
entanglement over all states from (C d ) 丢 n is rather close to
the upper bound n log2d such that the ratio S * (n,d)/n log2d
approaches unity for fixed d and sufficiently large n.
There is also one subtle point in the definition 共2兲 concerning its stability under extension of each party’s space of
states. Assume that each party sharing the state 兩 ⌿ 典
苸(C d ) 丢 n adds a new local degree of freedom to his part of
the system thus extending his space of states to C D , D⬎d.
The original space C d is somehow embedded into the extended space C D and the original state 兩 ⌿ 典 now is a vector
from (C D ) 丢 n . There are two ways to compute the entropy of
entanglement for the state 兩 ⌿ 典 : the parties may perform
complete von Neumann measurements either in the original
space C d or in the extended space C D . If the entropy of
entanglement is a well defined functional of a quantum state
then both computation procedures should yield the same
value of the entropy. If S 关 ⌿ 兴 is indeed invariant under such
local extensions we will call 兩 ⌿ 典 a stable state. In the bipartite case any state is stable because S 关 ⌿ 兴 is an invariant
functional of the single party’s density matrix, see Eqs. 共1兲
and 共6兲. The determinant state 兩 Detn 典 is also stable for any n.
Indeed, the formula 共11兲 which guarantees the equality
S 关 Detn 兴 ⫽log2(n!) remains valid even if we allow the states
兩 ␾ i 典 to be chosen from the extended space: the maximum is
obviously achieved when all 兩 ␾ i 典 ’s belong to the original
space C n . However, we were not able to prove that an arbitrary multipartite state is stable, so this question remains
open.
Also it would be interesting to check whether the generalized entropy of entanglement is monotonic under local
quantum operations and classical communication for n⬎2.
ACKNOWLEDGMENTS
I would like to acknowledge fruitful discussions with
Alexei Kitaev, John Preskill, and David DiVincenzo during
my visit to the Institute for Quantum Information, Caltech. I
would also like to thank Guifre Vidal for his contribution to
the investigation of the properties of the determinant states.
Financial support from NWO-Russia collaboration program
is also acknowledged.
any n⫺k bits is equal to 2 ⫺k ]. For example, the set P nn
consists of just one point—the absolutely uniform distribution of n bits p(x)⬅2 ⫺n , while the set P 0n involves all possible n-bit distributions. Take any p(x)苸 P kn and consider its
binary Fourier transform q(y):
q共 y 兲⫽
p 共 x 兲 ⫽2 ⫺n
n
共 ⫺1 兲 xy p 共 x 兲 ,
兺
y苸B n
共A1兲
共 ⫺1 兲 xy q 共 y 兲 .
Here B n denotes the set of all length n binary strings and
n
x i y i (mod 2). Denote wt(y)苸 关 0,n 兴 the number of
xy⫽ 兺 i⫽1
1’s in the binary string y苸B n . The definition of k-uniformity
can be rephrased in terms of Fourier components as
p 共 x 兲 苸 P kn iff
冦
q 共 y 兲 ⫽0
if 1⭐wt共 y 兲 ⭐k,
q 共 0, . . . ,0兲 ⫽1
for all x苸B n ,
兺
y苸B
n
共 ⫺1 兲 xy q 共 y 兲 ⭓0.
共A2兲
a convex set: if p ⬘ (x),p ⬙ (x)苸 P kn
have ␣ p ⬘ (x)⫹(1⫺ ␣ )p ⬙ (x)苸 P kn .
P kn
is
then
Note also that
for any ␣ 苸 关 0,1兴 we
It is
known that the Shannon entropy is a concave functional, i.e.,
S 关 ␣ p ⬘ 共 x 兲 ⫹ 共 1⫺ ␣ 兲 p ⬙ 共 x 兲兴 ⭓ ␣ S 关 p ⬘ 共 x 兲兴 ⫹ 共 1⫺ ␣ 兲 S 关 p ⬙ 共 x 兲兴 .
