k-stretchability of entanglement, and the duality of k-
separability and k-producibility
Szilárd Szalay
Strongly Correlated Systems “Lendület” Research Group, Wigner Research Centre for Physics, 29-33, Konkoly-Thege Miklós str., H-1121
Budapest, Hungary
The notions of k-separability and k-
producibility are useful and expressive tools
for the characterization of entanglement in
multipartite quantum systems, when a more
detailed analysis would be infeasible or simply
needless. In this work we reveal a partial
duality between them, which is valid also for
their correlation counterparts. This duality
can be seen from a much wider perspective,
when we consider the entanglement and cor-
relation properties which are invariant under
the permutations of the subsystems. These
properties are labeled by Young diagrams,
which we endow with a reﬁnement-like partial
order, to build up their classiﬁcation scheme.
This general treatment reveals a new property,
which we call k-stretchability, being sensitive
in a balanced way to both the maximal size of
correlated (or entangled) subsystems and the
minimal number of subsystems uncorrelated
with (or separable from) one another.
Contents
1 Introduction 1
2 Multipartite correlation and entangle-
ment 3
2.1 Level 0: subsystems . . . . . . . . . . 3
2.2 Level I: set partitions . . . . . . . . . . 3
2.3 Level II: multiple set partitions . . . . 5
2.4 Level III: classes . . . . . . . . . . . . 6
3 Multipartite correlation and entangle-
ment: permutation invariant properties 6
3.1 Level 0: subsystem sizes . . . . . . . . 7
3.2 Level I: integer partitions . . . . . . . 7
3.3 Level II: multiple integer partitions . . 8
3.4 Level III: classes . . . . . . . . . . . . 9
4 An alternative way of introducing per-
mutation invariance 11
5 k-partitionability, k-producibility and k-
stretchability 11
Szilárd Szalay: szalay.szilard@wigner.mta.hu
5.1 k-partitionability and k-producibility
of correlation and entanglement . . . . 11
5.2 Duality by conjugation . . . . . . . . . 13
5.3 k-stretchability of correlation and en-
tanglement . . . . . . . . . . . . . . . 14
5.4 Relations among k-partitionability, k-
producibility and k-stretchability . . . 15
6 Summary, remarks and open questions 16
Acknowledgments 18
A On the structure of the classiﬁcation of
permutation invariant correlations 18
A.1 Coarsening a poset . . . . . . . . . . . 18
A.2 Down-sets and up-sets . . . . . . . . . 19
A.3 Embedding . . . . . . . . . . . . . . . 20
A.4 Application to the set and integer par-
titions . . . . . . . . . . . . . . . . . . 21
B Miscellaneous proofs 22
B.1 Monotonicity of correlation measures . 22
References 22
1 Introduction
The investigation of the correlations among the parts
of a composite physical system is an essential tool
in statistical physics. If the system is described by
quantum mechanics, then nonclassical forms of corre-
lations arise, the most notable is entanglement [14].
It is the main resource of quantum information the-
ory [57], and its nonclassical properties, playing im-
portant role also in many-body physics [810], make
it inﬂuential in the behavior and characterization of
strongly correlated systems [1114].
The correlation and entanglement between two
parts of a system is relatively well-understood [1517].
At least for pure states, the convertibility and the clas-
siﬁcation with respect to (S)LOCC ((stochastic) local
operations and classical communication [15, 1719])
shows a simple structure [20, 21], and there are ba-
sically unique correlation and entanglement measures
[2224]. The multipartite case is much more compli-
cated [19, 2528]. Even for pure states, the nonexis-
tence of a maximally entangled reference state [29, 30]
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and the involved nature of state-transformations in
general [31, 32] seem to make the (S)LOCC-based
classiﬁcation practically unaccomplishable, and the
standard (S)LOCC paradigm less enlightening so less
expressive. Taking into account partial entanglement
(partial separability) [3340], or partial correlations
[14, 41, 42] only, leads to a combinatoric [4345], dis-
crete classiﬁcation (based on the lattice of set parti-
tions [44]), endowed naturally with a well-behaving
set of correlation and entanglement measures, char-
acterizing the ﬁnite number of properties [14, 40].
