Maximum N -body correlations do not in general imply
genuine multipartite entanglement
Christopher Eltschka
1
and Jens Siewert
2,3
1
Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
2
Departamento de Química Física, Universidad del País Vasco UPV/EHU, E-48080 Bilbao, Spain
3
IKERBASQUE Basque Foundation for Science, E-48013 Bilbao, Spain
The existence of correlations between
the parts of a quantum system on the one
hand, and entanglement between them
on the other, are different properties.
Yet, one intuitively would identify
strong N-party correlations with N -party
entanglement in an N -partite quantum
state. If the local systems are qubits,
this intuition is confirmed: The state
with the strongest N-party correlations
is the Greenberger-Horne-Zeilinger
(GHZ) state, which does have genuine
multipartite entanglement. However, for
high-dimensional local systems the state
with strongest N-party correlations may
be a tensor product of Bell states, that
is, partially separable. We show this
by introducing several novel tools for
handling the Bloch representation.
1 Introduction
The expansion of the density operator in
terms of a matrix basis is called the Bloch
representation [1, 2, 3]. Technically, this
representation is rather demanding: A pure
state of N parties each of local dimension d
is characterized by 2d
N
− 2 real coefficients,
whereas the same state written in the Bloch
representation requires d
2N
− 1 parameters, just
as any mixed state. On the other hand, this
representation appears to be perfectly adapted to
studying the correlation properties of a quantum
system, because the above-mentioned expansion
corresponds to a decomposition of the state
into all possible correlation contributions (hence
the terms are also called correlation tensors).
Therefore, from a better understanding of the
technical characteristics of this expansion one
may expect significant insight into the physics of
correlated quantum systems.
The systematic investigation of the properties
of the Bloch representation for finite-dimensional
multi-party quantum systems is a relatively
recent subject [4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15], although many important results were
found earlier, mostly relating specific features of
the Bloch picture to the entanglement properties
of the state (e.g., [16, 17, 18, 19, 20, 21, 22,
23, 24, 26, 25]). Currently much activity is
devoted to working out the technical details
and properties for an easier use of the Bloch
representation to solving physics problems. An
essential part of this is to figure out smaller sets
of parameters that carry sufficient amounts of
information to facilitate the characterization of
relevant physical properties for a state given in
the Bloch representation. In this contribution,
we define such a set of parameters, which we call
the “sector distribution” and discuss some of its
key features. Moreover, we illustrate how the
properties of this distribution are reflected in the
correlation properties of the states.
To be more specific, let us preliminarily
introduce the Bloch representation; the precise
definition will be given below. If we enumerate
the parties of an N-party system (of equal local
dimension d) by {1, 2 . . . N} and A is a subset of
parties, then
ρ =
1
d
N
X
A
G
A
⊗ 1
¯
A
. (1)
Here, G
A
is a Hermitian operator that acts
nontrivially on the parties belonging to the subset
A, and 1
¯
A
is the identity operator for the
complementary set. Consider now the sum of
all those terms in Eq. (1) that act on the same
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arXiv:1908.04220v2 [quant-ph] 6 Feb 2020
number k of parties,
S
k
≡
1
d
N
X
|A|=k
G
A
⊗ 1
¯
A
. (2)
We call S
k
the “k-sector” of ρ and the (squared)
Hilbert-Schmidt length of S
k
the “k-sector
length” S
k
[27]
S
k
≡ d
N
Tr
S
†
k
S
k
. (3)
The k-sector length S
k
is a natural quantifier for
the k-party correlations in a state [28]. Clearly,
for an N-partite state there are N sector lengths
(S
0
= 1 for all normalized states). Sector
lengths were discussed earlier [16, 17, 20, 4,
5, 6, 8, 7, 9, 10, 14, 15]. In particular the
N-sector was intuitively linked with the N -party
quantum correlations. Therefore it came as a
surprise that there exist mixed states that are
N-party entangled but do not possess N-party
correlations [29, 30, 31, 32]. Later it was
realized [4] that, in order to witness genuine
multipartite entanglement, it may be necessary
to consider a collection of the highest sector
lengths S
N
, S
N−1
, . . . rather than just S
N
. In the
present work we systematically study the set of all
sector lengths {S
k
}. As we will demonstrate the
distribution {S
1
, S
2
. . . S
N
} represents a reduced
set of parameters in the spirit described above
(linear in the system size instead of exponential)
that carries substantial information regarding
some of the correlation properties of the state.
