Distribution of entanglement and correlations in all finite
dimensions
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
May 16, 2018
The physics of a many-particle system
is determined by the correlations in its
quantum state. Therefore, analyzing these
correlations is the foremost task of many-
body physics. Any ‘a priori’ constraint
for the properties of the global vs. the lo-
cal states—the so-called marginals—would
help in order to narrow down the wealth of
possible solutions for a given many-body
problem, however, little is known about
such constraints. We derive an equal-
ity for correlation-related quantities of any
multipartite quantum system composed of
finite-dimensional local parties. This rela-
tion defines a necessary condition for the
compatibility of the marginal properties
with those of the joint state. While the
equality holds both for pure and mixed
states, the pure-state version containing
only entanglement measures represents a
fully general monogamy relation for entan-
glement. These findings have interesting
implications in terms of conservation laws
for correlations, and also with respect to
topology.
Correlations between different parts of many-
particle quantum systems are manifestly different
from their classical counterparts. The hallmark of
the latter is that, if the global state is completely
known, so are the local states, that is, the indi-
vidual states of each particle. For quantum states
this may be different: Even if the global state is
known with certainty, there may be no informa-
tion whatsoever regarding the local states such
as, for example, in maximally entangled states of
two d-dimensional systems [1, 2].
Often, different degrees of departure from
the classical behavior are distinguished, namely
entanglement, steerability, and nonlocality [3].
While all these correlation types are based on en-
tanglement, nonlocal correlations are regarded as
the strongest. This is because all nonlocal states
are entangled, but only those for which there
are no local models may violate a Bell inequal-
ity [4, 5]. Already such basic considerations reveal
that there must exist mathematical tools to char-
acterize quantitative aspects of correlations in a
multipartite quantum state. Entanglement mea-
sures [6, 7] constitute one example of such tools,
but there are many others [8, 9] and to date it is
not clear how to exhaustively characterize quan-
tum correlations, not even for finite-dimensional
systems.
Such a characterization can be achieved via
mathematical constraints for different correla-
tion functions and/or quantifiers. A Bell in-
equality is precisely one such constraint: It re-
lates the existence of a local model to the maxi-
mum absolute value of a certain linear combina-
tion of correlation functions. Another example
are so-called monogamy inequalities for entangle-
ment [10, 11, 12, 13, 14] or nonlocality [16, 17]
which establish constraints between the correla-
tions that may coexist in different subsystems of
a multipartite quantum system.
It is remarkable that there exist exact relations,
that is, equalities between certain entanglement
measures in qubit systems. Coffman et al. [10]
discovered the now famous relation for pure states
ψ
3
of three qubits (denoted by A, B, and C)
τ
3
(ψ
3
) = τ
A
− C
2
AB
− C
2
AC
, (1)
where τ
A
denotes the linear entropy of the re-
duced state ρ
A
= Tr
BC
|ψ
3
ihψ
3
|, C
AB
is the con-
currence of the two-qubit reduced state ρ
AB
=
Tr
C
|ψ
3
ihψ
3
| (analogously for C
AC
), and τ
3
(ψ
3
) is
the absolute value of a polynomial local SL (LSL)
invariant that depends on the state coefficients of
ψ
3
. It turned out only recently [18] that simi-
lar relations exists for any N-qubit system, the
Accepted in Quantum 2018-05-07, click title to verify 1
arXiv:1708.09639v2 [quant-ph] 15 May 2018
simplest being
2|H(ψ
N
)|
2
=
N
X
k=1
(−1)
k+1
X
j
1
<...<j
k
τ
A
j
1
...A
j
k
, (2)
where now τ
A
j
1
...A
j
k
is the linear entropy of the re-
duced state of qubits A
j
1
, . . . , A
j
k
and H(ψ
N
) =
hψ
∗
N
|σ
⊗N
2
|ψ
N
i is another polynomial LSL invari-
ant [20, 7] (here |ψ
∗
N
i is the complex conjugate of
|ψ
N
i and σ
2
is the second Pauli matrix). Strik-
ingly, all of the quantities in Eqs. (1), (2) are mea-
sures for entanglement, either of the individual
party (the linear entropies and the concurrences)
or in the global state (the polynomial LSL invari-
ants). These relations impose rigid constraints
between the properties of the global and the re-
duced states, and are therefore intimately con-
nected with the quantum marginal problem [19].
However, since their validity appears to be re-
stricted to qubit systems, it is not clear whether
they bear any relevance to a general physical set-
ting.
This is the motivation for the present work:
We show that the analogue of Eq. (2) holds for
any multipartite system of finite-dimensional con-
stituents. In passing, by this relation a local uni-
tary invariant of the pure state is introduced,
which we term distributed concurrence, because
it describes the distribution of bipartite entan-
glement in the pure state and is a natural gener-
alization both of the bipartite concurrence and of
|H(ψ
N
)| in Eq. (2).
1 Universal state inversion
1.1 Definition
The key ingredient in our investigation is the op-
erator obtained by applying the so-called univer-
sal state inversion [21, 22], see also Refs. [23, 24,
25, 26]. Consider a d-dimensional Hilbert space
H and B(H), the set of bounded positive semidef-
inite operators on H. Horodecki et al. [21] defined
the operator
S
d
(O) = Tr(O) 1 − O , O ∈ B(H) (3)
which was used by Rungta et al. [22] to write the
concurrence of a state ψ ∈ H
A
⊗ H
B
as
C(ψ) =
q
hψ|S
d
A
⊗ S
d
B
(|ψihψ|) |ψi (4)
=
q
2 − Tr
A
ρ
2
A
− Tr
B
ρ
2
B
, (5)
(where d
j
= dim H
j
, j = A, B; ρ
A
= Tr
B
|ψihψ|,
ρ
B
= Tr
A
|ψihψ| are the reduced states for the
parties). They termed the operator S
d
the uni-
versal state inverter. We will show below that,
for non-Hermitian operators O, the appropriate
definition of the single-system state inverter is
˜
O = Tr
O
†
1 − O
†
(see Section 1.4).