Even the partial correlation and partial entangle-
ment properties are getting too involved rapidly, with
the increasing of the number of subsystems. Sin-
gling out particularly expressive properties, we con-
sider the k-partitionability and k-producibility of cor-
relation and of entanglement. For n subsystems, k
ranges from 1 to n, so the number of these proper-
ties scales linearly with the number of subsystems,
moreover, their structures are the simplest possi-
ble ones, chains. (In the case of entanglement, k-
partitionability is called k-separability [35, 37], while
k-producibility [4649] is also called entanglement
depth [5052]. Here we use both concepts for cor-
relation and also for entanglement, this is why we
use the naming k-partitionability of correlation” and
k-producibility of correlation”, k-partitionability of
entanglement” and k-producibility of entanglement”
[14, 40, 41].) These characterize the strength of two
diﬀerent (one-parameter-) aspects of multipartite cor-
relation and entanglement: those which cannot be
restricted inside at least k parts, and those which
cannot be restricted inside parts of size at most k,
respectively. The concepts of k-partitionability and
k-producibility of entanglement found application in
spin chains [47, 48], appeared in quantum nonlo-
cality [53], leading to device independent certiﬁca-
tion of them [54, 55], and were also demonstrated in
quantum optical experiment [56]. k-producibility also
plays particularly important role in quantum metrol-
ogy [57]. k-producibly entangled states for larger k
lead to higher sensitivity, so better precision in phase
estimation, which has been illustrated in experiments
[5860], and which also leads to k-producibility en-
tanglement criteria [61].
k-partitionability and k-producibility are special
cases of properties invariant under the permutation
of the subsystems. The permutation invariant cor-
relation properties are based on the integer partitions
[45, 62], also known (represented) as Young diagrams.
For the description of the structure of these, we intro-
duce a new order over the integer partitions, called
reﬁnement, induced by the reﬁnement order over set
partitions, used for the description of the structure
of the partial correlation or entanglement properties
[40]. The structure of the permutation invariant prop-
erties contains the chains of k-partitionability and
k-producibility, and it is simpler than the structure
describing all the properties. The number of these
scales still rapidly with the number of subsystems, but
slower than the number of all the partial correlation
or entanglement properties.
The general treatment of the permutation invari-
ant properties reveals a partial duality between k-
partitionability and k-producibility, which is the man-
ifestation of a duality on a deeper level, relating im-
portant properties of Young diagrams, by which k-
partitionability and k-producibility are formulated.
The general treatment of the permutation invari-
ant properties reveals also a particularly expressive
new property, which we call k-stretchability, leading to
the deﬁnitions of k-stretchability of correlation” and
k-stretchability of entanglement”. It combines the
Namely, k-partitionability is about the number of sub-
systems uncorrelated with (or separable from) one
another, and not sensitive to the size of correlated
(or entangled) subsystems [47]; and k-producibility
is about the size of the largest correlated (or entan-
gled) subsystem, and not sensitive to the number of
subsystems uncorrelated with (or separable from) one
another; while k-stretchability is sensitive to both of
these, in a balanced way. The price to pay for this is
that we have roughly twice as many k-stretchability
properties, k goes from (n1) to n1. However, k-
stretchability is linearly ordered with k, while the re-
lations between k-partitionability and k-producibility
are far more complicated.
The organization of this work is as follows. In Sec-
tion 2, we recall the structure of multipartite correla-
tion and entanglement. In Section 3, we construct the
parallel structure for the permutation invariant case.
In Section 4, we show another introduction of the per-
mutation invariant classiﬁcation, which can although
be considered simpler, but less transparent. In Sec-
tion 5, we recall k-partitionability and k-producibility,
introduce k-stretchability, and show how these prop-
erties are related to each other. In Section 6, sum-
mary, remarks and open questions are listed. In Ap-
pendix A, we work out the “coarsening” step and some
other tools, used in the main text. In Appendix B, we
present the proofs of some further propositions given
in the main text.
In the course of the presentation, the abstract
mathematical structure of the correlation and entan-
glement properties (given in terms of lattice theoretic
constructions) is well-separated from the concrete hi-
erarchies of the state sets and measures. This leads
to a transparent construction, which is easy to re-
strict to the permutation invariant case later. For the
convenience of the reader, we provide a short sum-
mary on the notations. The abstract correlation and
entanglement properties are labeled by Greek letters
ξ, υ. On the three levels of the construction, these are
typesetted as ξ, ξ, ξ
for the general case, and as
ˆ
ξ,
ˆ
ξ,
ˆ
ξ
for the permutation invariant case. On the ﬁrst two
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levels of the construction, there are sets of quantum
states of given correlation and entanglement proper-
ties, typesetted as D
ξ-unc
, D
ξ-sep
, D
ξ-unc
, D
ξ-sep
for
the general case, and D
ˆ
ξ-unc
, D
ˆ
ξ-sep
, D
ˆ
ξ-unc
, D
ˆ
ξ-sep
for
the permutation invariant case. On the ﬁrst two lev-
els of the construction, there are LO(CC)-monotonic
measures of given correlation and entanglement prop-
erties, typesetted as C
ξ
, E
ξ
, C
ξ
, E
ξ
for the general
case, and C
ˆ
ξ
, E
ˆ
ξ
, C
ˆ
ξ
, E
ˆ
ξ
for the permutation invariant
case. On the third level of the construction, there are
classes (disjoint sets) of quantum states of given cor-
relation and entanglement properties, typesetted as
C
ξ-unc
, C
ξ-sep
for the general case, and C
ˆ
ξ-unc
, C
ˆ
ξ-sep
for
the permutation invariant case. The abstract correla-
tion and entanglement properties are ordered, which
is denoted by on the three levels of the construc-
tion in both the general and the permutation invari-
ant cases. On the ﬁrst two levels of the construction,
thanks to the monotonicity properties, these mani-
fest themselves as and for the state sets and for
the measures, respectively. On the third level of the
construction, these manifest themselves as LO(CC)
conversion results, for which we do not introduce no-
tation.