Often it has little meaning to study the sector
lengths S
k
individually; rather, there exists a
variety of strict relations between them that
determine the entire distribution.
We introduce several novel technical concepts,
most importantly the N-sector projector. We use
this toolbox to prove the long-standing conjecture
that for any number N of qubits the GHZ state
maximizes the N-sector length. Subsequently
we analyze the sector distribution for few-party
systems of higher local dimension d > 2. Here we
prove that for higher local dimension the state
with maximum N-sector length may be partially
separable. Moreover, we provide a comprehensive
discussion for the behavior of the N-sector with
increasing number of parties as well as growing
local dimension.
2 Definitions and preliminaries
For the Bloch representation of an N-partite state
with all local dimensions equal to d = 2 the
operators G
A
in Eq. (1) are expanded in local
operators, i.e., a basis of traceless matrices {g
j
},
1 5 j 5 d
2
− 1, g
0
≡ 1, with normalization
Tr
g
†
j
g
k
= dδ
jk
,
ρ =
1
d
N
X
A
X
j
l
: l∈A
r
j
1
···j
N
g
j
1
⊗ ··· ⊗ g
j
N
⊗ 1
¯
A
,
(4)
and
r
j
1
···j
N
= Tr
h
g
†
j
1
⊗ ··· ⊗ g
†
j
N
⊗ 1
¯
A
i
ρ
. (5)
Here, all indices j
m
, m ∈
¯
A are set to 0. With
this, the k-sector length simply becomes
S
k
=
X
j
l
: l∈A,|A|=k
|r
j
1
···j
N
|
2
, (6)
where |A| is the number of elements in A. All
the sector lengths are local unitary invariants of
the state. As the actual sector lengths
√
S
k
equal
the Hilbert-Schmidt norms of S
k
, they obey the
triangle inequality. Because of this, the N-sector
length of a mixture of pure states can never
exceed the largest N-sector length of any of the
pure states. Therefore, throughout this article we
focus on pure states Π, that is, for the purity we
have Tr Π
2
= Tr Π = 1 and, hence, for the sum
of all sector lengths,
P
N
k=0
S
k
= d
N
.
Consider the simplest case of all, that is,
product states
prod
N
j
E
= |ji
⊗N
; here |ji
denotes a state of the computational basis,
j = 0, 1, . . . , (d − 1). It is easy to see that
S
k
(prod
N
j
) =
N
k
(d −1)
k
. Remarkably, it was
shown by Tran et al. [6] that among the pure
states only product states have the minimum
N-sector length
min S
N
= (d −1)
N
,
that is, for d > 2 the N-sector is always on the
order of d
N
. It turned out that the opposite
question regarding the states with maximum
N-sector is considerably more complex. A
relevant state for this discussion is the GHZ state
for N parties of local dimension d
GHZ
N
d
E
=
1
√
d
d−1
X
j=0
|ji
⊗N
. (7)
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For N -qudit GHZ states we find the sector
distribution (see Appendix)
S
k
(GHZ
N
d
) = (d − 1)d
N−1
δ
kN
+
+
N
k
!
(d −1)
k
+ (−1)
k
(d −1)
d
.
(8)
Tran et al. [6] showed that for odd party
number N of qubits the GHZ state has maximum
N-sector length. They conjectured that this
statement holds also for even N. For d > 2 it
is not clear which state has maximum N -party
correlations. In the following we will prove the
conjecture for even-N qubit GHZ states.
