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 [1–4].
It is the main resource of quantum information the-
ory [5–7], and its nonclassical properties, playing im-
portant role also in many-body physics [8–10], make
it influential in the behavior and characterization of
strongly correlated systems [11–14].
The correlation and entanglement between two
parts of a system is relatively well-understood [15–17].
At least for pure states, the convertibility and the clas-
sification with respect to (S)LOCC ((stochastic) local
operations and classical communication [15, 17–19])
shows a simple structure [20, 21], and there are ba-
sically unique correlation and entanglement measures
[22–24]. The multipartite case is much more compli-
cated [19, 25–28]. 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) [33–40], or partial correlations
[14, 41, 42] only, leads to a combinatoric [43–45], 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 [46–49] is also called entanglement
depth [50–52]. 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
[58–60], 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
advantages of k-partitionability and k-producibility.
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 −(n−1) to n−1. 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
i∈X
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, 63–65]
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, 64–68]
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
X∈ξ
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
X∈ξ
%
X
o
;
(3a)
the others are ξ-correlated states. With respect to the
finest partition ξ = ⊥, we call ⊥-uncorrelated states
N
i∈L
%
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
X∈ξ
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
[1–3, 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
X∈ξ
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
:=
[
ξ∈ξ
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
ξ∈ξ
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
:=
\
ξ∈ξ
D
ξ-unc
∩
\
ξ∈ξ
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
:=
\
ξ∈ξ
D
ξ-sep
∩
\
ξ∈ξ
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
Accepted in Quantum 2019-11-05, click title to verify. Published under CC-BY 4.0. 6
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)
Accepted in Quantum 2019-11-05, click title to verify. Published under CC-BY 4.0. 7
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)
Accepted in Quantum 2019-11-05, click title to verify. Published under CC-BY 4.0. 8
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.
⇐⇒ ∃υ ∈ s
−1
(
ˆ
υ), ξ ∈ s
−1
(
ˆ
ξ) s.t. υ ξ
⇐⇒
ˆ
υ ⊆
ˆ
ξ,
(24c)
which turns out to be the inclusion, being the natu-
ral partial order for ideals. This might not seem to
be obvious, this is given in (14), proven in Corollary
17 in Appendix A.4. (For illustrations, see Figure 4.)
Because of these,
ˆ
P
II
with this partial order is a lat-
tice. 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 partition ideals
ˆ
ξ ∈
ˆ
P
II
,
we can define the partial correlation and entanglement
properties, as well as the measures quantifying these.
The
ˆ
ξ-uncorrelated states are those which are ξ-
uncorrelated (3a) with respect to a partition ξ of type
ˆ
ξ for a
ˆ
ξ ∈
ˆ
ξ, which is actually a Level II state set (8a)
labeled by the ideal ξ = ∨s
−1
(
ˆ
ξ) =
W
ˆ
ξ∈
ˆ
ξ
↓ s
−1
(
ˆ
ξ),
D
ˆ
ξ-unc
: =
[
ξ:s(ξ)=
ˆ
ξ
D
ξ-unc
= D
∨s
−1
(
ˆ
ξ)-unc
=
[
ξ:s(ξ)∈
ˆ
ξ
D
ξ-unc
= D
W
ˆ
ξ∈
ˆ
ξ
↓s
−1
(
ˆ
ξ)-unc
,
(25a)
(for the proof, see (73b) in Appendix A.4); the oth-
ers are
ˆ
ξ-correlated states. (We use the notation
∨A :=
W
a∈A
a for any subset A of a lattice.) The
ˆ
ξ-uncorrelated states are exactly those which can be
prepared from uncorrelated states by ξ-LO for a par-
tition ξ of a type
ˆ
ξ ∈
ˆ
ξ. 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) =
W
ˆ
ξ∈
ˆ
ξ
↓ s
−1
(
ˆ
ξ),
D
ˆ
ξ-sep
:= Conv D
ˆ
ξ-unc
= D
∨s
−1
(
ˆ
ξ)-sep
, (25b)
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 different types
ˆ
ξ ∈
ˆ
ξ. We also have
that D
ˆ
ξ-unc
is closed under LO, D
ˆ
ξ-sep
is closed under
LOCC, because these hold for the general Level II
state sets D
∨s
−1
(
ˆ
ξ)
and D
∨s
−1
(
ˆ
ξ)
, see in Section 2.3.
It also follows that these properties show the same
lattice structure as the integer partition ideals,
ˆ
P
II
,
that is,
ˆ
υ
ˆ
ξ ⇐⇒ D
ˆ
υ-unc
⊆ D
ˆ
ξ-unc
, (26a)
ˆ
υ
ˆ
ξ ⇐⇒ D
ˆ
υ-sep
⊆ D
ˆ
ξ-sep
. (26b)
(These come by using (74b) and (73b) in Ap-
pendix A.4, then (9) and (25).)
In accordance with the general case in Section 2.3,
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 II 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)
(%). (27a)
With the same reasoning as in Section 2.3, the
ˆ
ξ-
entanglement of a pure state is
E
ˆ
ξ
(π) := C
ˆ
ξ
pure
(π) = E
∨s
−1
(
ˆ
ξ)
(π), (27b)
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)
(%). (27c)
These inherit the properties of of the general case in
Section 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 lattice structure as the integer partition ideals,
ˆ
P
II
, that is,
ˆ
υ
ˆ
ξ ⇐⇒ C
ˆ
υ
≥ C
ˆ
ξ
, (28a)
ˆ
υ
ˆ
ξ ⇐⇒ E
ˆ
υ
≥ E
ˆ
ξ
, (28b)
which we call multipartite monotonicity for Level II of
the permutation invariant case. (These come by using
(74b) and (73b) in Appendix A.4, then (11) and (27).)
3.4 Level III: classes
Let the map s in (23) act elementwisely on the parti-
tion ideal filters ξ ∈ P
III
,
s(ξ) :=
s(ξ)
∀ξ ∈ ξ
, (29)
where on the right-hand side a set stays. We call s(ξ)
the type of the set partition ideal filter ξ. This is a
set of integer partition filters, which is denoted as
ˆ
ξ =
ˆ
ξ
1
,
ˆ
ξ
2
, . . . ,
ˆ
ξ
|
ˆ
ξ|
⊆
ˆ
P
II
, (30a)
Accepted in Quantum 2019-11-05, click title to verify. Published under CC-BY 4.0. 9
which is a nonempty filter of ideals of integer parti-
tions, the set of which is denoted as
ˆ
P
III
:= s(P
III
) = O
↑
(
ˆ
P
II
) \ {∅}. (30b)
This might not seem to be obvious, this is given in
(14), proven in Corollary 17 in Appendix A.4. Based
on the refinement of the set partition ideal filters
(12c), we define a partial order over the integer parti-
tion ideal filters, which we call refinement again, and
denote with again, given as
ˆ
υ
ˆ
ξ
def.
⇐⇒ ∃υ ∈ s
−1
(
ˆ
υ), ξ ∈ s
−1
(
ˆ
ξ) s.t. υ ξ
⇐⇒
ˆ
υ ⊆
ˆ
ξ,
(30c)
which turns out to be the inclusion, being the natural
partial order for filters. This might not seem to be
obvious, this is given in (14), proven in Corollary 17 in
Appendix A.4. (For illustrations, see Figure 4.) Note
that here, contrary to Level I and II, we call
ˆ
ξ finer
and
ˆ
υ coarser. Because of these,
ˆ
P
III
with this partial
order is a lattice. 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 partition ideal filters
ˆ
ξ ∈
ˆ
P
III
, we can define the strict partial correlation
and entanglement properties.