2 Multipartite correlation and entan-
glement
Here we recall the structure of the classiﬁcation and
quantiﬁcation of multipartite correlation and entan-
glement [14, 40, 41]. Our goal is to do this in the way
suﬃcient to see how the permutation invariant prop-
erties can be formulated parallel to this in the next
section.
2.1 Level 0: subsystems
The classiﬁcation scheme we present here is rather
general. The elementary and composite subsystems
can be any discrete ﬁnite systems possessing prob-
abilistic description, supposed that the joint sys-
tems can be represented by the use of tensor prod-
ucts, which is the basic tool in the constructions.
Such systems can be distinguishable quantum sys-
tems, second quantized bosonic systems, second quan-
tized fermionic systems with fermion number parity
superselection rule imposed, or even classical systems,
with signiﬁcant simpliﬁcation in the structure in the
latter case [41].
Let L be the set of the labels of |L| = n elementary
subsystems. All the (possibly composite) subsystems
are then labeled by the subsets X L, the set of
which, P
0
:= 2
L
, naturally possesses a Boolean lat-
tice structure with respect to the inclusion , which
is now denoted with . For every elementary sub-
system i L, let the Hilbert space H
i
be associated
with it, where 1 < dim H
i
< . From these, for ev-
ery subsystem X P
0
, the Hilbert space associated
with it is H
X
=
N
iX
H
i
. (For the trivial subsystem
X = , we have the one-dimensional Hilbert space
H
= Span{|ψi}
=
C [40].) The states of the subsys-
tems X P
0
are given by density operators (positive
semideﬁnite operators of trace 1) acting on H
X
, the
sets of those are denoted with D
X
.
The mixedness of a state %
X
D
X
of a subsystem
X can be characterized by the von Neumann entropy
[7, 6365]
S(%
X
) = Tr %
X
ln %
X
; (1a)
and the distinguishability of two states %
X
, σ
X
D
X
of subsystem X can be characterized by the Umegaki
relative entropy [6, 7, 6468]
D(%
X
kσ
X
) = Tr %
X
(ln %
X
ln σ
X
). (1b)
The von Neumann entropy is monotone increasing
in bistochastic quantum channels, and the relative
entropy is monotone decreasing in quantum chan-
nels [6, 65, 67] and S(%
X
) = 0 % = |ψihψ|,
D(%
X
kσ
X
) = 0 %
X
= σ
X
.
2.2 Level I: set partitions
For handling the diﬀerent possible splits of a compos-
ite system into subsystems, we need to use the math-
ematical notion of (set) partition [44] of the system
L. The partitions of L are sets of subsystems,
ξ = {X
1
, X
2
, . . . , X
|ξ|
} Π(L), (2a)
where the parts, X ξ, are nonempty disjoint sub-
systems, for which
S
Xξ
X = L. The set of the par-
titions of L is denoted with
P
I
:= Π(L). (2b)
Its size is given by the Bell numbers [69], which are
rapidly growing with n. There is a natural partial
order over the partitions, which is called reﬁnement
, given as
υ ξ
def.
Y υ, X ξ s.t. Y X. (2c)
(For illustrations, see Figure 1.) This grabs our natu-
ral intuition of comparing splits of systems. Note that
the reﬁnement is a partial order only, there are pairs
of partitions which cannot be ordered, for example,
12|3 13|2 and 12|3 13|2, see Figure 1. (We use
a simpliﬁed notation for the partitions, for example,
12|3 {{1, 2}, {3}}, where this does not cause confu-
sion.) Note that P
I
with the reﬁnement is a lattice,
with minimal and maximal elements = {{i} | i
L} and > = {L}, and the least upper and greatest
lower bounds, and , can be constructed [44, 45].
With respect to the partitions ξ P
I
, we can deﬁne
the partial correlation and entanglement properties,
as well as the measures quantifying them.