3 The N -sector projector
For the proof we need some mathematical
tools based on universal state inversion [35,
36, 37, 9, 12]. First we define the projection
(super-)operator [38] onto the last (k = N) sector
(or N -sector projector for short whenever there
are no ambiguities)
P(ρ) =
N
Y
j=1
id −
1
d
Tr
j
(·) ⊗1
j
ρ . (9)
It is easy to check by writing ρ in the Bloch
representation that P indeed realizes a projection
onto the N-sector, S
N
. The map (9) belongs to
the class of generalized universal state inversions
discussed in Refs. [37, 12] that have the form
I
{α
j
,β
j
}
=
Q
N
j=1
α
j
Tr
j
(·) ⊗ 1
j
− β
j
id
, where
id denotes the identity map and α
j
, β
j
are real
numbers. With definition (9) we get immediately
S
N
(ρ) = d
N
Tr [ρ P(ρ)]
= d
N
X
A
−
1
d
|A|
Tr
ρ
2
¯
A
, (10)
where A, as before, runs through all subsets
of {1 . . . N} and ρ
¯
A
= Tr
A
ρ is the reduced
state on the subset of parties
¯
A. The equality
Tr
h
(Tr
A
Π)
2
i
= Tr
h
(Tr
¯
A
Π)
2
i
for pure states Π
motivates the definition of another operator
Q(ρ) =
N
Y
j=1
Tr
j
(·) ⊗1
j
−
1
d
id
ρ , (11)
so that
S
N
= d
N
Tr [Π P(Π)]
= d
N
Tr [Π Q(Π)] = 0 . (12)
Because of the projector property of P and
the Cauchy-Schwarz inequality, kMk
P
≡
q
Tr [M
†
P(M)] defines a seminorm for operators
M (while the analogous statement does not hold
for Q). By considering the action of Q(Π) in the
Bloch representation relation (12) gives rise to the
astounding equality
d
N
S
N
(Π) =
N
X
k=0
(−1)
k
(d
2
− 1)
N−k
S
k
(Π) ,
(13)
which links the last sector length S
N
with all
others.
Finally we rewrite the well-known purity
condition for reductions of pure states
Tr
h
(Tr
A
|ψihψ|)
2
i
= Tr
h
(Tr
¯
A
|ψihψ|)
2
i
in
terms of sector lengths. This is achieved by
symmetrizing the purity conditions for fixed
|A| and accomplishing the combinatorial
accounting. We find for the k-purity
relation (k = 0, 1, . . . , b
N−1
2
c with the floor
function b·c) [33]
d
N−2k
k
X
m=0
N − m
k − m
!
S
m
=
N−k
X
n=0
N − n
k
!
S
n
.
(14)
For k = 0 this gives the well-known condition
d
N
=
P
N
0
S
n
. We explicitly write the relations
for k = 1 and k = 2 as they will turn out useful
later:
d
N−2
[N + S
1
] = N + (N − 1)S
1
+ . . . + 2S
N−2
+ S
N−1
, (15a)
d
N−4
"
N
2
!
+ (N −1)S
1
+ S
2
#
=
N
2
!
+
N − 1
2
!
S
1
+ . . . +
3
2
!
S
N−3
+ S
N−2
. (15b)
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Interestingly, Eqs. (14) elucidate the role of the
N-sector for pure states: All sector lengths have
to be adjusted so as to obey the k-purity relations
(k = 1 . . . b
N−1
2
c) between the reduced states of
non-empty complementary partitions; note that
the last sector is excluded from establishing this
balance. The last sector serves to fill up the
total length d
N
of the Bloch vector (the “0-purity”
relation).
4 The N -qubit GHZ state maximizes
the N -sector
For odd N we know max S
N
= S
N
(GHZ) =
2
N−1
, cf. Ref. [6]. We recall this proof in the
Appendix. We are prepared now to show for even
N qubits that
max S
N
= S
N
(GHZ) = 2
N−1
+ 1 .
As here the GHZ state has only even-numbered
sectors, Eq. (13) would imply for a hypothetical
state Π
x
= |xihx| with larger N-sector than GHZ
that S
2m
>
N
2m
for some m < N/2. In order
to obtain information regarding the distribution
of the even-numbered sectors we consider an R
matrix (analogous to Refs. [34, 5, 9]) of ρ
[1]
≡
Tr
{1}
Π
x
after tracing the first party,
R
[1]
≡ ρ
[1]
I
−
(ρ
[1]
) = ρ
[1]
X
A
(−1)
|A|
Tr
¯
A
ρ
[1]
⊗ 1
¯
A
,
(16)
where I
−
(σ) ≡
Q
M
j=1
[Tr
j
(·) ⊗1
j
− id] σ denotes
the standard universal state inversion for an
M-partite state σ (cf. Refs. [36, 9, 11]). If
we symmetrize over the traced party we can
establish a relation between
P
j
Tr R
[j]
and the
sector lengths S
k
(Π
x
) of Π
x
,
0 5 d
N−1
N
X
j=1
Tr R
[j]
=
N−1
X
k=0
(−1)
k
(N − k)S
k
.