The strictly
ˆ
ξ-uncorrelated states are those which
are uncorrelated with respect to all
ˆ
ξ ∈
ˆ
ξ, and corre-
lated with respect to all
ˆ
ξ ∈
ˆ
ξ =
ˆ
P
II
\
ˆ
ξ, so the class of
these (permutation invariant partial correlation class)
is
C
ˆ
ξ-unc
:=
\
ˆ
ξ∈
ˆ
ξ
D
ˆ
ξ-unc
∩
\
ˆ
ξ∈
ˆ
ξ
D
ˆ
ξ-unc
. (31a)
The strictly
ˆ
ξ-separable states are those which are sep-
arable with respect to all
ˆ
ξ ∈
ˆ
ξ, and entangled with
respect to all
ˆ
ξ ∈
ˆ
ξ, so the class of these (permutation
invariant partial separability class, or permutation in-
variant partial entanglement class) is
C
ˆ
ξ-sep
:=
\
ˆ
ξ∈
ˆ
ξ
D
ˆ
ξ-sep
∩
\
ˆ
ξ∈
ˆ
ξ
D
ˆ
ξ-sep
. (31b)
The meaning of the permutation invariant Level III
hierarchy can also be clarified. If there exists a
% ∈ C
ˆ
υ-unc
and an LO mapping it into C
ˆ
ξ-unc
, then
ˆ
υ
ˆ
ξ; and if there exists a % ∈ C
ˆ
υ-sep
and an LOCC
mapping it into C
ˆ
ξ-sep
, then
ˆ
υ
ˆ
ξ. In this sense, the
permutation invariant Level III hierarchy compares
the strength of correlation and entanglement among
the permutation invariant classes. (These come by the
analogue result for the general case in Section 2.4, for
the coarsened classification based on the permutation
invariant properties P
II*
= {∨s
−1
(
ˆ
ξ) |
ˆ
ξ ∈
ˆ
P
II
}, see
in the next section.)
7−→ 7−→
7−→ 7−→
7−→ 7−→
7−→
7−→
Figure 4: The posets of the three-level structure of permuta-
tion invariant multipartite correlation and entanglement for
n = 2, 3, 4 and 5. Only the maximal elements of the down-
sets of
ˆ
P
I
are shown (with different colors) in
ˆ
P
II
, while only
the minimal elements of the up-sets of
ˆ
P
II
are shown (side
by side) in
ˆ
P
III
. The partial orders (18c), (24c) and (30c) are
represented by consecutive arrows.
Accepted in Quantum 2019-11-05, click title to verify. Published under CC-BY 4.0. 10
4 An alternative way of introducing
permutation invariance
The Level II structure P
II
encodes the different par-
tial correlation and entanglement properties. In Sec-
tions 2 and 3, we have built up the structure of the
classification, parallel for the general, and for the
permutation invariant cases. Here we show a more
compact, but less transparent treatment of the same
structure, by embedding both constructions into P
II
.
The three-level building of the correlation and en-
tanglement classification was given by the construc-
tion
P
I
7→ P
II
= O
↓
(P
I
) \ {∅} 7→ P
III
= O
↑
(P
II
) \ {∅}
in Section 2. Because of (4), one may embed P
I
into
P
II
, using the principal ideals in P
II
,
P
I
∼
=
P
II pr.
:=
↓{ξ}
ξ ∈ P
I
⊆ P
II
, (32a)
by noting that
υ ξ ⇐⇒ ↓{υ} ↓{ξ}. (32b)
In this way, P
I
is a sublattice of P
II
.
The same can be done for the permutation invariant
correlation and entanglement classification given by
the construction
ˆ
P
I
7→
ˆ
P
II
= O
↓
(
ˆ
P
I
) \ {∅} 7→
ˆ
P
III
= O
↑
(
ˆ
P
II
) \ {∅}
in Section 3. In the same way, because of (20), one
may embed
ˆ
P
I
into
ˆ
P
II
, using the principal ideals in
ˆ
P
II
,
ˆ
P
I
∼
=
ˆ
P
II pr.
:=
↓{
ˆ
ξ}
ˆ
ξ ∈
ˆ
P
I
⊆
ˆ
P
II
, (33a)
by noting that
ˆυ
ˆ
ξ ⇐⇒ ↓{ˆυ} ↓{
ˆ
ξ}. (33b)
In this way,
ˆ
P
I
is a sublattice of
ˆ
P
II
.
The third point here is that, noting that ↓ s
−1
(
ˆ
ξ) =
∨s
−1
(↓{
ˆ
ξ}) (for the proof, see (73a) in Appendix A.4),
one may also embed
ˆ
P
II
(thus also
ˆ
P
I
) into P
II
, using
ˆ
P
I
∼
=
∨s
−1
(↓{
ˆ
ξ})
ˆ
ξ ∈
ˆ
P
I
⊂ P
II
, (34a)
ˆ
P
II
∼
=
∨s
−1
(
ˆ
ξ)
ˆ
ξ ∈
ˆ
P
II
=: P
II pinv.
⊂ P
II
, (34b)
by noting that
ˆυ
ˆ
ξ ⇐⇒ ∨s
−1
(↓{ˆυ}) ∨s
−1
(↓{
ˆ
ξ}), (34c)
ˆ
υ
ˆ
ξ ⇐⇒ ∨s
−1
(
ˆ
υ) ∨s
−1
(
ˆ
ξ). (34d)
(For the proof, see (74a) and (74b) in Appendix A.4.)
In this way,
ˆ
P
I
and
ˆ
P
II
are sublattices of P
II
.
A ξ ∈ P
II
is permutation invariant, if and only if
∨s
−1
(s(ξ)) = ξ. (Then ξ ∈ P
II
describes the same
property as
ˆ
ξ := s(ξ) ∈
ˆ
P
II
.) So P
II pinv.
can be given
directly as
P
II pinv.
=
ξ ∈
ˆ
P
II
∨ s
−1
(s(ξ)) = ξ
, (35)
and the permutation invariant classification can be
described as a coarsened classification
ˆ
P
III
∼
=
P
III*
=
O
↑
(P
II*
) \ {∅} with respect to the permutation invari-
ant properties P
II*
= P
II pinv.
.
We note that P
II
could also be embedded into P
III
by principal filters (and similarly
ˆ
P
II
into
ˆ
P
III
), how-
ever, the construction has led to a different direction.
5 k-partitionability, k-producibility
and k-stretchability
In this section, we consider k-partitionability and k-
producibility as particular cases of permutation in-
variant properties of partial correlation and entangle-
ment, and elaborate a duality between them. Our
point of view makes possible to reveal a new prop-
erty, which we call k-stretchability, combining some
advantages of k-partitionability and k-producibility.
We also investigate the relations among these three
properties.
5.1 k-partitionability and k-producibility of
correlation and entanglement
k-partitionability and k-producibility are permuta-
tion invariant Level II properties. A partition is k-
partitionable, if the number of its parts is at least k,
while it is k-producible, if the sizes of its parts are at
most k. (For illustration, see Figure 5.) These are en-
coded by the ideals of k-partitionable and k-producible
partitions,
ξ
k-part
:=
ξ ∈ P
I
|ξ| ≥ k
∈ P
II
, (36a)
ξ
k-prod
:=
ξ ∈ P
I
∀X ∈ ξ : |X| ≤ k
∈ P
II
, (36b)
for k = 1, 2, . . . , n, forming chains in the lattice P
II
,
ξ
l-part
ξ
k-part
⇐⇒ l ≥ k, (37a)
ξ
l-pro d
ξ
k-prod
⇐⇒ l ≤ k. (37b)
Since k-partitionability and k-producibility are per-
mutation invariant properties, it is enough to consider
their types (23)
ˆ
ξ
k-part
:= s(ξ
k-part
)
=
ˆ
ξ ∈
ˆ
P
I
|
ˆ
ξ| ≥ k
∈
ˆ
P
II
,
(38a)
ˆ
ξ
k-prod
:= s(ξ
k-prod
)
=
ˆ
ξ ∈
ˆ
P
I
∀x ∈
ˆ
ξ : x ≤ k
∈
ˆ
P
II
,
(38b)
forming chains in the lattice
ˆ
P
II
,
ˆ
ξ
l-part
ˆ
ξ
k-part
⇐⇒ l ≥ k, (39a)
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1
2
2
1
k-partitionability
k-producibility
1
2
3
3
2
1
k-partitionability
k-producibility
1
2
3
4
4
3
2
1
k-partitionability
k-producibility
Figure 5: The down-sets corresponding to k-partitionability and k-producibility, illustrated on the lattices P
I
for n = 2, 3 and
4. (The two kinds of down-sets contain the elements below the specific green and yellow dashed lines.)