It is straightforward to generalize universal
state inversion toward density matrices ρ of N-
partite systems, i.e., ρ ∈ B(H
1
⊗ . . . ⊗ H
N
),
X ∈ {1, . . . , N}
˜ρ = S
d
1
S
d
2
. . . S
d
n
(ρ) (6)
=
h
N
Y
X=1
Tr
X
(·) ⊗ 1
X
− 1
i
ρ , (7)
where 1
X
denotes the identity on a single subsys-
tem X, as opposed to 1, the identity on the full
system. We use the tilde notation for the inverted
state ˜ρ in order to make reference to Wootters’
notation in Ref. [27], Eq. (5), because that defi-
nition for two qubits is naturally generalized by
Eq. (7), as we will see below. Note that if the
dimension of any party is larger than 2, the trace
of ˜ρ is larger than 1. Nonetheless we will call ˜ρ
the ‘tilde state’ if no ambiguity is possible.
The operator ˜ρ appears quite frequently in
algebraic considerations regarding properties of
quantum states, in particular, entanglement. Yet
it is not obvious to attribute to it a physical signif-
icance beyond the interpretation of Eq. (3) given
in Ref. [22]: It takes a pure state to the maxi-
mally mixed state in the subspace orthogonal to
the pure state, multiplied by the dimension of
that subspace. The latter statement can be re-
expressed in the following way: Geometrically,
the single-system inverter maps a pure state of a
d-dimensional system to the barycenter of the hy-
persurface opposite to it, that is, the convex hull
of those pure states which, together with the orig-
inal state, form an orthonormal basis. The result
is then scaled with a prefactor (d −1), cf. Fig. 1.
This geometrical interpretation can be ex-
tended to the multipartite case. To this end,
consider a state of the computational basis
|j
1
. . . j
N
i ∈ H
1
⊗ ··· ⊗H
N
. Then we have
^
|j
1
. . . j
N
ihj
1
. . . j
N
| =
N
O
k=1
1
k
− |j
k
ihj
k
|
, (8)
which corresponds to the barycenter of the hy-
persurface formed by all those states of the mul-
tipartite computational basis whose k-th tensor
Accepted in Quantum 2018-05-07, click title to verify 2
factor is different from |j
k
i. This state is multi-
plied by a factor
Q
N
k=1
(d
k
− 1) that keeps track
of the total dimension of that hypersurface.
Figure 1: Illustration of the action of the state inverter
on a pure state ρ in dimension d = 3. The orange trian-
gle is a section through the qutrit Bloch body contain-
ing the pure state ρ at one corner (orange sphere). The
other two corners of the triangle are pure states of which
˜ρ is a linear combination. The vertical axis denotes the
multiples of the identity matrix determining the trace.
All density matrices have trace 1, thus the Bloch body
is located at a constant height. The axis crosses the
Bloch body in the completely mixed state. The state
inverter maps the pure state to the state lying furthest
on the opposite site of the completely mixed state, as
shown by the blue arrow crossing the vertical axis. This
state is an equal mixture of the other two pure states in
the triangle. At the same time, it scales the vector by
the factor (d − 1) = 2, as indicated by the second blue
arrow. This scaling maps the Bloch body to a copy at
height (d − 1), shown as semitransparent yellow trian-
gle. The resulting operator ˜ρ is again marked by a small
orange sphere.
1.2 Representation in terms of reduced states
There are several noteworthy ways to explicitly
write the state ˜ρ. The first and obvious possibil-
ity is to expand the operator product in Eq. (7)
which leads to an alternating sum over all bipar-
titions A [23]
˜ρ =
X
A
(−1)
|A|
(Tr
¯
A
ρ) ⊗ 1
¯
A
. (9)
Here A denotes a subset of {1, . . . , N}, |A| is the
number of elements in A,
¯
A is the complement
of the subset A, and ρ
A
= Tr
¯
A
ρ is the reduced
density matrix of the parties in the subset A.
1.3 Relation to the Bloch representation of the
original state
A second important way to write ˜ρ is its Bloch
representation. In order to obtain it we first con-
sider the Bloch representation of ρ itself. An N -
qudit state ρ can be written as
ρ =
1
d
1
···d
N
X
j
1
...j
N
r
j
1
...j
N
g
j
1
⊗ ··· ⊗g
j
N
(10)
=
1
d
tot
(1 + P
1
+ P
2
+ . . . + P
N
) . (11)
Here, {g
j
k
} are sets of d
2
k
−1 traceless generators
of SU(d
k
), g
j
k
= 1
k
for j
k
= 0, r
j
1
...j
N
∈ C.
Moreover, d
tot
= d
1
···d
N
is the total dimension
of the Hilbert space. We use the symbol P
q
for
the sum of all those terms in Eq. (10) that contain
the same number q of notrivial generators.
The Bloch representation of ˜ρ is particularly
nice if all subsystem dimensions are equal, i.e.,
d
1
= d
2
= . . . = d
N
≡ d and d
tot
= d
N
. In
order to avoid cumbersome notation we restrict
ourselves to this case. Then it is easy to see by
directly applying the product form of the state
inversion operator Eq. (7) to each term in the
Bloch representation Eq. (11) separately that
˜ρ =
1
d
N
h
(d − 1)
N
1 − (d − 1)
N−1
P
1
+
+ (d − 1)
N−2
P
2
− + . . . +
+ (−1)
N−1
(d − 1)P
N−1
+ (−1)
N
P
N
i
.
(12)
It is obvious from Eq. (12) that for d > 2 the
universal state inverter is not a trace-preserving
map. Further, it shows that only for d = 2 the
tilde state of a pure state Π
Nqubit
is pure as well,
because in that case we have for the purity of the
tilde state
Tr
˜
Π
Nqubit
2
= Tr
˜
Π
2
Nqubit
=
1
2
N
X
j
1
...j
N
r
j
1
...j
N
2
= 1 .