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The ξ-uncorrelated states are those which are prod-
ucts with respect to the partition ξ,
D
ξ-unc
:=
n
%
L
D
L
X ξ, %
X
D
X
: %
L
=
O
Xξ
%
X
o
;
(3a)
the others are ξ-correlated states. With respect to the
ﬁnest partition ξ = , we call -uncorrelated states
N
iL
%
i
simply uncorrelated states. The expectation
values of all ξ-local observables factorize if and only if
the state is ξ-uncorrelated. The ξ-uncorrelated states
are exactly those which can be prepared from uncor-
related states by ξ-local operations. (This is abbre-
viated as ξ-LO. With respect to the ﬁnest partition
ξ = , we write simply LO.) The ξ-separable states
are convex combinations (or statistical mixtures) of
ξ-uncorrelated states,
D
ξ-sep
:= Conv D
ξ-unc
n
X
j
p
j
%
j
%
j
D
ξ-unc
o
;
(3b)
the others are ξ-entangled states. (Here, and in the
entire text, {p
j
| j = 1, 2, . . . , m} is a probability dis-
tribution, p
j
0 and
P
j
p
j
= 1.) The ξ-separable
states are exactly those which can be prepared from
uncorrelated states by ξ-local operations and classi-
cal communication among the parts X ξ. (This
is abbreviated as ξ-LOCC. With respect to the ﬁnest
partition , we write simply LOCC.) Another point of
view is that ξ-separable states are exactly those which
can be prepared from uncorrelated states by mixtures
of ξ-LOs. We also have that D
ξ-unc
is closed under ξ-
LO, D
ξ-sep
is closed under ξ-LOCC [40, 41]. Clearly, if
a state is product with respect to a partition, then it
is also product with respect to any coarser partition,
it is always free to forget about some tensor prod-
uct signs. This means that these properties show the
same lattice structure as the partitions [40, 41], P
I
,
that is,
υ ξ D
υ-unc
D
ξ-unc
, (4a)
υ ξ D
υ-sep
D
ξ-sep
. (4b)
One can deﬁne the corresponding (information-
geometry based) correlation and entanglement mea-
sures [14, 40] for all ξ-correlation and ξ-entanglement.
These are the most natural generalizations of the mu-
tual information [6, 7], the entanglement entropy [16],
and the entanglement of formation [17] for Level I of
the multipartite case.
The ξ-correlation of a state % is its distinguisha-
bility by the relative entropy (1b) from the ξ-
uncorrelated states [4, 14, 40, 70],
C
ξ
(%) := min
σ∈D
ξ-unc
D(%||σ) =
X
Xξ
S(%
X
) S(%), (5a)
given in terms of the von Neumann entropies (1a)
of the reduced, or marginal states %
X
= Tr
L\X
% of
7− 7−
7−
7−
Figure 1: The lattices of the three-level structure of multipar-
tite correlation and entanglement for n = 2 and 3. Only the
maximal elements of the down-sets of P
I
are shown (with
diﬀerent colors) in P
II
, while only the minimal elements of
the up-sets of P
II
are shown (side by side) in P
III
. The partial
orders (2c), (7c) and (12c) are represented by consecutive
arrows.
subsystems X ξ. Pure states of classical systems are
always uncorrelated, the correlation in pure states is
of quantum origin, this is what we call entanglement
[13, 14, 15, 40, 41]. So, for pure states, the measure
of entanglement should be that of correlation. The
ξ-entanglement of a pure state π = |ψihψ| D
L
is its
ξ-correlation,
E
ξ
(π) := C
ξ
pure
(π) =
X
Xξ
S(π
X
), (5b)
with π
X
= Tr
L\X
π, and for mixed states, one can use,
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for example, the convex roof extension [17, 71, 72] to
deﬁne the ξ-entanglement of formation
E
ξ
(%) :=
C
ξ
pure
(%)
min
n
X
j
p
j
C
ξ
pure
(π
j
)
X
j
p
j
π
j
= %
o
,
(5c)
where
P
j
p
j
π
j
is a pure decomposition of the state
[40]. We have that C
ξ
is a correlation monotone (not
increasing with respect to ξ-LO, for the proof see
Appendix B.1), E
ξ
is a strong entanglement mono-
tone (convex and not increasing on average with re-
spect to selective ξ-LOCC [22, 23, 40]), and both
of these are faithful, C
ξ
(%) = 0 % D
ξ-unc
,
E
ξ
(%) = 0 % D
ξ-sep
[40], moreover, they show
the same lattice structure as the partitions, P
I
, that
is,
υ ξ C
υ
C
ξ
, (6a)
υ ξ E
υ
E
ξ
, (6b)
which is called multipartite monotonicity [14, 40].