ˆ
ξ
l-part
ˆ
ξ
k-prod
⇐⇒ l ≤ k. (39b)
The corresponding k-partitionably uncorrelated and
k-producibly uncorrelated states (25a) are
D
k-part unc
:= D
ξ
k-part
-unc
= D
ˆ
ξ
k-part
-unc
=
n
O
X∈ξ
%
X
∀ξ ∈ P
I
s.t. |ξ| ≥ k
o
,
(40a)
D
k-prod unc
:= D
ξ
k-prod
-unc
= D
ˆ
ξ
k-prod
-unc
=
n
O
X∈ξ
%
X
∀ξ ∈ P
I
s.t. ∀X ∈ ξ : |X| ≤ k
o
,
(40b)
(with %
X
∈ D
X
), which are products of density op-
erators of at least k subsystems, and of density op-
erators of subsystems containing at most k elemen-
tary subsystems, respectively. The corresponding k-
partitionably separable (also called k-separable [35,
37, 47]) and k-producibly separable (also called k-
producible [46, 47, 49]) states (25b) are
D
k-part sep
:= D
ξ
k-part
-sep
= D
ˆ
ξ
k-part
-sep
= Conv D
k-part unc
=
n
X
j
p
j
O
X∈ξ
j
%
X,j
∀j, ∀ξ
j
∈ P
I
s.t. |ξ
j
| ≥ k
o
,
(40c)
D
k-prod sep
:= D
ξ
k-prod
-sep
= D
ˆ
ξ
k-prod
-sep
= Conv D
k-prod unc
=
n
X
j
p
j
O
X∈ξ
j
%
X,j
∀j,
∀ξ
j
∈ P
I
s.t. ∀X ∈ ξ
j
: |X| ≤ k
o
,
(40d)
which can be decomposed into k-partitionably, or k-
producibly uncorrelated states, respectively. Because
of (26a), these properties show the same chain struc-
ture as the chains of the corresponding partition ideals
(39), that is,
l ≥ k ⇐⇒ D
l-part unc
⊆ D
k-part unc
, (41a)
l ≤ k ⇐⇒ D
l-pro d unc
⊆ D
k-prod unc
, (41b)
l ≥ k ⇐⇒ D
l-part sep
⊆ D
k-part sep
, (41c)
l ≤ k ⇐⇒ D
l-pro d sep
⊆ D
k-prod sep
, (41d)
that is, if a state is l-partitionably uncorrelated (sep-
arable) then it is also k-partitionably uncorrelated
(separable) for all l ≥ k, and if a state is l-producibly
uncorrelated (separable) then it is also k-producibly
uncorrelated (separable) for all l ≤ k.
The corresponding k-partitionability correlation
and k-producibility correlation (27a) are [14]
C
k-part
:= C
ξ
k-part
= C
ˆ
ξ
k-part
= min
ξ:|ξ|≥k
C
ξ
, (42a)
C
k-prod
:= C
ξ
k-prod
= C
ˆ
ξ
k-prod
= min
ξ:∀X∈ξ:|X|≤k
C
ξ
.
(42b)
The corresponding k-partitionability entanglement
and k-producibility entanglement (27c) are [40]
E
k-part
:= E
ξ
k-part
= E
ˆ
ξ
k-part
, (42c)
E
k-prod
:= E
ξ
k-prod
= E
ˆ
ξ
k-prod
. (42d)
Because of (28), these measures show the same chain
structure as the chains of the corresponding partition
ideals (39), that is,
l ≥ k ⇐⇒ C
l-part
≥ C
k-part
, (43a)
l ≤ k ⇐⇒ C
l-pro d
≥ C
k-prod
, (43b)
l ≥ k ⇐⇒ E
l-part
≥ E
k-part
, (43c)
l ≤ k ⇐⇒ E
l-pro d
≥ E
k-prod
, (43d)
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that is, l-partitionability correlation (entanglement)
is always stronger than k-partitionability correlation
(entanglement) for all l ≥ k, and l-producibility cor-
relation (entanglement) is always stronger than k-
producibility correlation (entanglement) for all l ≤ k,
which is the multipartite monotonicity of these mea-
sures.
For the labeling of the strict partitionability and
producibility properties, we have the ideal filters
ˆ
ξ
k-part
:= ↑
ˆ
ξ
k-part
=
ˆ
ξ
l-part
l ≤ k
, (44a)
ˆ
ξ
k-prod
:= ↑
ˆ
ξ
k-prod
=
ˆ
ξ
l-pro d
l ≥ k
, (44b)
by (36) and (37). The corresponding strictly k-
partitionably uncorrelated and strictly k-producibly
uncorrelated states (31a) are
C
k-part unc
:= C
ˆ
ξ
k-part
-unc
= D
k-part unc
\ D
(k+1)-part unc
,
(45a)
C
k-prod unc
:= C
ˆ
ξ
k-prod
-unc
= D
k-prod unc
\ D
(k−1)-prod unc
,
(45b)
which are products of nonproduct density operators of
exactly k subsystems, and of nonproduct density op-
erators of subsystems containing exactly k elementary
subsystems, respectively. The corresponding strictly
k-partitionably separable and strictly k-producibly sep-
arable states (31b) are
C
k-part sep
:= C
ˆ
ξ
k-part
-sep
= D
k-part sep
\ D
(k+1)-part sep
,
(45c)
C
k-prod sep
:= C
ˆ
ξ
k-part
-sep
= D
k-prod sep
\ D
(k−1)-prod sep
,
(45d)
which can be decomposed into k-partitionably but
not (k + 1)-partitionably, and k-producibly but not
(k − 1)-producibly uncorrelated states, respectively.
(D
(n+1)-part unc/sep
= ∅ and D
0-prod unc/sep
= ∅ are
understood.)
5.2 Duality by conjugation
Looking at the k-partitionability and k-producibility
properties in Figure 5, and also at their definitions
(36a) and (36b), it is not clear how these are related
to each other.
In Section 3, the permutation invariant correlation
properties were described by the use of integer par-
titions, represented by Young diagrams in the fig-
ures. Important properties of Young diagrams are
their height, width and rank, given for an integer par-
tition
ˆ
ξ ∈
ˆ
P
I
as
h(
ˆ
ξ) := |
ˆ
ξ|, (46a)
w(
ˆ
ξ) := max
ˆ
ξ, (46b)
r(
ˆ
ξ) := w(
ˆ
ξ) − h(
ˆ
ξ), (46c)
and also for set partition ξ ∈ P
I
as h(ξ) := h(s(ξ)) =
|ξ|, w(ξ) := w(s(ξ)) = max
X∈ξ
|X|, r(ξ) := r(s(ξ)) =
w(ξ) − h(ξ). It is enlightening now to arrange the
poset
ˆ
P
I
according to the height and width of the par-
titions, in the way as can be seen in Figure 6, i.e., the
height is increasing downwards, the width is increas-
ing to the right, then the rank is increasing up-right.
It is easy to see that the height is strictly decreas-
ing, the width is increasing, and the rank is strictly
increasing monotone with respect to the refinement
(18c),
ˆυ ≺
ˆ
ξ =⇒
h(ˆυ) > h(
ˆ
ξ), w(ˆυ) ≤ w(
ˆ
ξ), r(ˆυ) < r(
ˆ
ξ).
(47)
(For illustration, see Figure 6: the arrows point always
upwards, and possibly to the up-right, never to the
up-left, horizontally or downwards.)
Now the partitionability and producibility proper-
ties (36a) and (36b) can be formulated with the height
and width as
ˆ
ξ
k-part
=
ˆ
ξ ∈
ˆ
P
I
h(
ˆ
ξ) ≥ k
, (48a)
ˆ
ξ
k-prod
=
ˆ
ξ ∈
ˆ
P
I
w(
ˆ
ξ) ≤ k
, (48b)
so k-partitionability is the minimal height, k-
producibility is the maximal width of the partition. In
the height-width based arrangement of
ˆ
P
I
in Figure 6,
the k-partitionable partitions (48a) are below the k-
th rows (counting from the top), and k-producible
partitions (48b) are to the left from the k-th column
(counting from the left). We can also introduce an-
other property, called k-stretchability, as
ˆ
ξ
k-str
=
ˆ
ξ ∈
ˆ
P
I
r(
ˆ
ξ) ≤ k
, (48c)
for k = −(n − 1), −(n − 2), . . . , n − 2, n − 1, so k-
stretchability is the maximal rank of the partition. It
is easy to see that
ˆ
ξ
k-str
∈
ˆ
P
II
, that is, it is a down-set
for all k, by the third inequality in (47). We elaborate
on the arising correlation and entanglement properties
later, in Section 5.3.
On the set of Young diagrams, an involution arises
naturally, called conjugation, being the reflection of
the diagram with respect to its “diagonal” [45]. Con-
sidering the integer partitions themselves, the conju-
gation is given as follows. For decreasingly ordered
values (x
1
, x
2
, . . . , x
|
ˆ
ξ|
) of the elements of
ˆ
ξ, the num-
ber of elements x
0
that equal to i is x
i
− x
i+1
in the
conjugated partition (setting x
i+1
= 0 for i = h(
ˆ
ξ)).