(17)
The reasoning is exactly the same as the one to
obtain the 1-purity, Eq. (15a). By adding the
latter equation and Eq. (17) (and dividing by
2) we obtain a relation for the even-numbered
sectors,
d
N−2
2
N + S
1
+
X
j
d Tr R
[j]
= N + (N −2)S
2
+ (N −4)S
4
+ . . . + 4S
N−4
+ 2S
N−2
. (18)
We observe that on the right-hand side (r.h.s.)
the prefactors increase with decreasing index,
this is analogous to Eq. (13), only that here
the prefactors increase linearly. Also here the
even-sector distribution of Π
x
would exceed
the result of the GHZ state. Relation (18)
gives us the possibility to directly check the
achievable maximum of the r.h.s. for pure states
by maximizing the terms on the left-hand side.
For qubits this is straightforward and shows that
the maximum is achieved for the GHZ state (we
present this calculation in the Appendix). Hence,
there is no state Π
x
with larger N-sector.
5 Few parties of higher local dimension
The obvious guess from the results so far is that
GHZ
N
d
E
maximizes the N -sector length also for
d > 2. It will turn out that this can only partially
be true. To this end, let us investigate states
with up to six parties. The following results are
obtained by using Eqs. (13), (14) for k = 0, 1, 2,
and increasingly tedious algebra.
N = 2 : We have d
2
= 1 + S
1
+ S
2
, so that
max S
2
= d
2
− 1
for S
1
= 0, that is, the Bell state
Φ
+
d
E
≡
GHZ
2
d
E
maximizes the 2-sector.
N = 3: Here,
S
3
= (d −1)
2
(d + 2) −(d −1)S
1
,
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so that S
1
= 0 leads to max S
3
= (d + 2)(d −1)
2
,
which again is realized by the GHZ state.
N = 4: In this case, there remains more than one
parameter undetermined
S
4
= (d
2
− 1)
2
−
1
2
h
(d
2
− 1)S
1
+ S
3
i
,
but since S
1
, S
3
= 0, the 4-sector gets maximized
for S
1
= S
3
= 0, so that max S
4
= (d
2
− 1)
2
.
That is, for four-party states the N-sector is not
maximized by the GHZ state, but by a tensor
product of Bell states, i.e., a biseparable state.
Curiously, the case d = 2 is right on the edge,
because the tensor product of a pair of two-qubit
Bell states and the four-qubit GHZ state have the
same 4-sector length, S
4
= 9.
N = 5: Here we find
(d −3)S
5
= (d − 1)
3
(d + 2)(d
2
− 2d − 4)−
− (d − 1)
2
(d
2
− d − 3)S
1
+ (d − 1)S
3
.
(19)
This suggests again S
1
= 0 for the N-sector
maximum, however, now the sign of S
3
is
reversed. We note that the maximum k-sector
length of an N-party system is on the order of
N
k
d
k
, so that the 3-sector length S
3
∼ O(d
3
),
and hence for large local dimension d 1
max S
5
(d −1)
3
(d + 1)(d + 2)
−→ 1 + O(d
−2
) ,
which indicates that the maximum N-sector
is approximated with better than first-order
accuracy for growing d by the polynomial in
the denominator. The latter corresponds to the
tensor product of a Bell state and a three-party
GHZ state,
Φ
+
d
E
⊗
GHZ
3
d
E
. Consequently, for
large d also here the state with maximum N-body
correlations may be biseparable. The case d = 3
is special: S
3
= 20 gives the largest 3-sector.
The five-qutrit GHZ state is compatible with
this [cf. Eq. (8)] and has larger 5-sector than
Φ
+
3
E
⊗
GHZ
3
3
E
(172 vs. 160). However, in
principle, there might be a state with S
1
= 0,
S
3
= 20 and even larger 5-sector.
N = 6: This case has similar features as N = 5.