1.4 Explicit generator representation
Finally, we want to find a representation of ˜ρ
in terms of generators of the unitary group. To
Accepted in Quantum 2018-05-07, click title to verify 3
this end, we use the generalized Gell-Mann ma-
trices [28] in d dimensions
x
(d)
kl
=
s
d
2
(|kihl| + |lihk|) (13a)
y
(d)
kl
=
s
d
2
(−i |kihl| + i |lihk|) (13b)
z
(d)
l
=
s
d
l(l + 1)
−l |lihl| +
l−1
X
j=0
|jihj|
(13c)
where 0 5 k < l < d. The normalization is cho-
sen such that
Tr(x
kl
x
mn
) = Tr(y
kl
y
mn
) = d δ
km
δ
ln
,
Tr(z
l
z
n
) = d δ
ln
.
(14)
Note that, up to a factor, these are the usual d-
dimensional Gell-Mann matrices.
Just as above in Eq. (10) it is useful to include
the identity matrix, as this gives a complete basis
for the Hermitian matrices. Sometimes we will
use a uniform labeling with only one index and
therefore introduce also the alternative form
h
(d)
0
= 1 (15a)
h
(d)
l
2
+2k
= x
kl
(15b)
h
(d)
l
2
+2k+1
= y
kl
(15c)
h
(d)
l
2
+2l
= z
l
(15d)
where the index corresponds to the usual num-
bering of the Gell-Mann matrices. Relations (14)
then simplify to
Tr(h
k
h
l
) = d δ
kl
. (16)
For the following considerations we need two for-
mulae for the single-system density matrix ρ ∈
B(H) with dim H = d (a proof is given in Ap-
pendix A)
(Tr ρ)1 =
1
d
d
2
−1
X
k=0
h
k
ρh
k
(17)
ρ
T
=
1
d
d
2
−1
X
k=0
h
T
k
ρh
k
. (18)
By applying these relations to Eq. (3) we find for
the tilde state of a single system
˜ρ = (Tr ρ)1 − ρ
=
h
(Tr ρ)1 − ρ
T
i
∗
=
1
d
d
2
−1
X
k=0
h
h
k
ρh
k
− h
T
k
ρh
k
i
∗
. (19)
To proceed from here, we note that the genera-
tors y
kl
change their sign on transposition, while
all other generators remain unchanged under that
operation. Therefore the sum simplifies to
˜ρ =
2
d
d−2
X
k=0
d−1
X
l=k+1
(y
kl
ρ y
kl
)
∗
=
2
d
d−2
X
k=0
d−1
X
l=k+1
y
kl
ρ
∗
y
kl
. (20)
It is straightforward to extend the result (20)
to states of multipartite systems ρ ∈ B(H
1
⊗···⊗
H
N
). For this purpose, we introduce the simpli-
fied notation
k = (k
1
, . . . , k
N
)
y
kl
= y
k
1
l
1
⊗ y
k
2
l
2
⊗ ··· ⊗y
k
N
l
N
d−1
X
k<l
≡
d
1
−2
X
k
1
=0
d
1
−1
X
l
1
=k
1
+1
···
d
N
−2
X
k
N
=0
d
N
−1
X
l
N
=k
N
+1
and can then write
˜ρ =
2
N
d
tot
d−1
X
k<l
y
kl
ρ
∗
y
kl
. (21)
This is a remarkable result, not only because of its
simplicity. It makes explicit that universal state
inversion is an antilinear operation. In fact, ˜ρ in
Eq. (21) is in all respects the natural generaliza-
tion of Wootters’ tilde state. We note also that
for pure states ρ
ψ
= |ψihψ| ≡ Π
ψ
the tilde state
˜
Π
ψ
in general is mixed, only for d
k
= 2 the gener-
ator sum contains a single term, and
˜
Π
ψ
becomes
pure as well.
Finally, it is immediately clear from Eq. (21)
that the tilde state ˜ρ is positive: With ρ, also ρ
∗
is positive and therefore ρ
∗
=
√
ρ
∗
·
√
ρ
∗
, so that
˜ρ =
2
N
d
tot
d−1
X
k<l
y
kl
p
ρ
∗
p
ρ
∗
y
kl
=
2
N
d
tot
d−1
X
k<l
p
ρ
∗
y
kl
†
p
ρ
∗
y
kl
. (22)
That is, ˜ρ is a sum of positive operators and there-
fore also positive. An important consequence of
Eq. (22) is
Tr (ρ˜ρ) = Tr
p
˜ρ
√
ρ
†
p
˜ρ
√
ρ
= 0 .
(23)
Accepted in Quantum 2018-05-07, click title to verify 4
2 Distributed concurrence
2.1 New concurrence-type invariant
By using the notation |ψihψ| ≡ Π
ψ
we can ex-
press the concurrence Eq. (5) for the pure state of
a bipartite system as C(ψ) =
q
Tr
Π
ψ
˜
Π
ψ
. It is
straightforward to extend this definition to pure
states of multipartite systems, ψ ∈ H
1
⊗ ···H
N
.
We try
C
D
(ψ)
2
= Tr
Π
ψ
˜
Π
ψ
=
2
N
d
tot
d−1
X
k<l
Tr
Π
ψ
y
kl
Π
∗
ψ
y
kl
= (−1)
N
2
N
d
tot
d−1
X
k<l
|hψ
∗
|y
kl
|ψi|
2
.
Here we see that for odd N, the quantity C
D
(ψ)
vanishes identically, for any local dimension,
in analogy with the behavior of the N-qubit
LSL invariant H(φ
N
) = hφ
∗
N
|σ
⊗N
2
|φ
N
i (where
φ
N
∈ C
⊗N
2
). This is because hφ
∗
N
|y
kl
|φ
N
i =
hφ
N
|y
∗
kl
|φ
∗
N
i
∗
= hφ
∗
N
|(−y
kl
) |φ
N
i ≡ 0 for an odd
number N of factors in the tensor product y
kl
.
We can define now the distributed concurrence
C
D
(ψ) =
q
Tr
Π
ψ
˜
Π
ψ
(24)
=
v
u
u
t
2
N
d
tot
d−1
X
k<l
|hψ
∗
|y
kl
|ψi|
2
. (25)
Since the universal state inverter Eq. (6) com-
mutes with local unitary operations [22], the dis-
tributed concurrence is a local unitary invariant.