With multisets, the conjugation is given as the map
ˆ
P
I
→
ˆ
P
I
,
ˆ
ξ 7→
ˆ
ξ
†
=
|{x ∈
ˆ
ξ | x ≥ i}|
i = 1, 2, . . . , w(
ˆ
ξ)
,
(49)
where both sets on the right-hand side are multisets.
The partial order (18c) does not show nice proper-
ties with respect to the conjugation. (For illustration,
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2
1
1
2
k-partitionability
k-producibility
3
1
2
2
1
3
k-partitionability
k-producibility
4
1
3
2
2
3
1
4
k-partitionability
k-producibility
5
1
4
2
3
3
2
4
1
5
k-partitionability
k-producibility
6
1
5
2
4
3
3
4
2
5
1
6
k-partitionability
k-producibility
Figure 6: The down-sets corresponding to k-partitionability and k-producibility, illustrated on the lattices
ˆ
P
I
for n = 2, 3, 4, 5
and 6. (The two kinds of down-sets contain the elements below the green and to the left from the yellow dashed lines.) For
the first three cases, compare with P
I
in Figure 5.
see the height-width based arrangement in Figure 6.)
The conjugation is not an anti-automorphism, if ˆυ
ˆ
ξ
then it does not follow that ˆυ
†
ˆ
ξ
†
(the first exam-
ple is for n = 4, where we have {2, 1, 1} {2, 2} and
{2, 1, 1}
†
= {3, 1} 6 {2, 2} = {2, 2}
†
). The conjuga-
tion is neither an automorphism of course, if ˆυ
ˆ
ξ
then it does not follow that ˆυ
†
ˆ
ξ
†
(the first exam-
ple is for n = 2, where we have {1, 1} {2} and
{1, 1}
†
= {2} 6 {1, 1} = {2}
†
). An integer parti-
tion
ˆ
ξ ∈
ˆ
P
I
and its conjugate cannot be ordered by
refinement (the first example is for n = 6, where we
have {2, 2, 2} and {3, 3}, which are conjugates of each
other, and cannot be ordered).
The conjugation interchanges the height and width,
and multiplies the rank with −1,
h(
ˆ
ξ
†
) = w(
ˆ
ξ), w(
ˆ
ξ
†
) = h(
ˆ
ξ), r(
ˆ
ξ
†
) = −r(
ˆ
ξ). (50)
(For illustration, see the height-width based arrange-
ment in Figure 6: the conjugation brings to the posi-
tion mirrored with respect to the diagonal w(
ˆ
ξ) =
h(
ˆ
ξ).) Note that, although the conjugation inter-
changes the height and width, it does not inter-
change the partitionability (48a) and producibility
(48b) properties, since these are given in different
ways by height and width, namely, by lower and upper
bounds.
5.3 k-stretchability of correlation and entan-
glement
In Section 5.2, we introduced the permutation invari-
ant property k-stretchability (48c) for k = −(n −
1), −(n − 2), . . . , n − 2, n − 1. Because of the third
inequality in (47), these form a chain in the lattice
ˆ
P
II
,
ˆ
ξ
l-str
ˆ
ξ
k-str
⇐⇒ l ≤ k. (51)
(For illustration, see the height-width based arrange-
ment in Figure 6.)
The corresponding k-stretchably uncorrelated states
(25a) are
D
k-str unc
:= D
ˆ
ξ
k-str
-unc
=
n
O
X∈ξ
%
X
∀ξ ∈ P
I
s.t. max
X∈ξ
|X| − |ξ| ≤ k
o
,
(52a)
which are products of density operators of as few
subsystems as possible, of size as large as possible,
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of difference upper-bounded by k. In this sense,
this is a combination of k-partitionability and k-
producibility. The corresponding k-stretchably sep-
arable states (25b) are
D
k-str sep
:= D
ˆ
ξ
k-str
-sep
= Conv D
k-str unc
=
n
X
j
p
j
O
X∈ξ
j
%
X,j
∀j,
∀ξ
j
∈ P
I
s.t. max
X∈ξ
j
|X| − |ξ
j
| ≤ k
o
,
(52b)
which can be decomposed into k-stretchably uncorre-
lated states. Because of (26a), these properties show
the same chain structure as the chains of the corre-
sponding partition ideals (51), that is,
l ≤ k ⇐⇒ D
l-str unc
⊆ D
k-str unc
, (53a)
l ≤ k ⇐⇒ D
l-str sep
⊆ D
k-str sep
, (53b)
that is, if a state is l-stretchably uncorrelated (sepa-
rable) then it is also k-stretchably uncorrelated (sep-
arable) for all l ≤ k.
The corresponding k-stretchability correlation (27a)
is
C
k-str
:= C
ξ
k-str
= C
ˆ
ξ
k-str
= min
ξ:max
X∈ξ
|X|−|ξ|≤k
C
ξ
. (54a)
The corresponding k-stretchability entanglement
(27c) is
E
k-str
:= E
ξ
k-str
= E
ˆ
ξ
k-str
. (54b)
Because of (28), these measures show the same chain
structure as the chain of the corresponding partition
ideals (51), that is,
l ≤ k ⇐⇒ C
l-str
≥ C
k-str
, (55a)
l ≤ k ⇐⇒ E
l-str
≥ E
k-str
, (55b)
that is, l-stretchability correlation (entanglement) is
always stronger than k-stretchability correlation (en-
tanglement) for all l ≤ k, which is the multipartite
monotonicity of these measures.
For the labeling of the strict stretchability proper-
ties, we have the ideal filters
ˆ
ξ
k-str
:= ↑
ˆ
ξ
k-str
=
ˆ
ξ
l-str
l ≥ k
, (56)
by (48c) and (51). The corresponding strictly k-
stretchably uncorrelated states (31a) are
C
k-str unc
:= C
ˆ
ξ
k-str
-unc
= D
k-str unc
\ D
(k−1)-str unc
,
(57a)
which are products of nonproduct density operators
of as few subsystems as possible, of size as large as
possible, of difference exactly k. The corresponding
strictly k-stretchably separable states (31b) are
C
k-str sep
:= C
ˆ
ξ
k-str
-sep
= D
k-str sep
\ D
(k−1)-str sep
,
(57b)
which can be decomposed into k-stretchably
but not (k − 1)-stretchably uncorrelated states.
(D
(−n)-str unc/sep
= ∅ is understood.)
k-partitionability (48a), k-producibility (48b) and
k-stretchability (48c) give, of course, a rather coars-
ened description of permutation invariant correla-
tion or entanglement properties. Because of this,
they must show some disadvantages. First, k-
partitionability (48a) takes into account only the
number of subsystems uncorrelated with (or separa-
ble from) one another, and does not distinguish be-
tween the cases when subsystems of roughly equal
sizes are uncorrelated (or separable), and when some
elementary subsystems are uncorrelated with (or sep-
arable from) the rest of a large system, although
these two cases represent highly different situations
form a resource-theoretical point of view. (For il-
lustration, see the rows in Figure 6.) Second, k-
producibility (48b) takes into account only the size
of the largest subsystem uncorrelated with (or sep-
arable from) the other part of the system, and does
not distinguish between the cases when a large subsys-
tem is uncorrelated with (or separable from) a slightly
smaller subsystem, or many elementary subsystems,
although these two cases represent again highly dif-
ferent situations form a resource-theoretical point of
view. (For illustration, see the columns in Figure 6.)
Third, k-stretchability (48c) combines the advantages
of the previous two properties in a balanced way, it
takes into account a kind of “difference” of them, and
does not distinguish among cases with rather different
types. (For illustration, see the lines parallel to the
diagonal in Figure 6.)