The 6-sector obeys
2(d
2
− 4)S
6
= 2(d − 2)(d
2
− 1)
3
(d + 2) −
− (d
2
− 1)
2
(d
2
− 3)S
1
+
+ (d
2
− 1)S
3
− (d
2
− 3)S
5
.
(20)
Again we see that for increasing d 1 the
6-sector (max S
6
)/(d
2
− 1)
3
−→ 1 + O(d
−3
)
because of the scaling of the sector lengths with
d; the corresponding state is
Φ
+
d
E
⊗3
. Note that
already for d = 3 the 6-sector of
Φ
+
3
E
⊗3
beats
the length of the six-qutrit GHZ state (512 vs.
508).
We summarize the results of this section in
Tables 1–3.
Table 1: Maximum of the N-sector for 2, 3, 4 parties.
N max N-sector S
N
state maximizing S
N
2 d
2
− 1
Φ
+
d
E
3 (d − 1)
2
(d + 2)
GHZ
3
d
E
4 (d
2
− 1)
2
Φ
+
d
E
⊗2
Table 2: Comparison of N-sectors for N = 5.
S
5
GHZ
5
d
E
GHZ
3
d
E
⊗
Φ
+
d
E
d (d − 1)
2
× (d − 1)
3
×
(d
3
+ 2d
2
− 2d + 4) (d + 1)(d + 2)
3 172 > 160
4 828 > 810
5 2704 > 2688
6 7000 = 7000
7 15516 < 15552
Table 3: Comparison of N-sectors for N = 6.
S
6
GHZ
6
d
E
Φ
+
d
E
⊗3
d
d−1
d
d
6
+ (d − 1)
5
+ 1
(d
2
− 1)
3
2 33 > 27
3 508 < 512
4 3255 < 3375
6 Maximum N -sector for large d and
large N
We can investigate the dominance of N-sectors
numerically. On increasing d, the partially
separable states—that is, a tensor product of Bell
states (even N) or Bell states and a three-party
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Figure 1: N-sector length difference
S
N
(GHZ
N
d
) − S
N
(Bell
N
d
). The border between
GHZ-dominated and Bell-dominated is given by a
straight line d ' 0.6275 · N (see text); however, note
the pronounced even-odd effect. For N = 2 and N = 3,
GHZ and Bell are the same state, therefore these cases
have to be counted as ‘undecided’. (a) Small scale
N, d 5 10; (b) larger scale N, d 5 100. Note that the
color scale is logarithmic.
GHZ state (odd N)—appear to dominate (we
will call these states “Bell”). In the opposite
case, the GHZ state has larger N-sector. This
behavior is shown in Fig. 1, where the difference
of N-sectors S
N
(GHZ
N
d
)−S
N
(Bell
N
d
) is displayed
in a (d, N) plane. We note the formal analogy of
our problem of finding the maximum N-sector
with that of deciding the existence of absolutely
maximimally entangled (AME) states [21, 39,
40, 8, 11] that is suggested by the structure of
Fig. 1. The analogy arises because also the
N-sector problem seems to have two solutions
whose validity regions in the (d, N) plane are
connected and separated by a single line. The
region of GHZ dominance corresponds to ‘AME
state does not exist’, while that of Bell dominance
relates to ‘AME does exist’. The line separating
the two corresponds to the Scott bound [21, 11].
An accurate analytical approximation for this
line is found by equating S
N
(GHZ
N
d
) in Eq. (8)
with S
N
(Bell
N
d
) = (d
2
− 1)
N/2
(for even N ).
Assuming d = γ
−1
N, this leads to an equation
that determines the γ parameter, e
−γ
= 1 −
γ
2
,
from which
d ' 0.6275 · N . (21)
Close to this line there may be exceptions from
the rule, just as in the case of AME states.