We will show below that C
D
(ψ) in certain cases
is also an entanglement monotone.
Again, the quantity C
D
(ψ) generalizes the
known concurrence-type quantities in the most
natural way: For N-qubit states φ
N
the LSL in-
variant |H(φ
N
)| is obtained, in particular for N =
2 we get Wootters’ concurrence |hφ
∗
2
|σ
2
⊗σ
2
|φ
2
i|.
On the other hand, for two parties in dimensions
d
1
= d
2
> 2 we get the standard d × d concur-
rence, Eq. (5) (cf. also Ref. [29, 30, 31]).
Concluding this section, we note that the op-
erator Π
ψ
˜
Π
ψ
Π
ψ
is proportional to Π
ψ
because of
the projector property of Π
ψ
. Then, by virtue of
Eq. (24), we have
Π
ψ
˜
Π
ψ
Π
ψ
= C
D
(ψ)
2
Π
ψ
. (26)
2.2 Distributed concurrence is an entangle-
ment monotone for local dimensions smaller
than four
In order to investigate the monotone property
of the distributed concurrence we need to study
whether or not, for pure states, C
D
(ψ) can in-
crease under the action of separable Kraus opera-
tors O
j
= A
(j)
1
⊗. . . ⊗A
(j)
N
(where
P
j
O
†
j
O
j
= 1),
which represent stochastic local operations and
classical communication (SLOCC) [32, 33]. If C
D
is an entanglement monotone for pure states, it
can be extended to the mixed states, as usual,
through the convex roof [34].
As the distributed concurrence acts symmetri-
cally on the parties it suffices to consider actions
on the first party only, A
j
= A
j
⊗1
2
⊗. . . ⊗1
N
.
We may assume that the number of parties N
is even and that the reduced density matrix of
the first party has rank r
1
> 1 because otherwise
the monotone property is trivially valid. Further,
since all generalized measurements may be com-
posed from two-outcome measurements, we will
consider Kraus operators with A
†
1
A
1
+A
†
2
A
2
= 1.
Then, for ψ ∈ H
1
⊗ ··· ⊗ H
N
we need to show
C
D
(ψ) = p
1
C
D
A
1
ψ
√
p
1
!
+ p
2
C
D
A
2
ψ
√
p
2
!
= C
D
(A
1
ψ) + C
D
(A
2
ψ) (27)
with p
j
= Tr
A
j
Π
ψ
A
†
j
. In the second line we
have used the property C
D
(αψ) = |α|
2
C
D
(ψ).
We now rewrite Eq (27) by using the decom-
position Π
ψ
=
P
jk
|e
j
f
j
ihe
k
f
k
|, where {|e
j
i} are
orthonormal vectors that span the Hilbert space
of the first party and are chosen such that they di-
agonalize the operator A
†
1
A
1
. The vectors {|f
j
i}
span the Hilbert space of the last (N − 1) par-
ties in ψ and are not necessarily orthogonal or
normalized, but they obey
P
j
hf
j
|f
j
i = 1. We
obtain
tr
Π
ψ
˜
Π
ψ
= 2
X
jk
F
kjjk
tr
A
1
Π
ψ
A
†
1
^
A
1
Π
ψ
A
†
1
= 2
X
kl
F
kjjk
D
j
D
k
where we have used the definitions
D
k
δ
jk
≡ he
k
|A
†
1
A
1
|e
j
i , 0 5 D
k
5 1 ,
F
klmj
≡ hf
k
|
^
|f
l
ihf
m
|
|f
j
i .
Accepted in Quantum 2018-05-07, click title to verify 5
Note that F
klmj
= F
lkjm
= −F
lkmj
, which follows
with the explicit inversion formula (21). Finally,
p
1
= Tr
A
1
Π
ψ
A
†
1
=
P
j
D
j
hf
j
|f
j
i and p
2
=
Tr
A
2
Π
ψ
A
†
2
=
P
j
(1 − D
j
) hf
j
|f
j
i.
With all this, we can transform Eq. (27) into
the equivalent inequality
s
X
jk
F
jkkj
=
s
X
jk
F
jkkj
D
j
D
k
+
+
s
X
jk
F
jkkj
(1 − D
j
)(1 − D
k
) .
(28)
Now it is straightforward to prove the monotone
property for the well-known cases d = 2, N ar-
bitrary, and N = 2, d arbitrary, as we show in
Appendix B. For the general case, we continue
transforming Eq. (28) by subtracting the second
term on the right, squaring and simplifying the
result, to obtain
X
jk
w
jk
D
j
2
=
X
jk
w
jk
D
j
D
k
, (29)
with the weights w
jk
≡ F
jkkj
/ (
P
lm
F
lmml
). Note
that, if the denominator here were zero, the
monotone property would be trivially satisfied.
It is possible to construct examples of {D
k
} and
{F
lmml
} such that inequality (29) is violated for
dim H
1
= 4 (see Appendix C). Therefore, C
D
is
not an entanglement monotone if a party has local
dimension d
j
= 4 and the party number N = 4.
Similarly, such sets can be found to show that C
2
D
is not a monotone for d
j
= 3, N = 4.
Remarkably, however, Eq. (29) does hold for
dim H
1
= 3, that is, C
D
is an entanglement
monotone for any number of parties if all local
dimensions are not larger than three. This be-
havior resembles that of LSL invariants for which
it is known that the monotone property depends
on the homogeneity degree [35] and, in fact, also
on the local dimension.