5.4 Relations among k-partitionability, k-
producibility and k-stretchability
The height h(
ˆ
ξ), width w(
ˆ
ξ) and rank r(
ˆ
ξ) (46) of the
integer partitions
ˆ
ξ ∈
ˆ
P
I
of n ∈ N are bounded by one
another as
n/w ≤ h ≤ n + 1 − w, (58a)
n/h ≤ w ≤ n + 1 − h, (58b)
n/h − h ≤ r ≤ n + 1 − 2h, (58c)
−(n + 1) + 2w ≤ r ≤ w − n/w, (58d)
1
2
p
r
2
+ 4n − r
≤ h ≤
1
2
(n + 1 − r), (58e)
1
2
p
r
2
+ 4n + r
≤ w ≤
1
2
(n + 1 + r). (58f)
The first inequality in (58a) can be seen by noting that
n ≤ hw, because the right-hand side is the area of the
smallest rectangle into which the Young diagram of n
boxes fits. The second inequality in (58a) can be seen
by noting that h + w ≤ n + 1, because the left-hand
side is the half circumference of the smallest rectan-
gle into which the Young diagram of n boxes fits. The
inequalities in (58b) come by the same reasoning, or
Accepted in Quantum 2019-11-05, click title to verify. Published under CC-BY 4.0. 15
by conjugation (50) in (58a). The inequalities in (58c)
come by subtracting h from (58b). The inequalities in
(58d) come by an analogous reasoning, or by conjuga-
tion (50) in (58c). The inequalities in (58e) come by
solving the respective inequalities in (58c) for h. The
inequalities in (58f) come by an analogous reasoning,
or by conjugation (50) in (58e).
The first inequality in (58d) and the second inequal-
ities in the others in (58) are saturated for the parti-
tion
ˆ
ξ = {m, 1, 1, . . . , 1}, where the integer 1 occurs
n − m times. The second inequality in (58d) and the
first inequalities in the others in (58) are saturated for
the partition
ˆ
ξ = {m, m, . . . , m, n − (dn/me − 1)m},
where the integer m occurs dn/me − 1 times. In the
cases where noninteger value stays on one side of an
inequality, saturation is understood for the inequality
strengthened by the ceiling or floor functions dxe =
min{m ∈ Z | m ≥ x} or bxc = max{m ∈ Z | m ≤ x};
that is, if q ≤ m ∈ N, then also dqe ≤ m, and if
q ≥ m ∈ N, then also bqc ≥ m. (For illustration, see
Figure 6. The second inequalities in (58a) and (58b)
express that the arrangement of Young diagrams in
Figure 6 is “skew upper triangular”, while the first in-
equalities in (58a) and (58b) describe the “hyperbolic”
shape of the upper boundary. The other inequali-
ties in (58) give the boundaries in the rank-height or
width-rank plane.)
Since the partitionability is related to lower-
bounding the height (48a), the producibility is related
to upper-bounding the width (48b), and the stretch-
ability is related to their difference (48c), the bounds
in (58) lead to the following relations between these
properties
ˆ
ξ
k-part
ˆ
ξ
(n+1−k)-prod
, (59a)
ˆ
ξ
k-part
ˆ
ξ
(n+1−2k)-str
, (59b)
ˆ
ξ
k-prod
ˆ
ξ
(dn/ke)-part
, (59c)
ˆ
ξ
k-prod
ˆ
ξ
(k−dn/ke)-str
, (59d)
ˆ
ξ
k-str
ˆ
ξ
1
2
(d
√
k
2
+4ne−k)-part
, (59e)
ˆ
ξ
k-str
ˆ
ξ
1
2
(n+1+k)-prod
. (59f)
The relations in (59a) and (59b) can be seen by using
the second inequality in (58b) and the second inequal-
ity in (58c), respectively, with (48a). The relations in
(59c) and (59d) can be seen by using the first inequal-
ity in (58a) and the second inequality in (58d), respec-
tively, with (48b). (For noninteger values, strength-
ening by the ceiling function was also exploited, as
before.) The relations in (59e) and (59f) can be seen
by using the first inequality in (58e) and the second
inequality in (58f), respectively, with (48c).
From the relations (59), for the inclusion of the
state spaces (40), we have
D
k-part unc
⊆ D
(n+1−k)-prod unc
, (60a)
D
k-part unc
⊆ D
(n+1−2k)-str unc
, (60b)
D
k-prod unc
⊆ D
dn/ke-part unc
, (60c)
D
k-prod unc
⊆ D
(k−dn/ke)-str unc
, (60d)
D
k-str unc
⊆ D
1
2
(d
√
k
2
+4ne−k)-part unc
, (60e)
D
k-str unc
⊆ D
1
2
(n+1+k)-prod unc
, (60f)
D
k-part sep
⊆ D
(n+1−k)-prod sep
, (60g)
D
k-part sep
⊆ D
(n+1−2k)-str sep
, (60h)
D
k-prod sep
⊆ D
dn/ke-part sep
, (60i)
D
k-prod sep
⊆ D
(k−dn/ke)-str sep
, (60j)
D
k-str sep
⊆ D
1
2
(d
√
k
2
+4ne−k)-part sep
, (60k)
D
k-str sep
⊆ D
1
2
(n+1+k)-prod sep
, (60l)
by the order isomorphisms (26); and for the bounds
of the measures (42), we have
C
k-part
≥ C
(n+1−k)-prod
, (61a)
C
k-part
≥ C
(n+1−2k)-str
, (61b)
C
k-prod
≥ C
dn/ke-part
, (61c)
C
k-prod
≥ C
(k−dn/ke)-str
, (61d)
C
k-str
≥ C
1
2
(d
√
k
2
+4ne−k)-part
, (61e)
C
k-str
≥ C
1
2
(n+1+k)-prod
, (61f)
E
k-part
≥ E
(n+1−k)-prod
, (61g)
E
k-part
≥ E
(n+1−2k)-str
, (61h)
E
k-prod
≥ E
dn/ke-part
, (61i)
E
k-prod
≥ E
(k−dn/ke)-str
, (61j)
E
k-str
≥ E
1
2
(d
√
k
2
+4ne−k)-part
, (61k)
E
k-str
≥ E
1
2
(n+1+k)-prod
, (61l)
by the multipartite monotonicity (28).
6 Summary, remarks and open ques-
tions
In this work we investigated the partial correlation
and entanglement properties which are invariant un-
der the permutations of the subsystems. The set parti-
tion based three-level structure, describing the classi-
fication of partial correlation and entanglement (Sec-
tion 2) was mapped to a parallel, integer partition
based three-level structure, describing the permuta-
tion invariant case (Section 3). This mapping is easy
to understand on Level I of the construction, however,
to see that it is working well through the whole con-
struction, a formal proof was given (Appendix A).
The construction can be made more compact, al-
though less transparent (Section 4), which can be used
as a starting point of more advanced investigations.
We also investigated k-partitionability and k-
producibility, fitting naturally into the structure of
permutation invariant properties. A kind of combina-
tion of these two gives k-stretchability, which is sen-
sitive in a balanced way to both the maximal size of
correlated (or entangled) subsystems and the minimal
Accepted in Quantum 2019-11-05, click title to verify. Published under CC-BY 4.0. 16
number of subsystems uncorrelated with (or separable
from) one another. We studied their relations, and a
duality, connecting the former two (Section 5).
In the following, we list some remarks and open
questions.
The first point to note is that we have followed a
treatment of entanglement, which is somewhat dif-
ferent than the standard LOCC paradigm [15]. The
LOCC paradigm grabs the essence of entanglement as
a correlation which cannot be created or increased by
classical communication (classical interaction). This
point of view leads to a classification too detailed and
practically unaccomplishable for the multipartite sce-
nario. Our treatment is rooted more in statistics, by
noticing that (i) pure states of classical systems are al-
ways uncorrelated, so in pure states, correlations are
of quantum origin, and this is what we call entangle-
ment [1]; and (ii) mixed states of classical systems
can always be formed by forgetting about the identity
of pure (hence uncorrelated) states, so in mixed states,
correlations which cannot be described in this way are
of quantum origin, and this is what we call entan-
glement. These principles are working painlessly in
the multipartite scenario, entangled states and cor-
relation based definitions of entanglement measures
were defined based on these in the first two levels of
the structure of multipartite entanglement. In par-
ticular, from (i) it follows that entanglement in pure
states should be measured by correlation (see (5b) and
(10b), and the corresponding quantities for the per-
mutation invariant case, and the notes in the end of
Section 2.2). We emphasize that this statistical point
of view is fully compatible with LOCC, and has led
to a much coarser, finite classification.
Being uncertain (or forgetting) about the identity
of the state of the system is a guiding principle in
the definition of the different aspects of multipartite
entanglement in Section 2 (see also in point (ii) in
Section VII.A. in [40]).
States in D
L
: We are uncertain about the (pure)
state, by which the system is described (Section 2.1).