In the following we provide arguments why
GHZ and Bell are, if not the dominating, at
least close to the states with dominating N-sector
in the limits of large N and d. Consider first
fixed even N and d 2, N. Our reasoning
is based on the purity relations Eqs. (15) and
on the consideration that to leading order the
maximum k-sector is given by S
k
∼
N
k
d
k
. From
the 0-purity relation S
N
= d
N
−S
N−1
−. . . −1 it
follows that the dominating terms S
N−1
+ S
N−2
need to be as small as possible in order to obtain
max S
N
. We observe that Eq. (15a) dictates that
S
N−2
and S
N−1
cannot both vanish, and their
sum needs to be at least of order Nd
N−2
. As
S
1
> 0 would only increase the r.h.s., S
1
= 0
is the sensible choice. Moreover, we see that
S
N−2
+ S
N−1
≈
1
2
d
N−2
N + S
N−1
, so that
the subleading correction becomes smallest for
S
N−1
= 0 and S
N−2
≈
N
2
d
N−2
. Substituting
this result into Eq. (15b) leads to S
2
≈
N
2
d
2
. In
particular the latter requirement together with
S
1
= 0 can be fulfilled if the state is a Bell tensor
product.
For the opposite limit, N d > 2, general
statements are more difficult to make, because
the sector sum does not correspond to a power
expansion in d any longer. We can discuss at least
the case of states that are more entangled than
GHZ, that is, m-uniform states [21, 39]. A state
is called m-uniform if S
1
= S
2
= ··· = S
m
= 0,
with the extreme case of AME states (m =
bN/2c). For AME states, S
N
' d
N
(1 −
1
d
2
)
N
is
a fair approximation that applies to some extent
also to m . N/2 if N does not exceed d
2
. Then,
for large d approximately S
N
∼ d
N
e
−
N
d
2
, which
shows that a substantial fraction of the Bloch
vector length is not in the N -sector, making these
states bad candidates for the maximum S
N
.
On the other hand, for the GHZ state we have
S
N
(GHZ
N
d
) ' d
N
1 −
1
d
+ d
N−1
e
−
N
d
. (22)
That is, S
N
is essentially given by the first term
in Eq. (22) and the relative error shrinks with
increasing N . This is the expected behavior
for the dominating state, because d
N
1 −
1
d
is
the absolute maximum the traceless part of a
pure state |ψihψ| can achieve: An offdiagonal
element consisting of orthogonal product states
gives k|jj . . . jihkk . . . k|k
P
= 1, which is the
maximum among all rank-1 operators. An
N-qudit state of local dimension d can have
at most Schmidt rank d in a bipartition
of a single party against the rest. This
amounts to a maximum offdiagonal contribution
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of
1
d
2
d(d − 1)d
N
to S
N
– which is precisely
the GHZ result. Consequently, for large N the
GHZ state N-sector approaches the maximum for
any rank-1 operator. Evidently, this discussion
cannot exclude the existence of a state that
approaches this maximum even faster.
7 Conclusions
We have analyzed the Bloch sector distribution
for multipartite pure quantum states of N d-level
systems, in particular the properties of the
N-sector. We have demonstrated that the sectors
must not be considered individually; rather, there
are numerous interdependencies that determine
the distribution. One of our main results
based on this insight is the proof that for
qubits the GHZ state has maximum N-sector
also for even N. We have given an extensive
characterization of the N-sector behavior for
arbitrary N and d, which can be viewed as
an algebraic problem analogous to that of the
existence of AME states. Most importantly,
we find that strong N-party correlations (viz
maximum N-sector) do not necessarily imply
genuine multipartite entanglement. Apart from
our physics results, our work provides several
novel technical tools for analyzing the Bloch
representation of pure states and thereby shows
that this is a powerful approach to obtain
new insight into the mathematical properties of
many-body quantum states.
Acknowledgments
This work was funded by the German
Research Foundation Project EL710/2-1
(C.E., J.S.), by Grant PGC2018-101355-B-100
(MCIU/FEDER/UE) and Basque Government
Grant IT986-16 (J.S.). The authors would like
to thank Marcus Huber and Nikolai Wyderka
for stimulating discussions. C.E. and J.S.
acknowledge Klaus Richter’s support of this
project.
Appendix
Sector distribution of GHZ state, Eq. (8)
In order to obtain the sector lengths for the GHZ
state it is not necessary to explicitly calculate the
Bloch representation. Yet we quickly do it for the
qubit example to demonstrate how simple it is.