For the proof of Eq. (29) in the case dim H
1
= 3
we collect only the D
2
j
terms on the left and
multiply both sides by 4. Moreover, we use
2(w
01
+ w
02
+ w
12
) = 1 and obtain
(1 − 2w
12
)
2
D
2
0
+ (1 − 2w
02
)
2
D
2
1
+ (1 − 2w
01
)
2
D
2
2
=D
0
q
2(2w
01
) − (1 − 2w
12
)(1 − 2w
02
)
2
D
1
+
+ D
0
q
2(2w
02
) − (1 − 2w
12
)(1 − 2w
01
)
2
D
2
+
+ D
1
q
2(2w
12
) − (1 − 2w
01
)(1 − 2w
02
)
2
D
2
+
+ D
1
q
2(2w
01
) − (1 − 2w
12
)(1 − 2w
02
)
2
D
0
+
+ D
2
q
2(2w
12
) − (1 − 2w
02
)(1 − 2w
01
)
2
D
1
+
+ D
2
q
2(2w
02
) − (1 − 2w
12
)(1 − 2w
01
)
2
D
0
. (30)
The peculiar way of writing Eq. (30) indicates
how one can see that this is precisely a Cauchy-
Schwarz inequality, and therefore is valid. This
concludes the proof that Eq. (29) holds for
dim H
1
= 3.
To summarize this section (see also Fig. 2),
we have found that both C
D
and C
2
D
are entan-
glement monotones only in the well-known cases
and, trivially, for odd number of parties N . For
local dimensions d
j
5 3 and even party number
N, C
D
is an entanglement monotone, but not C
2
D
.
For local dimensions d
j
= 4 neither C
D
nor C
2
D
is
an entanglement monotone for any even N > 2.
3 Distribution of correlations and
monogamy of entanglement
In Ref. [18], the monogamy relation Eq. (2) for
N qubits was derived. The strategy there was to
write down a local SL invariant for an N-qubit
state ρ
N
via combining ρ
N
and ˜ρ
N
, and, by ex-
Accepted in Quantum 2018-05-07, click title to verify 6
Figure 2: The local unitary invariant C
D
, cf. Eq. (24),
termed distributed concurrence, naturally generalizes the
well-known cases of bipartite concurrence d = 2, N = 2
[22, 29] and H invariant for N -qubit system [20]. While
in those cases both C
D
and C
2
D
are monotones, the
general case d > 2, N > 2 is different: C
D
(but not C
2
D
)
is an entanglement monotone only if all local dimensions
d 5 3. If there is one subsystem of dimension at least
four, neither C
D
nor C
2
D
are entanglement monotones.
panding that expression in terms of Bloch com-
ponents to obtain an identity which connects the
invariant (for the global state) with combinations
of local quantities (linear entropies). Now we ask
whether one can obtain a new result by generaliz-
ing Eq. (2) in the frame of the present formalism.
The answer is affirmative, as we will show.
We consider a state ρ of a finite-dimensional
multipartite quantum system ρ ∈ B(H
1
⊗ . . . ⊗
H
N
). Clearly, Tr (ρ˜ρ) is a local unitary invari-
ant. We expand this expression as a sum over all
bipartite splits A|
¯
A by applying Eq. (9)
Tr (ρ˜ρ) = 1 +
X
|A|>0
(−1)
|A|
Tr
h
ρ Tr
¯
A
ρ ⊗ 1
¯
A
i
= 1 +
X
|A|>0
(−1)
|A|
Tr
A
h
Tr
¯
A
ρ
i
2
=
1
2
X
|A|>0
(−1)
|A|+1
2
1 − Tr ρ
2
A
By using the definition of the linear entropy on
the partition A, τ
A
= 2(1 − Tr ρ
2
A
) we obtain
0 5 2 Tr (ρ˜ρ) =
X
|A|>0
(−1)
|A|+1
τ
A
. (31)
This is an equality for the distribution of corre-
lations and holds both for pure and for mixed
states of any finite-dimensional multipartite sys-
tem. The general validity of Eq. (31) is quite
remarkable: In Ref. [18] it was derived by using
a qubit-specific formalism, therefore its full gen-
erality could not be expected.
It is in order here to recall the quantum
marginal problem [19], which poses the question
whether or not a given set of marginal states is
compatible with a (possibly unique) global state.
The equality (31) represents a simple necessary
(but not sufficient) condition for a favorable an-
swer to that question.
It is interesting here to consider the special case
of mixed states ρ
ABC
of three-parties A, B, and
C. From Eq. (31) and Tr(ρ˜ρ) = 0 we find
τ
A
+ τ
B
+ τ
C
+ τ
ABC
= τ
AB
+ τ
AC
+ τ
BC
.
(32)
This reminds of the strong subadditivity inequal-
ity for the von-Neumann entropy S [37], which
reads S
ABC
+ S
B
5 S
AB
+ S
BC
. However, in-
equality (32) is symmetrized between the parties
and has the opposite direction.
We emphasize that the τ
A
in Eq. (31) are sim-
ply the linear entropies of the reduced states ρ
A
,
so they are not entanglement measures. However,
if we consider pure states |ψihψ| ≡ Π
ψ
, we note
τ
A
(ψ) = 0 for |A| = N, and the equality reads
0 5 2C
2
D
(ψ) =
X
|A|>0
(−1)
|A|+1
τ
A
(ψ)
=
X
N>|A|>0
(−1)
|A|+1
C
2
A|
¯
A
(ψ) .
(33)
Here we denote by C
A|
¯
A
(ψ) the bipartite concur-
rence on the split A|
¯
A according to Eq. (5). We
see that Eq. (33) represents a true monogamy re-
lation, because all the terms on the right-hand
side are entanglement measures. If the local di-
mensions are not larger than three, also C
D
is
an entanglement measure, so that we even have
a monogamy equality. Thus, Eq. (33) is a con-
straint for the distribution of entanglement across
all bipartitions of an arbitrary multipartite pure
state of finite dimensionality. For odd N the
equality is trivial, because the purities on com-
plementary partitions A and
¯
A enter it with a
different sign, confirming that C
D
(ψ) = 0 in this
case. We note that aspects of the results dis-
cussed here and summarized in Eqs. (31), (33)
were obtained in Ref. [36].
It is an interesting feature of Eqs. (33), (32)
that they describe all local parties in a symmetric
manner, whereas the Coffman-Kundu-Wootters
identity Eq. (1) or the strong subadditivity in-
equality single out one party. Our way of deriv-
ing the general monogamy relation (33) seems to
Accepted in Quantum 2018-05-07, click title to verify 7
suggest that the symmetric form has a peculiar
role, just because the tilde state ˜ρ is symmetric
in the local parties as well.