States in D
ξ-unc
(D
ξ-sep
): We are uncertain about the
pure state, by which the system is described, but we
are certain about the partition with respect to which
the state is uncorrelated (separable) (Section 2.2).
States in D
ξ-unc
(D
ξ-sep
): We are uncertain about the
pure state, by which the system is described, and we
are also uncertain about the partition with respect to
which the state is uncorrelated (separable), but we
are certain about the possible partitions with respect
to which the state is uncorrelated (separable) (Sec-
tion 2.3).
States in C
ξ-unc
(C
ξ-sep
): We are uncertain about the
pure state, by which the system is described, and we
are also uncertain about the partition with respect to
which the state is uncorrelated (separable), but we
are certain about the possible partitions with respect
to which the state is uncorrelated (separable), and
we are also certain about the possible partitions with
respect to which the state is correlated (entangled)
(Section 2.4).
We also mention here that for the state sets of
given multipartite correlation (entanglement) proper-
ties D
ξ-unc
(D
ξ-sep
), except from the fully uncorrelated
(or fully separable) case ξ = {⊥}, we cannot formu-
late semigroups of quantum channels which could play
the role of “free operations” for these “free state” sets
in a usual resource-theoretical scenario [75]. Instead
of this, being uncertain (or forgetting) about the iden-
tity of the maps is the guiding principle here (see the
characterizations in Sections 2.2 and 2.3, as well as
in the permutation invariant case in Sections 3.2 and
3.3). Note that this is physical: only the fully uncor-
related (fully separable) states are for free, the others
are resource states, possibly of different “value.”
Note that, after the general case (Section 2), we
considered the permutation invariant (correlation or
entanglement) properties (from Section 3). The use of
these is not restricted to permutation invariant states:
these are simply the permutation invariant properties
of also non-permutation-invariant states. If the states
considered are permutation invariant (for example, in
the first quantization of bosonic or fermionic systems),
then these are the only relevant properties.
The permutation invariant properties were de-
scribed by the use of integer partitions, for which
a refinement-like order was given, which is just the
coarsening of the refinement order of set partitions.
In the literature, there are several partial orders con-
structed for integer partitions, such as the dominance
order [76] (based on majorization, leading to a lat-
tice), the Young order [45, 62] (based on diagram con-
tainment, given for partitions of all integers, leading
to Young’s lattice, here partitions of the same inte-
ger cannot be ordered, and the order is invariant to
the conjugation), or the reverse lexicographic order
(leading to a chain). The refinement order for inte-
ger partitions, introduced in (18c) through the refine-
ment order for set partitions (2c), is different from
the above orders, and, to our knowledge [45, 62], was
not considered upon its merits in the literature be-
fore. Although Birkhoff mentioned this partial order
in his book [77] as an example, it was not used for
any reasonable purpose. The reason for this might
be that the resulting poset shows properties not so
nice or powerful as those shown by the others. How-
ever, this order is what needed is in the classification
problem of permutation invariant properties, that is
reflected also by the monotonicities (47).
k-partitionability is about the natural gradation of
the lattice of set partitions P
I
or of the poset of integer
partitions
ˆ
P
I
, but it was not clear, how k-producibility
can be understood. The conjugation of integer par-
titions has explained the role of the latter, by es-
tablishing a kind of duality between the two prop-
erties. Note, however, that this duality is only par-
Accepted in Quantum 2019-11-05, click title to verify. Published under CC-BY 4.0. 17
tial: although the conjugation of integer partitions
(49) interchanges the height and the width (50), by
which the k-partitionability and k-producibility prop-
erties are given, but the latter properties are given
in an opposite way by height and width, see (48a)
and (48b). So the conjugation does not interchange
k-partitionability and k-producibility, it establishes a
connection on the deeper level of integer partitions
only. This is also reflected in the relations among the
different k-partitionability and k-producibility prop-
erties, given in (59a) and (59c).
For n ≤ 6, the height and width determine
ˆ
ξ
uniquely. For n = 7, we have {3, 2, 2} and {3, 3, 1},
being of height 3 and width 3. Note that two differ-
ent integer partitions of the same height and width
can never be ordered. (Ordered pairs are of differ-
ent height, because of the strict monotonicity of the
height (47).)
The property of k-stretchability (48c) was intro-
duced by the rank of the partition. We note that
the rank of an integer partition (46c) was defined and
used originally in a very different context in number
theory and combinatorics. It was introduced by Free-
man Dyson [78] in his investigations of Ramanujan’s
congruences in the partition function [74].
For n subsystems, the number of nontrivial
k-partitionability and k-producibility properties is
n − 1 in both cases (1-partitionability and n-
producibility are trivial), while the number of non-
trivial k-stretchability properties is 2(n − 1) ((n − 1)-
stretchability is trivial). All of these three prop-
erties form chains, see (39a), (39b) and (51), and
the relations among them are given in (59). Note
that k-stretchability combines the advantages of k-
partitionability and k-producibility, in the sense that
it rewards large correlated (or entangled) subsystems,
while it punishes the larger number of subsystems
uncorrelated (or separable) with one another. Also,
while the relations between k-partitionability and k-
producibility are highly nontrivial (59a), (59c), the
k-stretchability properties form a chain (51). The
price to pay for this is that sometimes there are rather
different properties not distinguished by stretchabil-
ity, for example, {3, 3} and {4, 1, 1} are both 1-
stretchable. Taking stretchability seriously leads to
a highly nontrivial, balanced comparison of permu-
tation invariant multipartite correlation or entangle-
ment properties.
Acknowledgments
Discussions with Géza Tóth and Mihály Máté are
gratefully acknowledged. This research was finan-
cially supported by the National Research, Devel-
opment and Innovation Fund of Hungary within
the Researcher-initiated Research Program (project
Nr: NKFIH-K120569) and within the Quantum
Technology National Excellence Program (project
Nr: 2017-1.2.1-NKP-2017-00001), the Ministry for
Innovation and Technology within the ÚNKP-19-4
New National Excellence Program, and the Hungar-
ian Academy of Sciences within the János Bolyai Re-
search Scholarship and the “Lendület” Program.
A On the structure of the classification
of permutation invariant correlations
A.1 Coarsening a poset
Let us have a finite set P endowed with a binary rela-
tion , and a function f , mapping P to P
0
:= f (P ).
(In the appendix we consider finite sets only, even if it
is not mentioned explicitly.) The same function trans-
forms also the relation naturally as follows. A binary
relation can be considered as a set of ordered pairs
in P , written as ≡ {(b, a) ∈ P × P | b a}, then
0
:= (f × f)() = {(f(b), f(a)) | ∀(b, a) ∈ } by the
elementwise action of f × f. We use the shorthand
notation
0
= f() for this. The meaning of this is
that
∀a
0
, b
0
∈ P
0
, b
0
0
a
0
⇐⇒
∃a ∈ f
−1
(a
0
), ∃b ∈ f
−1
(b
0
) s.t. b a.
(62)
(Here f
−1
(a
0
) = {a ∈ P | f(a) = a
0
} ⊆ P denotes the
inverse image of the singleton {a
0
}.)
If the binary relation is a partial order [43, 45],
the transformed one
0
is not necessarily that. To
make it a partial order, we need some constraints im-
posed on f. Let us introduce the following three con-
ditions:
∀a
0
, b
0
∈ P
0
, if b
0
0
a
0
then
∀a ∈ f
−1
(a
0
), ∃b ∈ f
−1
(b
0
), s.t. b a;
(63a)
∀a
0
, b
0
∈ P
0
, if b
0
0
a
0
then
∀b ∈ f
−1
(b
0
), ∃a ∈ f
−1
(a
0
), s.t. b a;
(63b)
∀a, b, c ∈ P, if c b a
and f(c) = f(a) then f (b) = f(a).
(63c)
These conditions, although being rather strong, hold
in the construction in which we need them (see Ap-
pendix A.4, and Lemma 16). They turn out to be
sufficient for
0
to be a partial order, as the following
lemma states.
Lemma 1. Let (P, ) be a poset, then (P
0
,
0
) =
(f(P ), f()) is a poset if (63a) and (63c) hold, or if
(63b) and (63c) hold.
Proof. We need to prove that
0
is a partial order in
these cases.