Note that, for two subsystems, for the only nontriv-
ial partition 1|2, we have that E
1|2
(π) = 2S(π
1
) =
2S(π
2
) is just two times the usual entanglement en-
tropy. However, we have a way of derivation com-
pletely diﬀerent than the usual, based on asymptotic
LOCC convertibility from Bell-pairs [16]. The usual
way cannot be generalized to the multipartite scenario
(there is no “reference state”, which was the Bell-pair
in the bipartite scenario), while our correlational, or
statistical physical approach above could straightfor-
wardly be generalized to the multipartite scenario not
only here, but also for the Level II properties in the
next subsystem.
2.3 Level II: multiple set partitions
In multipartite entanglement theory, it is necessary
to handle mixtures of states uncorrelated with re-
spect to diﬀerent partitions [35, 37, 40]. For ex-
ample, there are tripartite states which cannot be
written as a mixture of a given kind of, e.g., 12|3-
uncorrelated states, that is, not 12|3-separable, while
can be written as a mixture of 12|3-uncorrelated and
13|2-uncorrelated states. Such states should not be
considered fully tripartite-entangled, since there is no
need for genuine tripartite entangled states in the
mixture [35, 37, 39, 40]. Also, if a state is 12|3-
separable and also 13|2-separable, it is not necessarily
1|2|3-separable. On the other hand, the order iso-
morphisms (4a)-(4b) tell us that if we consider states
uncorrelated (or separable) with respect to a parti-
tion, then we automatically consider states uncorre-
lated (or separable) with respect to all ﬁner partitions.
To embed these requirements in the labeling of the
multipartite correlation and entanglement properties,
we use the nonempty down-sets (nonempty ideals) of
partitions [40], which are sets of partitions
ξ = {ξ
1
, ξ
2
, . . . , ξ
|ξ|
} P
I
(7a)
which are closed downwards with respect to . That
is, if ξ ξ, then for all partitions υ ξ we have υ ξ.
The set of the nonempty partition ideals is denoted
with
P
II
:= O
(P
I
) \ {∅}. (7b)
This possesses a lattice structure with respect to the
standard inclusion as partial order, which we call re-
ﬁnement again, and denote with again,
υ ξ
def.
υ ξ. (7c)
(For illustrations, see Figure 1.) The least upper and
greatest lower bounds, and , are then the union
and the intersection, and .
With respect to the partition ideals ξ P
II
, we
can deﬁne the partial correlation and entanglement
properties, as well as the measures quantifying these.
The ξ-uncorrelated states are those which are ξ-
uncorrelated (3a) with respect to a ξ ξ,
D
ξ-unc
:=
[
ξξ
D
ξ-unc
; (8a)
the others are ξ-correlated states. The ξ-uncorrelated
states are exactly those which can be prepared from
uncorrelated states by ξ-LO for a partition ξ ξ.
The ξ-separable states are convex combinations of ξ-
uncorrelated states,
D
ξ-sep
:= Conv D
ξ-unc
; (8b)
the others are ξ-entangled states. The ξ-separable
states are exactly those which can be prepared from
uncorrelated states by mixtures of ξ-LOs for diﬀerent
partitions ξ ξ. (Note that such transformations do
not form a semigroup.) We also have that D
ξ-unc
is
closed under LO, D
ξ-sep
is closed under LOCC [40, 41].
It follows from (4) that these properties show the same
lattice structure as the partition ideals [40], P
II
, that
is,
υ ξ D
υ-unc
D
ξ-unc
, (9a)
υ ξ D
υ-sep
D
ξ-sep
. (9b)
One can deﬁne the corresponding (information-
geometry based) correlation and entanglement mea-
sures [14, 40] for all ξ-correlation and ξ-entanglement.
These are the most natural generalizations of the mu-
tual information [6, 7], the entanglement entropy [16],
and the entanglement of formation [17] for Level II of
the multipartite case.
The ξ-correlation of a state % is its distinguisha-
bility by the relative entropy (1b) from the ξ-
uncorrelated states [14, 40],
C
ξ
(%) := min
σ∈D
ξ-unc
D(%||σ) = min
ξξ
C
ξ
(%). (10a)
With the same reasoning as in Section 2.2, the ξ-
entanglement of a pure state is
E
ξ
(π) := C
ξ
pure
(π), (10b)
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and for mixed states, one can use the convex roof
extension to deﬁne the ξ-entanglement of formation
E
ξ
(%) :=
C
ξ
pure
(%). (10c)
We have that C
ξ
is a correlation monotone (for the
proof, see Appendix B.1), E
ξ
is a strong entanglement
monotone [40], and both of these are faithful, C
ξ
(%) =
0 % D
ξ-unc
, E
ξ
(%) = 0 % D
ξ-sep
[40],
moreover, they show the same lattice structure as the
partition ideals, P
II
, that is,
υ ξ C
υ
C
ξ
, (11a)
υ ξ E
υ
E
ξ
, (11b)
which is called multipartite monotonicity for Level II
[14, 40].