It means that the minimum of S 关 p(x) 兴 over all p(x)苸 P kn is
achieved when p(x) is an extremal point of P kn . Because P kn
is specified by a finite number of linear equalities and inequalities, it has only a finite number of extremal points. In
principle, we could find all of them, compute S 关 p(x) 兴 for
each point and choose the minimal value. Unfortunately, the
set P 36 which we are interested in has too many extremal
points and such method does not work in practice.
Instead we will proceed as follows. Consider the six-bit
probability distribution p(x) defined by Eq. 共20兲. An explicit
verification shows that p(x)苸 P 36 共as it should be, because
any measurement in a product basis on the state 兩 H 典 produces a probability distribution from P 36 ) and that S 关 p(x) 兴
⫽4. It tells us that
inf S 关 p 共 x 兲兴 ⭐4.
共A3兲
3
p(x)苸 P 6
Also note that
inf S 关 p 共 x 兲兴 ⭐ inf S 关 p 共 x 兲兴 .
3
p(x)苸 P 5
APPENDIX
The purpose of this section is to prove the equality 共24兲.
We will consider probability distributions p(x) of n classical
bits, i.e., x⫽(x 1 , . . . ,x n ), x i ⫽0,1. By analogy with the set
P 36 we will consider the sets P kn of k-uniform distributions of
n bits. By definition, p(x)苸 P kn iff any k of n bits have the
absolutely uniform distribution 关i.e., if the sum of p(x) over
兺
x苸B
共A4兲
3
p(x)苸 P 6
Indeed, choose any p(x)苸 P 36 and take the average over the
sixth bit. Then, by definition, the distribution of the bits
1,2, . . . ,5 is 3-uniform: 兺 x 6 p(x)苸 P 35 . The entropy of a partial distribution cannot exceed the entropy of a joint distribution so that S 关 兺 x 6 p(x) 兴 ⭐S 关 p(x) 兴 which implies Eq.
共A4兲. Now if we will manage to prove that
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ENTANGLEMENT ENTROPY OF MULTIPARTITE PURE STATES
inf S 关 p 共 x 兲兴 ⫽4,
共A5兲
3
p(x)苸 P 5
then the work is done because Eqs. 共A3兲, 共A4兲, and 共A5兲
imply Eq. 共24兲. In the remaining part of the text we prove the
equality 共A5兲.
The proof is based on the extremal point analysis as it was
suggested above. Let us parametrize p(x)苸 P 35 using its Fourier transform q(y), see Eqs. 共A1兲 and 共A2兲. The nonzero
components of q(y) are listed below:
q 共 1,1,1,1,1 兲 ⬅q,
q 共 1,1,0,1,1 兲 ⬅q 3 ,
q 共 0,1,1,1,1 兲 ⬅q 1 ,
q 共 1,1,1,0,1 兲 ⬅q 4 ,
q 共 1,0,1,1,1 兲 ⬅q 2 ,
q 共 1,1,1,1,0 兲 ⬅q 5 .
再
p 共 x 兲 ⫽ 共 1/32兲 1⫹ 共 ⫺1 兲
冋
兺 q i共 ⫺1 兲
i⫽1
xi
册冎
e (2)
i ⫽ 兵 q i ⭓0,q j ⫽⫺q i if j⫽i 其 ,
共A7兲
The positivity constraint p(x)⭓0, x苸B specifies the convex set P 35 in the space of q,q 1 , . . . ,q 5 . If p(x) is an extremal point of P 35 then there is at least one y苸B 5 such that
p(y)⫽0. By the symmetry, we can assume that
p(0,0,0,0,0)⫽0. So to find all extremal points of P 35 one
suffices to find all extremal points of the convex set
共A8兲
In terms of the variables q,q 1 , . . . ,q 5 the set P̃ 35 is described
by the following linear constraints:
P̃ 35 ⫽
冦
5
q⫽⫺1⫺
兺 qi ,
i⫽1
5
共A9兲
兺 q i ⭓⫺1,
i⫽1
q i ⫹q j ⭐0,
i苸 关 1,5兴 ,
共A11兲
i苸 关 1,5兴 .