Figure 3: Illustration of the N -qudit monogamy relation
Eq. (33). The left-hand side symbolizes C
2
D
while, on
the right-hand side, there are the representations of the
linear entropies of all possible bipartite splits in the set
of N qudits.
Top panel: Four qudits. Complementary partitions A
and
¯
A enter with the same sign.
Bottom panel: Five qudits. For odd N, the monogamy
relation is trivial because the linear entropy of a parti-
tion A equals that of its complementary partition
¯
A, but
contributes with opposite sign.
4 Other interesting aspects
4.1 Conservation laws for correlations
In the previous section we have emphasized that
in Eq. (31) all terms are local unitary invariants.
The left-hand side is locally invariant on each sin-
gle party, whereas the terms on the right-hand
side are locally invariant on their respective bi-
partition. We explain now that Eq. (31) gives
rise to nontrivial unitary invariants on an entire
subset of the parties.
To this end, let us consider an example of a
four-party state ρ
1234
where each index denotes
the respective party. The reduced state, e.g., of
parties {13} is ρ
13
= Tr
24
ρ
1234
. Correspondingly,
the linear entropy of parties {13} is τ
13
= 2(1 −
Tr
13
ρ
2
13
). Then we have
Tr ρ
1234
˜ρ
1234
= τ
1
+ τ
2
+ τ
3
+ τ
4
− τ
12
− . . . −
− τ
34
+ τ
123
+ τ
124
+ τ
134
+ τ
234
−
− τ
1234
. (34)
Assume now that the subsystem of parties {13}
is subject to unitary evolution U
13
(t) = 1
24
⊗
exp (−iH
13
t) generated by a (time-independent)
Hamiltonian H
13
. The full state ρ
1234
be mixed,
e.g., the thermal state at some temperature. Let
us collect all the terms in Eq. (34) in which both
of the parties 1 and 3, or none of them, are traced
out:
−τ
2
− τ
4
+ τ
13
+ τ
24
− τ
123
− τ
134
+ τ
1234
=
= − Tr ρ
1234
˜ρ
1234
+ τ
1
+ τ
3
− τ
12
− τ
14
−
− τ
23
− τ
34
+ τ
124
+ τ
234
(35)
The left-hand side of Eq. (35) is invariant un-
der the evolution U
13
(t) while each of the terms
on the right will vary in a nontrivial way, as the
time evolution generates or degrades correlations
of the corresponding parties with the other sub-
systems. This means Eq. (35) represents a con-
servation law for correlations. Analogous conser-
vation laws can be found for Hamiltonian subsys-
tem dynamics of any finite-dimensional multipar-
tite state.
4.2 Relation between distributed concurrence
and topology
A striking property of Eqs. (31) and (33) is their
apparent connection with topology. Just the
form of those equations hints at the so-called Eu-
ler characteristic [38], a number characterizing
the structure of a topological space. Yet in the
present case the nature of the topological entity,
described in particular by Eq. (33) is not really
clear. Another problem is that Eq. (33) in general
is not a relation between integers, which makes its
topological interpretation questionable.
However, there are more indications in the
properties of C
D
that there indeed exists a re-
lation to topology. First, it is known that a
closed topological space of odd dimension has
Euler characteristic 0. This coincides with the
property C
D
≡ 0 for systems with odd number
of parties N. Further, the Euler characteristic
of a product manifold equals the product of the
Euler characteristics of the factors. Again, it is
easy to see from the explicit definition of C
D
,
Eq. (25), that the distributed concurrence has the
same property. Finally, the Euler characteristic
bears a close relation with the inclusion-exclusion
principle. The derivation of Eq. (33) for qubits
in Ref. [18] via the Bloch representation is an ex-
plicit application of this principle.
We can only speculate what the topological na-
ture of Eq. (33) is. A tempting interpretation is
the following. We regard the N parties as vertices
of a standard (N −1)-simplex. In a simplex, each
Accepted in Quantum 2018-05-07, click title to verify 8
vertex is connected to each other vertex. This ap-
pears to correspond to the distribution of entan-
glement in a genuinely entangled [33] pure state:
A given party is entangled with any other party
in such a pure state. In a strict sense there is no
distinction, such as ‘only pairwise entanglement’,
as is possible for mixed states. The boundary
of an (N − 1)-simplex is topologically equivalent
to a sphere S
N−2
and therefore has Euler char-
acteristic 1 + (−1)
N−2
which corresponds to the
behavior of C
D
(vanishing C
D
for odd N). Fi-
nally, if the pure state can be written as a prod-
uct of exactly two factors (with party numbers k
and N − k), the topology would be that of two
disjoint simplices, one with k and the other with
N − k vertices.
5 Conclusions
We have analyzed universal state inversion ap-
plied to multipartite quantum states with finite-
dimensional parties. This map takes pure
states to the opposite hypersurface of the state
space and therefore algebraically encodes ex-
tremal properties of quantum states. We made
explicit that this map is antilinear. In the frame-
work of our formalism we were able to derive a
universally valid equality, Eq. (31), that relates
a local unitary invariant depending on the global
state to the linear entropies of all possible reduced
states. Remarkably, if we apply this equality to
pure states we obtain the first known monogamy
relation for bipartite entanglement in a multipar-
tite state, Eq. (33), which is valid for any number
of parties and any finite local dimension. These
findings appear to have a close relation with the
topological properties of multipartite quantum
systems, as we have argued in Sec. 4.2. Moreover,
our correlation equalities give rise to conservation
laws for correlations, which are valid for any type
of unitary subsystem dynamics.