(i) Reflexivity (∀a
0
∈ P
0
, a
0
0
a
0
): for all a
0
∈ P
0
we
have f
−1
(a
0
) 6= ∅, since f is surjective, and we need
b ∈ f
−1
(a
0
) and a ∈ f
−1
(a
0
) for which b a by (62);
this holds for the choice b = a, since the partial order
is reflexive, a a. (This proof does not use any of
Accepted in Quantum 2019-11-05, click title to verify. Published under CC-BY 4.0. 18
the constraints (63).)
(ii) Antisymmetry (∀a
0
, b
0
∈ P
0
, if a
0
0
b
0
and b
0
0
a
0
,
then a
0
= b
0
): let b
0
0
a
0
, which means that there
exist a ∈ f
−1
(a
0
) and b ∈ f
−1
(b
0
) such that b a,
by (62). Fix such a pair b and a. Also, let a
0
0
b
0
,
applying (63a) for b, we have that there exists c ∈
f
−1
(a
0
), such that c b. Then, applying (63c) to
c b a, we have that b
0
= f (b) = f(a) = a
0
. (This
proof uses (63a) and (63c). A similar proof can be
given by using (63b) and (63c).)
(iii) Transitivity (∀a
0
, b
0
, c
0
∈ P
0
, if c
0
0
b
0
and b
0
0
a
0
, then c
0
0
a
0
): let b
0
0
a
0
, which means that there
exist a ∈ f
−1
(a
0
) and b ∈ f
−1
(b
0
) such that b a,
by (62). Fix such a pair b and a. Also, let c
0
0
b
0
,
applying (63a) for b, we have that there exists c ∈
f
−1
(c
0
), such that c b. Then, since the partial order
is transitive, we have that c a for an a ∈ f
−1
(a
0
)
and c ∈ f
−1
(c
0
), which means that c
0
0
a
0
by (62).
(This proof uses (63a). A similar proof can be given
by using (63b).)
The following construction would be much sim-
pler if the conditions (63) would also be necessary.
Note that this is not the case: if
0
is a partial or-
der, then (63c) holds, but (63a) and (63b) do not
hold. For example, for the poset P = {a
1
, a
2
, b} with
the only arrow b ≺ a
1
, and the function given as
f(a
1
) = f (a
2
) = a
0
6= f (b) = b
0
, we have b
0
≺
0
a
0
,
but (63a) does not hold.
If (63a) or (63b), and (63c) hold, then the new poset
(P
0
,
0
) can be considered as a coarsening of (P, ) by
f. (f : P → P
0
is surjective by definition. If it is also
injective, then (P, ) and (P
0
,
0
) are isomorphic, and
the coarsening is trivial.) Note that, because of the
construction, f : P → P
0
is automatically monotone,
b a =⇒ f (b)
0
f(a). (64)
(Indeed, a ∈ f
−1
(f(a)) and b ∈ f
−1
(f(b)), so (62)
holds.) Note also that, if the poset (P, ) is a lattice,
the transformed one (P
0
,
0
) is not necessarily that.
To make it a lattice, further conditions have to be
imposed; this problem is not addressed here.
A.2 Down-sets and up-sets
In a poset P , a down-set (order ideal) is a subset
a ⊆ P , which is closed downwards [43, 45],
a ∈ O
↓
(P )
def.
⇐⇒ ∀a ∈ a, ∀b a : b ∈ a; (65a)
while an up-set (order filter) is a subset a ⊆ P , which
is closed upwards [43, 45],
a ∈ O
↑
(P )
def.
⇐⇒ ∀a ∈ a, ∀b a : b ∈ a. (65b)
These form lattices with respect to the inclusion ⊆,
with the join ∨ (least upper bound), being the union
∪, and the meet ∧ (greatest lower bound), being the
intersection ∩.
As in the previous section, let us have the
poset (P, ), the function f, by which (P
0
,
0
) :=
(f(P ), f()), and for which (63a) or (63b), and (63c)
hold. Additionally, let P have bottom and top ele-
ments. (Then also P
0
has bottom and top elements.
Indeed, this is because the bottom and top elements
in P are mapped to the bottom and top elements in
P
0
, because of the (64) monotonicity of f.) Now let us
form from the posets (P, ) and (P
0
,
0
) the lattices
of nonempty down-sets
(Q, v) := (O
↓
(P ) \ {∅}, ⊆), (66a)
(Q
0
, v
0
) := (O
↓
(P
0
) \ {∅}, ⊆). (66b)
(The existence of the bottom elements in P and P
0
ensures that not only the down-sets but also the
nonempty down-sets form lattices in both cases [40].)
Then, by denoting the elementwise action of f with
g : 2
P
→ 2
P
0
, that is, g(a) = {f (a) | a ∈ a}, in
the following lemmas we will show that (Q
0
, v
0
) =
(g(Q), g(v)), that is, the diagram
(Q, v)
g
//
(Q
0
, v
0
)
(P, )
f
//
_
O
↓
\{∅}
OO
(P
0
,
0
)
_
O
↓
\{∅}
OO
(67)
commutes. Here g(v) is the partial order transformed
by g in the same way as was transformed by f in
(62), and we also have automatically that g is mono-
tone for v and g(v), in the same way as in (64).
The following two technical lemmas concerning the
sets
S
a
0
∈a
0
f
−1
(a
0
) ⊆ P will be used several times
later.
Lemma 2. In the above setting (and assuming (63)),
for all a
0
∈ Q
0
, we have g
S
a
0
∈a
0
f
−1
(a
0
)
= a
0
.
Proof. This is because g is the elementwise action of
f, so g
S
a
0
∈a
0
f
−1
(a
0
)
= g
S
a
0
∈a
0
{a ∈ P | f(a) =
a
0
}
=
S
a
0
∈a
0
g
{a ∈ P | f(a) = a
0
}
=
S
a
0
∈a
0
{a
0
} =
a
0
.
Lemma 3. In the above setting (and assuming (63)),
for all a
0
∈ Q
0
, we have
S
a
0
∈a
0
f
−1
(a
0
) ∈ Q, that is,
it is a nonempty down-set.
Proof. First, denote a :=
S
a
0
∈a
0
f
−1
(a
0
). On the one
hand, a 6= ∅, since a
0
6= ∅ and f is surjective. On the
other hand, if b a for an a ∈ a, then f(b)
0
f(a)
because of the monotonicity (64). Since f(a) ∈ a
0
by
the construction of a, and a
0
is a down-set (65a), we
have that f (b) ∈ a
0
. Then b ∈ f
−1
(f(b)) ⊆ a by the
construction of a = f
−1
(f(b))∪
S
a
0
∈a
0
,a
0
6=f(b)
f
−1
(a
0
),
and then b ∈ a, so a is a down-set. Altogether we have
that a ∈ Q.
The following two lemmas show (67).
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Lemma 4. In the above setting (and assuming (63)),
we have g(Q) = Q
0
.
Proof. We need to prove both inclusions.
(i) We need that g(Q) ⊆ Q
0
. Let a ∈ Q and a
0
:=
g(a). Then for all a
0
∈ a
0
let us have a ∈ f
−1
(a
0
) ∩ a.
(Note that f
−1
(a
0
)∩a 6= ∅ by construction.) Then let
b
0
0
a
0
, and applying (63a) for a, we have that there
exists b ∈ f
−1
(b
0
) such that b a. Since a ∈ a, and
a is a down-set (65a), we have that b ∈ a, and then
b
0
∈ a
0
, so a
0
is a down-set.
(ii) We also need that g(Q) ⊇ Q
0
. Let a
0
∈ Q
0
, then we
construct an a ∈ Q, for which a
0
= g(a). The element
a :=
S
a
0
∈a
0
f
−1
(a
0
) fulfills these criteria, we have a
0
=
g(a) by Lemma 2, and a ∈ Q by Lemma 3.
Lemma 5. In the above setting (and assuming (63)),
we have g(v) = v
0
.
Proof. We need to prove both directions.
(i) We need that for all a
0
, b
0
∈ Q
0
, if b
0
g(v)a
0
then
b
0
v
0
a
0
. b
0
g(v)a
0
means that ∃b ∈ g
−1
(b
0
) and ∃a ∈
g
−1
(a
0
) such that b v a. (Again, g
−1
(a
0
) = {a ∈
Q | g(a) = a
0
} is the inverse image of the singleton
{a
0
}.) In this case, b
0
= g(b) = {f (b) | b ∈ b} ⊆
{f(a) | a ∈ a} = g(a) = a
0
, so b
0
v
0
a
0
.