The density matrix of the N-qubit GHZ state is
Π
GHZ
N
2
=
1
2
|00 . . . 0ih00 . . . 0| +
+ |11 . . . 1ih11 . . . 1| +
+ |00 . . . 0ih11 . . . 1| +
+ |11 . . . 1ih00 . . . 0|
. (A1)
Each term here is a tensor product of N identical
rank-1 single-qubit operators,
|0ih0|
⊗N
=
1
2
N
(1 + Z)
⊗N
|1ih1|
⊗N
=
1
2
N
(1 − Z)
⊗N
|0ih1|
⊗N
=
1
2
N
(X + iY )
⊗N
|1ih0|
⊗N
=
1
2
N
(X − iY )
⊗N
,
where X ≡ σ
1
, Y ≡ σ
2
, Z ≡ σ
3
are the
Pauli matrices and 1 is the qubit identity matrix.
Hence
Π
GHZ
N
2
=
1
2
N
X
even#Z
ZZ . . . 11 +
+
X
even#Y
(−1)
#Y
2
X . . . Y Y . . . X . . . Y Y
,
(A2)
where the sums run over all combinations of
even numbers of Z occurrences padded with 1s
(diagonal) and even numbers of Y occurences
padded with Xs (offdiagonal); for simplicity we
omit the tensor product signs. The difference
between even and odd N is that for even N there
is one N -sector term ZZ . . . Z in the diagonal
part, whereas for odd N the N-sector exclusively
consists of offdiagonal terms. The GHZ state for
d > 2 can be built in an analogous manner.
In order to derive Eq. (8) we can take a shortcut
and use Eq. (10),
S
N
= d
N
X
A
−
1
d
|A|
Tr
ρ
2
¯
A
. (A3)
The GHZ state is particularly simple as all
reduced states are of rank d and completely
mixed on their span, so that Tr
ρ
2
A
=
1
d
for all
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|A| 6= 0, N. For |A| = 0 and |A| = N we have
Tr (ρ
A
)
2
= 1, so that
S
N
=d
N
1
d
1 −
1
d
N
+ d
N
1 −
1
d
−
− (−1)
N
d
N
1
d
N+1
−
1
d
N
=d
N−1
(d − 1) +
+
1
d
h
(d − 1)
N
+ (−1)
N
(d − 1)
i
. (A4)
For k < N, S
k
is given by the length of the
last sector of the reduced density matrix ρ
A
(|A| = k) times the number of such reduced
density matrices, S
k
= d
k
N
k
kρ
A
k
2
P
. In contrast
to the N-sector we need not include a correction
for the first term, so that
S
k
= d
k
1
d
1 −
1
d
k
N
k
!
−
− (−1)
k
d
k
1
d
k
−
1
d
k−1
=
N
k
!
1
d
h
(d − 1)
k
+ (−1)
k
(d − 1)
i
. (A5)
Proof maximum N sector of odd-N qubit
GHZ state
Here we show the proof that for odd N qubits,
the maximum N-sector length is max S
N
=
S
N
(GHZ) = 2
N−1
, which is realized by the GHZ
state [6].
First, we recall that for odd N qubit states
Π = |ψihψ| the degree-2 SL invariant
H = Tr
h
Π Y
⊗N
Π
∗
Y
⊗N
i
= 0 (A6)
always vanishes [5, 6, 10] (here, Y ≡ σ
2
is a
Pauli matrix and Π
∗
= |ψ
∗
ihψ
∗
|, where |ψ
∗
i is
the vector with complex conjugate components).
In terms of sector lengths Eq. (A6) reads [5, 6, 10]
0 =
P
N
k=0
(−1)
k
S
k
, so that
S
even
≡
N−1
2
X
k=0
S
2k
=
N−1
2
X
k=0
S
2k+1
≡ S
odd
,
(A7)
that is, the sum of the even-numbered sector
lengths S
even
always equals that of the
odd-numbered ones, S
odd
. Because of the
purity-0 constraint
P
N
k=0
S
k
= S
even
+ S
odd
= 2
N
this means that both even and odd sector
length sums are always equal to 2
N−1
, so that
the properties of a state are encoded in the
distribution of the even sector lengths among
themselves on the one hand, and separately the
odd ones, on the other hand.
It is quite obvious then that the GHZ state
(odd N) is the one with maximum N-sector:
Here, the entire odd sector length 2
N−1
is
shifted to the N-sector, and the other odd
sector lengths vanish. (The peculiarity is that
such a state actually does exist – this is by
no means guaranteed by Eq. (A7) and the
purity constraint.) Note also that for the GHZ
state, Eq. (A7) does not say anything about the
distribution of the even-numbered sectors.