The local unitary invariant mentioned above,
the distributed concurrence, describes the quan-
titative distribution of bipartite entanglement in
a multipartite pure state. It naturally general-
izes the concurrence in the well-known cases, that
is, for bipartite systems of arbitrary dimension
and for many-qubit systems. Thus, it provides
both the mathematical and conceptual link be-
tween these hitherto somewhat unrelated quan-
tities, see Eqs (2),(5). Intriguingly, for systems
of even party number N = 4 the distributed
concurrence is an entanglement monotone only
below a ‘critical’ local dimension d 5 3. Note,
however, that for its role as a ‘generator’ of the
monogamy inequality the monotone property of
the distributed concurrence is not essential.
Our work shows that, by means of algebraic
methods, it is possible to express the mathemat-
ical constraints to quantum states, in particu-
lar their positivity, in terms of physically rele-
vant quantities and principles. Importantly, our
results establish necessary relations between the
global and the local states which may be con-
sidered as a partial answer in the context of the
quantum-marginal problem. Clearly, our result
highlights only one specific aspect of correlations,
and the question is whether also other mathe-
matical constraints allow for a similar formulation
with direct access to their physical meaning.
Acknowledgments
This work was funded by the German Re-
search Foundation Project EL710/2-1 (C.E.),
by Basque Government grant IT986-16,
MINECO/FEDER/UE grants FIS2012-36673-
C03-01 and FIS2015-67161-P and UPV/EHU
program UFI 11/55 (J.S.). The authors would
like to thank Jaroslav Fabian, Felix Huber, Mar-
cus Huber, Peter D. Jarvis, and Denis Kochan
for stimulating discussions, and Klaus Richter
for supporting this project. F.H. pointed out to
us Refs. [23, 24, 36]. D.K. noticed the apparent
similarity between the monogamy relation and
the Euler characteristic.
A Proof of useful generator formulae
The interesting relations (17), (18) are closely
linked to the well-known completeness relation
for the set of generators {h
k
} (see, e.g., [28]). It
is instructive to derive all these relations in the
same framework, i.e., by consequently using the
terminology of quantum information.
An important ingredient is the SWAP operator.
Consider two states |ψi, |φi ∈ H. Then we define
SWAP · (|ψi ⊗ |φi) = |φi ⊗ |ψi . (36)
With the computational basis {|ji} of H it
is straightforward to show that SWAP =
Accepted in Quantum 2018-05-07, click title to verify 9
P
jk
|jkihkj|. Let us denote by Tr
[1]
the partial
trace over the first tensor factor in H ⊗ H, and
correspondingly by Tr
[2]
the partial trace over the
second factor. Then we find by explicit calcula-
tion in the computational basis that for two op-
erators A, B ∈ B(H)
Tr
[1]
(SWAP · (A ⊗ B)) = A · B (37)
Tr
[2]
(SWAP · (A ⊗ B)) = B · A (38)
and
Tr
[1]
((A ⊗ B) · SWAP) = B · A (39)
Tr
[2]
((A ⊗ B) · SWAP) = A · B . (40)
Important special cases of these relations are ob-
tained by setting, e.g., B = 1,
Tr
[1]
(SWAP · (A ⊗ 1)) = A . (41)
Now consider a set {h
j
}, j = 1 . . . (d
2
−1) of trace-
less Hermitian generators of SU(d), Tr (h
j
h
k
) =
dδ
jk
, h
0
≡ 1, so that we can expand
A =
1
d
d
2
−1
X
j=0
a
j
h
j
=
1
d
d
2
−1
X
j=0
Tr (h
j
A) h
j
(42)
= Tr
[1]
1
d
d
2
−1
X
j=0
h
j
⊗ h
j
·
A ⊗ 1
. (43)
Because A is an arbitrary operator here, com-
parison of Eqs. (41), (43) yields the completeness
relation for the generators h
j
SWAP =
1
d
d
2
−1
X
j=0
h
j
⊗ h
j
. (44)
With these prelimary considerations we can
turn to the proof of the operator relations in
Sec. 1.4. Inserting Eq. (44) into the identity
SWAP · (A ⊗ 1) · SWAP = 1 ⊗ A
and tracing over the second party gives
Tr (A) 1 =
1
d
2
d
2
−1
X
j,k=0
(h
j
Ah
k
) Tr (h
j
h
k
)
=
1
d
d
2
−1
X
j=0
h
j
Ah
j
, (45)
which proves Eq. (17).
From the explicit representation of SWAP in
the computational basis it is obvious that it is
related to the maximally entangled state
Φ
+
≡
1
√
d
P
j
|jji by virtue of partial transposition, e.g.,
on the first party, T
[1]
,
SWAP
T
[1]
=
X
jk
|jkihkj|
T
[1]
=
X
jk
|kkihjj|
= d
Φ
+
ED
Φ
+
. (46)
We conclude from Eqs. (44) and (46) that
SWAP
T
[1]
=
1
d
d
2
−1
X
j=0
h
T
j
⊗ h
j
. (47)
In analogy with Eqs. (37)–(40) we obtain now (by
explicit calculation in the computational basis)
Tr
[1]
d
Φ
+
ED
Φ
+
· (A ⊗ B)
= A
T
· B (48)
Tr
[2]
d
Φ
+
ED
Φ
+
· (A ⊗ B)
= B
T
· A (49)
Tr
[1]
(A ⊗ B) · d
Φ
+
ED
Φ
+
= B · A
T
(50)
Tr
[2]
(A ⊗ B) · d
Φ
+
ED
Φ
+
= A · B
T
. (51)
By using Eq. (48) with B = 1, the permutation
invariance of
Φ
+
, and Eq. (47) we find
A
T
= Tr
[1]
d
Φ
+
ED
Φ
+
· (A ⊗ 1)
= Tr
[1]
SWAP · d
Φ
+
ED
Φ
+
· (A ⊗ 1)
=
1
d
d
2
−1
X
j=0
Tr
[1]
SWAP ·
h
(h
T
j
A) ⊗ h
j
i
and, finally, with Eq. (37)
A
T
=
1
d
d
2
−1
X
j=0
h
T
j
Ah
j
, (52)
which proves Eq. (18).