2.4 Level III: classes
The partial correlation and entanglement properties
form an inclusion hierarchy (9a)-(9b), for example, if a
state is 1|2|3-separable, then it is also 12|3-separable.
We are interested in the labeling of the strict, or ex-
clusive properties, for example, those states which are
12|3-separable and not 1|2|3-separable. In general, we
would like to determine all the possible nonempty in-
tersections of the state sets D
ξ-unc
and D
ξ-sep
. We call
these intersections partial correlation and partial en-
tanglement classes, containing states of well-deﬁned
partial correlation and partial entanglement proper-
ties.
To embed these requirements in the labeling of the
strict properties of multipartite correlation and en-
tanglement, we use the nonempty up-sets (nonempty
ﬁlters) of nonempty down-sets (nonempty ideals) of
partitions [40], which are sets of down-sets of parti-
tions
ξ = {ξ
1
, ξ
2
, . . . , ξ
|ξ|
} P
II
(12a)
which are closed upwards with respect to . That is,
if ξ ξ, then for all partition ideals υ ξ we have
υ ξ. The set of the nonempty up-sets of nonempty
down-sets of partitions of L is denoted with
P
III
:= O
(P
II
) \ {∅}. (12b)
This possesses a lattice structure with respect to the
standard inclusion as partial order, which we call re-
ﬁnement again, and denote with again,
υ
ξ
def.
υ ξ. (12c)
(For illustrations, see Figure 1.) Note that here, con-
trary to Level I and II, we call ξ ﬁner and υ coarser.
In the generic case, if the inclusion of sets is described
by a poset P , then the possible intersections can be
described by O
(P ) (for the proof, see appendix A.2
in [41]). One may make the classiﬁcation coarser
by selecting a subposet P
II*
P
II
, with respect to
which the classiﬁcation is done, P
III*
:= O
(P
II*
)\{∅}
[40, 41].
With respect to the partition ideal ﬁlters ξ P
III*
,
we can deﬁne the strict partial correlation and entan-
glement properties.
The strictly ξ-uncorrelated states are those which
are uncorrelated with respect to all ξ ξ, and corre-
lated with respect to all ξ ξ = P
II*
\ ξ, so the class
of these (partial correlation class) is
C
ξ-unc
:=
\
ξξ
D
ξ-unc
\
ξξ
D
ξ-unc
. (13a)
The strictly ξ-separable states are those which are sep-
arable with respect to all ξ ξ, and entangled with
respect to all ξ ξ, so the class of these (partial sep-
arability class, or partial entanglement class) is
C
ξ-sep
:=
\
ξξ
D
ξ-sep
\
ξξ
D
ξ-sep
. (13b)
The meaning of the Level III hierarchy could also be
clariﬁed. If there exists a % C
υ-unc
and an LO map-
ping it into C
ξ-unc
, then υ ξ [40, 41]; and if there
exists a % C
υ-sep
and an LOCC mapping it into
C
ξ-sep
, then υ ξ [40]. In this sense, the Level III
hierarchy compares the strength of correlation and
entanglement among the classes.
The ﬁlters ξ are suﬃcient for the description of the
nontrivial classes (nonempty intersections) of the cor-
relation and entanglement properties, but they are not
necessary in general. For the strictly ξ-uncorrelated
states, the structure P
III*
simpliﬁes signiﬁcantly [41].
(For example, for the ﬁnest classiﬁcation P
II*
= P
II
,
we have that the structure of the partial correla-
tion classes is P
III*
=
P
I
, as the unique labeling of
the nontrivial partial correlation classes is given as
ξ = ↑{↓{ξ}} for the partitions ξ P
I
[41].) On the
other hand, it is still a conjecture that ξ-separability
is nontrivial for all ξ P
III
[40]. The conjecture holds
for three subsystems, for which explicit examples were
constructed [39, 73].
3 Multipartite correlation and entan-
glement: permutation invariant proper-
ties
Here we build up the structure of the classiﬁcation
and quantiﬁcation of the permutation invariant mul-
tipartite correlation and entanglement properties. We
do this from the point of view of the general proper-
ties in the previous section. To give a quick guidance
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on the construction, we show the diagram
(P
III
, )
s
//
(
ˆ
P
III
, )
(P
II
, )
s
//
_
O
\{∅}
OO
(
ˆ
P
II
, )
_
O
\{∅}
OO
(P
I
, )
s
//
_
O
\{∅}
OO
(
ˆ
P
I
, )
_
O
\{∅}
OO
(14)
in advance. This might not seem to be obvious, for
the proof, see Corollary 17 in Appendix A.4.