Now consider the PQ relation, Eq. (13) from
the main text,
S
N
=
1
2
N
N
X
k=0
(−1)
k
3
N−k
S
k
=
1
2
N
N−1
2
X
l=0
3
N−2l
S
2l
−
N−1
2
X
m=0
3
N−2m−1
S
2m+1
.
(A8)
We see that also from the point of view of
Eq. (A8) the maximum N-sector for the GHZ
state makes perfect sense: All odd sector
contributions are moved to S
N
where they cause
the ‘least damage’ for maximizing the r.h.s. of the
equation, because S
N
has the smallest prefactor.
Proof maximum l.h.s. of Eq. (18) in main text
In the following we demonstrate the last step
of the proof in the main text that max S
N
=
2
N−1
+ 1 for even-N qubit GHZ states. This step
consists in maximizing the left-hand side (l.h.s.)
of Eq. (18) of the manuscript,
S
1
+
X
j
2 Tr R
[j]
−→ max ,
where ρ
[j]
= Tr
{j}
Π
x
and Π
x
= |xihx| is a pure
state.
First, let us consider ρ
[1]
= Tr
{1}
Π
x
. We write
the Schmidt decomposition of |xi with respect to
the first qubit,
|xi =
p
λ
1
|0i|X
0
i +
p
1 − λ
1
|1i|X
1
i , (A9)
where {|0i, |1i} is the Schmidt basis on the first
qubit and |X
0
i, |X
1
i two orthogonal odd-(N −1)
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qubit states, the Schmidt vectors on qubits
{2 . . . N}. Hence,
ρ
[1]
= λ
1
|X
0
ihX
0
| + (1 − λ
1
) |X
1
ihX
1
| ,
(A10)
so that, as for k-qubit states |φi the inverted state
|
˜
φi = Y
⊗k
|φ
∗
i [cf. [5, 9, 10] and the discussion
below Eq. (A6)],
Tr R
[1]
= Tr
h
ρ
[1]
Y
⊗(N−1)
ρ
∗
[1]
Y
⊗(N−1)
i
= λ
2
1
|hX
0
|
˜
X
0
i|
2
+ (1 − λ
1
)
2
hX
1
|
˜
X
1
i|
2
+
+ λ
1
(1 − λ
1
)|hX
0
|
˜
X
1
i|
2
+
+ λ
1
(1 − λ
1
)|hX
1
|
˜
X
0
i|
2
= 2λ
1
(1 − λ
1
)|hX
0
|
˜
X
1
i|
2
= 2λ
1
(1 − λ
1
)|hX
0
|Y
⊗(N−1)
X
∗
1
i|
2
,
(A11)
because hφ|
˜
φi = 0 for odd-N qubit states [see also
Eq. (A6)]. For the matrix element in Eq. (A11)
we have
∆ = |hX
0
|Y
⊗(N−1)
X
∗
1
i| 5 1 , (A12)
since the operator Y
⊗(N−1)
has only eigenvalues
of modulus 1. Consequently we find, if we add
the 1-sector S
(1)
1
of the 1st qubit in Π
x
,
max
|xi
h
S
(1)
1
+ 2 Tr R
[1]
i
=
= max
λ
1
,∆
h
2λ
2
1
+ 2(1 − λ
1
)
2
− 1 + 4λ
1
(1 − λ
1
)∆
2
i
= 1 . (A13)
The states |X
0
i, |X
1
i that realize this maximum
are, e.g., |X
0
i = |0i
⊗(N−1)
and |X
1
i = |1i
⊗(N−1)
.
That is, an even-N qubit state that maximizes
the l.h.s. of Eq. (A13) is, e.g., the GHZ state |xi =
1
√
2
|0i
⊗N
+ |1i
⊗N
.
The same reasoning as above can be applied
for all qubits j = 1 . . . N, so that we find for the
symmetrized l.h.s. of Eq. (A13),
max
|xi
S
1
+
X
j
2 Tr R
[j]
= N , (A14)
with the GHZ state attaining the maximum.
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