B Proof of monotone property in the
known cases
For the sake of completeness, we present the proof
for the monotone property of both C
D
and C
2
D
in
the known cases in the frame of the formalism in
Sec. 2.2.
d = 2, N arbitrary: Here, there are
only two indices j ∈ {0, 1} and only one
Accepted in Quantum 2018-05-07, click title to verify 10
value of F
0110
= F
1001
, so that the inequal-
ity (28) becomes
√
2F
0110
=
√
2F
0110
D
0
D
1
+
p
2F
0110
(1 − D
0
)(1 − D
1
) from which we have
1 =
p
D
0
D
1
+
q
(1 − D
0
)(1 − D
1
)
and further
1 −
p
D
0
D
1
2
= (1 − D
0
)(1 − D
1
)
which is equivalent to
D
0
+ D
1
= 2
p
D
0
D
1
and always fulfilled because of the relation be-
tween arithmetic and geometric mean.
The monotone condition Eq. (27) for the square
of C
D
reads
C
2
D
(ψ) = p
1
C
2
D
A
1
ψ
√
p
1
!
+ p
2
C
2
D
A
2
ψ
√
p
2
!
=
C
2
D
(A
1
ψ)
p
1
+
C
2
D
(A
2
ψ)
p
2
. (53)
In the proof for C
2
D
we use the probability p
1
=
hf
0
|f
0
iD
0
+ hf
1
|f
1
iD
1
(recall that
P
j
hf
j
|f
j
i =
1), and we have to show that
1 =
D
0
D
1
p
1
+
(1 − D
0
)(1 − D
1
)
1 − p
1
.
This inequality is equivalent to
0 = (p
1
− D
0
)(p
1
− D
1
)
which is always fulfilled as is easily checked with
the explicit formula for p
1
.
N = 2, d arbitrary: In this case we have
F
jkkj
= hf
j
|
^
|f
k
ihf
k
|
|f
j
i
= hf
j
|
Tr
|f
k
ihf
k
|
1 − |f
k
ihf
k
|
|f
j
i
= hf
j
|f
j
ihf
k
|f
k
i − |hf
j
|f
k
i|
2
,
because the subnormalized vectors {|f
j
i} belong
to a single party. and inequality (28) takes the
form
s
1 −
X
jk
|hf
j
|f
k
i|
2
=
s
p
2
1
−
X
jk
|hf
j
|f
k
i|
2
D
j
D
k
+
+
s
(1 − p
1
)
2
−
X
jk
|hf
j
|f
k
i|
2
(1 − D
j
)(1 − D
k
) .
The right-hand side (rhs) of this relation can be upper-bounded by using the concavity of the
square root. Subsequent squaring yields
(rhs)
2
5 1 −
P
jk
|hf
j
|f
k
i|
2
D
j
D
k
p
1
−
P
jk
|hf
j
|f
k
i|
2
(1 − D
j
)(1 − D
k
)
1 − p
1
5 1 −
X
jk
|hf
j
|f
k
i|
2
p
1
(1 − p
1
) + (p
1
− D
j
)(p
1
− D
k
)
p
1
(1 − p
1
)
.
Hence, in order to prove the monotone property
it remains to show that
0 5
X
jk
|hf
j
|f
k
i|
2
(p
1
− D
j
)(p
1
− D
k
) .
This can be seen by noting that
X
jk
|hf
j
|f
k
i|
2
(p
1
− D
j
)(p
1
− D
k
) = Tr
W
†
W
= 0
Accepted in Quantum 2018-05-07, click title to verify 11
with the Hermitian operator
W =
X
j
|f
j
ihf
j
|(p
1
− D
j
) ,
which concludes the proof. We note that appli-
cation of the concavity of the square root above
implies that also C
2
D
is a monotone in this case.
C Examples for states violating the
monotone property
As we have mentioned in Sec. 2.2, it is possible to
find examples violating the monotone conditions,
Eqs. (28), (29), which implies that C
D
cannot
be an entanglement monotone for the local di-
mension in question, and all higher dimensions
as well.
A numerical strategy for finding such coun-
terexamples for d = 4 is the following. We
can restrict our attention to channels which
are diagonal in the Schmidt basis of the first
party. Then, the decomposition |ψi =
P
j
|e
j
f
j
i
from Sec. 2.2 becomes the Schmidt decompo-
sition |ψi =
P
j
p
λ
j
e
j
¯
f
j
E
with orthonormal
{
¯
f
j
E
} and
P
j
λ
j
= 1. A strategy to look for
parameters to violate Eq. (29) is to construct
a matrix F
kjjk
≡ λ
j
λ
k
¯
F
kjjk
(where
¯
F
kjjk
≡
D
¯
f
k
^
¯
f
j
ED
¯
f
j
!
¯
f
k
E
) with sufficiently extremal
entries, that is, with values close to 0 and oth-
ers of order 1.
A matrix
¯
F
kjjk
with such extremal entries can
be found as follows. First, we generate a state ψ
1
with random entries and apply the state inverter
to the projector Π
ψ
1
. Since we consider an odd
number of parties, the state ψ
1
is an eigenstate
of
˜
Π
ψ
1
with eigenvalue 0. In general, the other
eigenvalues of
˜
Π
ψ
1
are distributed between 0 and
numbers of order 1, so that we can choose three
of those other eigenstates of
˜
Π
ψ
1
with suffienctly
different eigenvalues. Once we have computed the
matrix
¯
F
kjjk
we can perform a random search
for Schmidt coefficients {λ
j
} and diagonal entries
of the channel {D
j
} and readily find numerical
examples that violate the monotone condition for
C
D
or C
2
D
. The same procedure can be applied
for d = 3 in order to find a counterexample for
the monotone condition of C
2
D
.
To conclude this part, we give an explicit an-
alytical counterexample for a four-party state
(with local dimensions 4, 2, 2, 2) that violates
the monotone condition both for C
D
and C
2
D
. It
reads
|ψ
4222
i =
1
√
8
(|0000i + |0011i + |1100i+
+ |1111i + |2000i − |2011i−
−|3100i + |3111i) .
The local channel acting on the first party is given
by the projectors
A
1
= diag (1, 1, 0, 0)
A
2
= diag (0, 0, 1, 1) .
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