In Section 2, for the description of the struc-
ture of multipartite correlations [14, 40, 41], the
natural language was that of set partitions [44].
For the permutation-invariant case, particularly k-
partitionability and k-producibility, it is natural to
use the simpler language of integer partitions [45, 62].
3.1 Level 0: subsystem sizes
Let us have the map, simply measuring the sizes of
the subsystems X P
0
,
s(X) := |X|, (15)
mapping P
0
ˆ
P
0
, which is simply
ˆ
P
0
:= s(P
0
) = {0, 1, 2, . . . , n}. (16)
This possesses the natural order , which is now de-
noted with , with respect to which s is monotone,
Y X s(Y ) s(X). We note for the sake of the
analogy with the construction in the subsequent levels
that one could deﬁne on
ˆ
P
0
as y x if there exist
Y s
1
(y) and X s
1
(x) such that Y X, from
which the monotonicity of s follows automatically (see
(64) in Appendix A.1).
3.2 Level I: integer partitions
Let the map s in (15) act elementwisely on the parti-
tions ξ P
I
,
s(ξ) :=
s(X)
X ξ
, (17)
where on the right-hand side a multiset stays, that
is, a set, which allows multiple instances of its el-
ements. (For example, for ξ = {{1}, {2, 4}, {3}},
s(ξ) = {1, 2, 1} {2, 1, 1}.) We call the integer parti-
tion s(ξ) the type of the set partition ξ. This is then
an integer partition of n [45, 62], which is denoted as
ˆ
ξ = {x
1
, x
2
, . . . , x
|
ˆ
ξ|
}, (18a)
where the parts x
ˆ
ξ are nonzero subsystem sizes, for
which
P
x
ˆ
ξ
x = n. The set of the integer partitions
of n is denoted with
ˆ
P
I
:= s(P
I
). (18b)
Its size is given by the partition numbers [74]. Based
on the reﬁnement of the set partitions (2c), we deﬁne
a partial order over the integer partitions, which we
call reﬁnement again, and denote with agai again,
given as
ˆυ
ˆ
ξ
def.
υ s
1
(ˆυ), ξ s
1
(
ˆ
ξ) s.t. υ ξ.
(18c)
(For illustrations, see Figures 2 and 3. The integer
partitions are represented by their Young diagrams,
where the rows of boxes are ordered decreasingly.)
This is a proper partial order (for the proof, see Ap-
pendix A.4). Note that the reﬁnement is a partial
order only, there are pairs of partitions which can-
not be ordered, for example, {2, 2} {3, 1} and
{2, 2} {3, 1}, see Figure 2. Note that
ˆ
P
I
with
this partial order is not a lattice, since there are no
unique least upper and greatest lower bounds with re-
spect to that. (The ﬁrst example is for n = 5, where
{2, 2, 1}, {3, 1, 1} {3, 2}, {4, 1}, and there is no inte-
ger partition between them, see Figure 3.) The reﬁne-
ment also admits a minimal and a maximal element
= {1, 1, . . . , 1} and > = {n}. By construction,
s is monotone with respect to these partial orders,
υ ξ s(υ) s(ξ) (see (64) in Appendix A.1).
With respect to the integer partitions
ˆ
ξ
ˆ
P
I
, we
can deﬁne the partial correlation and entanglement
properties, as well as the measures quantifying them.
The
ˆ
ξ-uncorrelated states are those which are ξ-
uncorrelated (3a) with respect to a partition ξ of type
ˆ
ξ, which is actually a Level II state set (8a) labeled
by the ideal ξ = s
1
(
ˆ
ξ),
D
ˆ
ξ-unc
:=
[
ξ:s(ξ)=
ˆ
ξ
D
ξ-unc
= D
s
1
(
ˆ
ξ)-unc
; (19a)
the others are
ˆ
ξ-correlated states. The
ˆ
ξ-uncorrelated
states are exactly those which can be prepared from
uncorrelated states by ξ-LO for a partition ξ of type
ˆ
ξ.
(Note that we consider arbitrary states here, not only
permutation invariant ones. The correlation property
deﬁned in this way is what permutation invariant is.)