.
5
P̃ 35 ⫽ 兵 p 共 x 兲 苸 P 35 :p 共 0,0,0,0,0 兲 ⫽0 其 .
共A10兲
It is also a convex set. One can easily show that Q has only
one extremal point q 1 ⫽•••⫽q 5 ⫽0 and two types of onedimensional edges with five edges of each type coming out
from this extremal point:
e (1)
i ⫽ 兵 q i ⭐0,q j ⫽0 if j⫽i 其 ,
5
q⫹
Q⫽ 兵 共 q 1 , . . . ,q 5 兲 :q i ⫹q j ⭐0,1⭐i⬍ j⭐5 其 .
共A6兲
The distribution p(x) can be written as
wt(x)
关fortunately, the inequalities p(x)⭓0 are not all independent兴. It is convenient to introduce one more auxiliary set Q
defined as
1⭐i⬍ j⭐5
关1兴 M. Nielsen and I. Chuang, Quantum Computation and Quantum Information 共Cambridge University Press, Cambridge,
2000兲.
关2兴 G. Vidal, J. Mod. Opt. 47, 355 共2000兲.
关3兴 C. Bennett, H. Bernstein, S. Popescu, and B. Schumacher,
Phys. Rev. A 53, 2046 共1996兲.
关4兴 S. Popescu and D. Rohrlich, Phys. Rev. A 56, R3319 共1997兲.
关5兴 K. Dennison and W. Wooters, e-print quant-ph/0106058.
关6兴 A. Calderbank, E. Rains, P. Shor, and N. Sloane, e-print
The extremal points of P̃ 35 are those extremal points of Q for
5
q i ⭓⫺1 and also the intersections of the onewhich 兺 i⫽1
5
dimensional edges of Q with the hyperplane 兺 i⫽1
q i ⫽⫺1.
Summarizing, there are only 11 extremal points of P̃ 35 :
共1兲 q⫽⫺1, q 1 ⫽•••⫽q 5 ⫽0.
共2兲 q⫽0, q i ⫽⫺1, q j ⫽0 if j⫽i; i苸 关 1,5兴 .
共3兲 q⫽0, q i ⫽1/3, q j ⫽⫺1/3 if j⫽i; i苸 关 1,5兴 .
Here the extremal points 共2兲 and 共3兲 represent the interand e (2)
correspondingly with the hyperplane
sections of e (1)
i
i
5
兺 i⫽1 q i ⫽⫺1, while 共1兲 is the extremal point of Q. Substituting them into Eq. 共A7兲 one can find the corresponding
distributions p(x). One can check that for the extremal
points 共1兲 and 共2兲 the probability p(x) takes only the values
0 and 1/16 thus having the entropy S 关 p(x) 兴 ⫽4. For the
extremal points 共3兲 the probability takes the values 0, 1/12,
and 1/24. Its entropy is S 关 p(x) 兴 ⫽17/6⫹log23⬇4.4. Thus
for all extremal points of P 35 we have S 关 p(x) 兴 ⭓4 and for
some extremal points S 关 p(x) 兴 ⫽4, which implies the equality 共A5兲.
quant-ph/9608006.
关7兴 A binary matrix M is nondegenerate if for any binary vector
x⫽0ជ we have M x⫽0ជ over the binary field. Equivalently M is
nondegenerate if det(M )⫽1(mod 2).
关8兴 According to Sec. III, it tells us that S 关 H 兴 ⭓3. This estimate
however does not take into account that ␳ w is absolutely uniform for any partition V⫽B艛W.
关9兴 Clearly the outcomes’ distribution has the same property for all
choices of the bases Bi , not only for the optimal one.
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