The
ˆ
ξ-separable states are convex combinations of
ˆ
ξ-
uncorrelated states, which is actually a Level II state
set (8b) labeled by the ideal ξ = s
1
(
ˆ
ξ),
D
ˆ
ξ-sep
:= Conv D
ˆ
ξ-unc
= D
s
1
(
ˆ
ξ)-sep
; (19b)
the others are
ˆ
ξ-entangled states. The
ˆ
ξ-separable
states are exactly those which can be prepared from
uncorrelated states by mixtures of ξ-LOs for diﬀerent
partitions ξ of type
ˆ
ξ. We also have that D
ˆ
ξ-unc
is
closed under LO, D
ˆ
ξ-sep
is closed under LOCC, be-
cause these hold for the general Level II state sets
D
s
1
(
ˆ
ξ)-unc
and D
s
1
(
ˆ
ξ)-sep
, see in Section 2.3. It
also follows that these properties show the same par-
tially ordered structure as the integer partitions,
ˆ
P
I
,
that is,
ˆυ
ˆ
ξ D
ˆυ-unc
D
ˆ
ξ-unc
, (20a)
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7−
7−
7−
Figure 2: The construction of the posets
ˆ
P
I
of permutation invariant multipartite correlation and entanglement properties
from the lattices P
I
for n = 2, 3 and 4. The integer partitions (17) are represented by their Young diagrams, the partial order
(18c) is represented by consecutive arrows.
Figure 3: The posets
ˆ
P
I
of permutation invariant multipar-
tite correlation and entanglement properties for n = 5 and
6. The notation is the same as in Figure 2.
ˆυ
ˆ
ξ D
ˆυ-sep
D
ˆ
ξ-sep
. (20b)
(These come by using (74a) and (73a) in Ap-
pendix A.4, then (9) and (19).)
In accordance with the general case in Section 2.2,
one can deﬁne the corresponding (information-
geometry based) correlation and entanglement mea-
sures for all
ˆ
ξ-correlation and
ˆ
ξ-entanglement. These
are the most natural generalizations of the mutual in-
formation [6, 7], the entanglement entropy [16], and
the entanglement of formation [17] for Level I of the
permutation invariant multipartite case.
The
ˆ
ξ-correlation of a state % is its distinguisha-
bility by the relative entropy (1b) from the
ˆ
ξ-
uncorrelated states, which is actually a Level II mea-
sure (10a) labeled by the ideal ξ = s
1
(
ˆ
ξ),
C
ˆ
ξ
(%) := min
σ∈D
ˆ
ξ-unc
D(%||σ) = C
s
1
(
ˆ
ξ)
(%). (21a)
With the same reasoning as in Section 2.2, the
ˆ
ξ-
entanglement of a pure state is
E
ˆ
ξ
(π) := C
ˆ
ξ
pure
(π) = E
s
1
(
ˆ
ξ)
(π), (21b)
and for mixed states, one can use the convex roof
extension, which is actually a Level II measure (10c)
labeled by the ideal ξ = s
1
(
ˆ
ξ),
E
ˆ
ξ
(%) :=
C
ˆ
ξ
pure
(%) = E
s
1
(
ˆ
ξ)
(%). (21c)
These inherit the properties of the general case in Sec-
tion 2.3. So we have that C
ˆ
ξ
is a correlation mono-
tone, E
ˆ
ξ
is a strong entanglement monotone, and
both of these are faithful, C
ˆ
ξ
(%) = 0 % D
ˆ
ξ-unc
,
E
ˆ
ξ
(%) = 0 % D
ˆ
ξ-sep
, moreover, they show the
same partially ordered structure as the integer parti-
tions,
ˆ
P
I
, that is,
ˆυ
ˆ
ξ C
ˆυ
C
ˆ
ξ
, (22a)
ˆυ
ˆ
ξ E
ˆυ
E
ˆ
ξ
, (22b)
which we call multipartite monotonicity for Level I of
the permutation invariant case. (These come by using
(74a) and (73a) in Appendix A.4, then (11) and (21).)
3.3 Level II: multiple integer partitions
Let the map s in (17) act elementwisely on the parti-
tion ideals ξ P
II
,
s(ξ) :=
s(ξ)
ξ ξ
, (23)
where on the right-hand side a set stays. We call s(ξ)
the type of the set partition ideal ξ. This is a set of
integer partitions, which is denoted as
ˆ
ξ =
ˆ
ξ
1
,
ˆ
ξ
2
, . . . ,
ˆ
ξ
|
ˆ
ξ|
ˆ
P
I
, (24a)
which is a nonempty ideal of integer partitions, the
set of which is denoted as
ˆ
P
II
:= s(P
II
) = O
(
ˆ
P
I
) \ {∅}. (24b)
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This might not seem to be obvious, this is given in
(14), proven in Corollary 17 in Appendix A.4. In
particular, all ξ ξ is of type
ˆ
ξ for a
ˆ
ξ s(ξ), on
the other hand, all
ˆ
ξ s(ξ) has representative ξ ξ
of type
ˆ
ξ. Based on the reﬁnement of the set partition
ideals (7c), we deﬁne a partial order over the integer
partition ideals, which we call reﬁnement again, and
denote with again, given as
ˆ
υ
ˆ
ξ
def.