(ii) We also need that for all a
0
, b
0
∈ Q
0
, if b
0
v
0
a
0
then b
0
g(v)a
0
. From b
0
and a
0
, let us form b :=
S
b
0
∈b
0
f
−1
(b
0
) and a :=
S
a
0
∈a
0
f
−1
(a
0
). We have that
b ∈ Q and a ∈ Q (they are nonempty down-sets) by
Lemma 3, also g(b) = b
0
and g(a) = a
0
by Lemma 2,
and if b
0
v
0
a
0
then b v a by construction. That is,
we have constructed b ∈ g
−1
(b
0
) and a ∈ g
−1
(a
0
), for
which b v a, so we have b
0
g(v)a
0
.
Summing up, Lemma 4 and Lemma 5 together state
that (67) commutes. Later we also need that the prop-
erties (63a), (63b) and (63c) are inherited.
Lemma 6. In the above setting (and assuming (63)),
we have
∀a
0
, b
0
∈ Q
0
, if b
0
v
0
a
0
then
∀a ∈ g
−1
(a
0
) ∃b ∈ g
−1
(b
0
) s.t. b v a;
(68a)
∀a
0
, b
0
∈ Q
0
, if b
0
v
0
a
0
then
∀b ∈ g
−1
(b
0
) ∃a ∈ g
−1
(a
0
) s.t. b v a;
(68b)
∀a, b, c ∈ Q, if c v b v a
and g(c) = g(a), then g(b) = g(a).
(68c)
Proof. (68a) can be proven by the explicit construc-
tion of b. Let b
0
v
0
a
0
, and a ∈ g
−1
(a
0
), then
b :=
S
b
0
∈b
0
f
−1
(b
0
)
∩ a. By construction, b v a.
Also, b ∈ Q, that is, it is a nonempty down-set, since
it is the intersection (meet) of two nonempty down-
sets: a ∈ Q, and
S
b
0
∈b
0
f
−1
(b
0
) ∈ Q by Lemma 3, and
Q is a lattice. In addition, we need that b ∈ g
−1
(b
0
),
that is, g(b) = b
0
. Since g(a) = a
0
, we have that
∀a
0
∈ a
0
, ∃a ∈ a such that f(a) = a
0
. Then also
∀b
0
∈ b
0
v
0
a
0
, ∃a ∈ a such that f (a) = b
0
, that is,
a ∈ f
−1
(b
0
), and also a ∈ a, so a ∈ b by construction
of b.
(68b) can be proven analogously.
(68c) is a simple consequence of the properties of v.
Let c v b v a, then g(c) v
0
g(b) v
0
g(a) by the
monotonicity of g, so by the transitivity and antisym-
metry of the partial order v
0
, if g(c) = g(a) then
g(b) = g(a).
Note that the conditions (63) were not used explicitly
in these proofs, they are assumed only for establishing
the partial order
0
for the set P
0
, see Lemma 1.
Lemma 7. Lemmas 2, 3, 4, 5 and 6, hold also if up-
sets are used instead of down-sets in the construction
(67).
Proof. All the proofs can be repeated with slight mod-
ifications, as interchanging the roles of (63a) and
(63b), down-closures and up-closures, and reversing
orders.
Thanks to Lemma 4, Lemma 5, Lemma 6 and
Lemma 7, the construction (67) can be repeated arbi-
trary times, with either up- or down-sets, if the con-
ditions (63) hold for the first level. Note that v and
v
0
are partial orders by construction anyway. The
inheritance of (63), shown in Lemma 6 is needed not
for that, but also for, e.g., Lemma 4, when applied for
the next level.
A.3 Embedding
Now, with the definition (66), let the function g
0
:
Q
0
→ Q be given as
g
0
(a
0
) := ∨g
−1
(a
0
), (69)
and its image Q
00
:= g
0
(Q
0
) ⊆ Q. We will show that
this is a subposet, and (Q
00
, v)
∼
=
(Q
0
, v
0
), and g
0
is
an embedding of Q
0
into Q. (We use the notation
∨A :=
W
a∈A
a for any subset A of a lattice.)
The following two technical lemmas concerning the
images g
0
(a
0
) will be used several times later.
Lemma 8. In the above setting (and assuming (63)),
for all a
0
∈ Q
0
, we have
g
0
(a
0
) = {a ∈ P | f(a) ∈ a
0
}, (70a)
g
0
(a
0
) =
[
a
0
∈a
0
f
−1
(a
0
). (70b)
Proof. (70a) is by definition g
0
(a
0
) ≡ ∨{a ∈
Q | g(a) = a
0
} = {a ∈ P | f(a) ∈ a
0
}, and
we need to prove both inclusions. First, ∨{a ∈
Q | g(a) = a
0
} ⊆ {a ∈ P | f(a) ∈ a
0
} holds,
since g is the elementwise action of f. Second, for
∨{a ∈ Q | g(a) = a
0
} ⊇ {a ∈ P | f (a) ∈ a
0
}, we need
to see that for all a ∈ P such that f (a) ∈ a
0
there
exists an a ∈ Q for which a ∈ a, while g(a) = a
0
. The
element a :=
S
a
0
∈a
0
f
−1
(a
0
) fulfills these criteria, we
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have a
0
= g(a) by Lemma 2, and a ∈ Q by Lemma 3;
while a ∈ a, since f (a) ∈ a
0
, so it appears in the union
S
a
0
∈a
0
f
−1
(a
0
) = f
−1
(f(a)) ∪
S
a
0
∈a
0
,a
0
6=f(a)
f
−1
(a
0
).
(70b) can be proven by noting that for the left-hand
side, we have by (70a) that g
0
(a
0
) =
a ∈ P
f(a) ∈
a
0
≡ {a ∈ P | ∃a
0
∈ a
0
s.t. f (a) = a
0
}, which is the
same as the right-hand side
S
a
0
∈a
0
{a ∈ P | f(a) =
a
0
}.
Lemma 9. In the above setting (and assuming (63)),
for all a
0
∈ Q
0
, we have g(g
0
(a
0
)) = a
0
.
Proof. Using (70b), we have that g(g
0
(a
0
)) =
g
S
a
0
∈a
0
f
−1
(a
0
)
, then Lemma 2 leads to the
claim.
The following lemma shows that g
0
: Q
0
→ Q
00
⊆ Q
is an embedding of Q
0
into Q.
Lemma 10. In the above setting (and assuming
(63)), (Q
00
, v)
∼
=
(Q
0
, v
0
).
Proof. We need to prove that for all a
0
, b
0
∈ Q
0
,
b
0
v
0
a
0
⇔ g
0
(b
0
) v g
0
(a
0
), then, from the antisym-
metry of the partial order, we have that g
0
is bijective.
To see the “if” direction, we have g
0
(b
0
) v g
0
(a
0
) then
g(g
0
(b
0
)) v
0
g(g
0
(a
0
)) by the monotonicity of g, simi-
larly to (64), then b
0
v
0
a
0
by Lemma 9.
To see the “only if” direction, we have b
0
v
0
a
0
, then
{b ∈ P | f(b) ∈ b
0
} v {a ∈ P | f(a) ∈ a
0
}, then
g
0
(b
0
) v g
0
(a
0
) by (70a).
In the following four lemmas, we provide some sim-
ple tools, concerning principal ideals.
Lemma 11. In the above setting (and assuming
(63)), for all a ∈ P , we have g(↓{a}) = ↓{f(a)}.
Proof. For the left-hand side we have g(↓{a}) =
g({b ∈ P | b a}) = {b
0
∈ P
0
| ∃b ∈ P s.t. f (b) =
b
0
, b a} = {b
0
∈ P
0
| ∃b ∈ f
−1
(b
0
) s.t. b a},
while for the right-hand side we have ↓{f(a)} = {b
0
∈
P
0
| b
0
0
f(a)}, by definitions. So we need that for
all a ∈ P and b
0
∈ P
0
, (∃b ∈ f
−1
(b
0
) s.t. b a) ⇔
b
0
0
f(a).
To see the “if” direction, we have that if b
0
0
f(a)
then for a ∈ f
−1
(f(a)) there exists b ∈ f
−1
(b
0
) such
that b a by (63a).
To see the “only if” direction, we have that if ∃b ∈
f
−1
(b
0
) such that b a, then f(b)
0
f(a) by
the monotonicity (64) of f, then b
0
0
f(a), since
f(b) = b
0
by the assumption b ∈ f
−1
(b
0
).
Lemma 12. In the above setting (and assuming
(63)), for all a
0
∈ P
0
, we have g
0
(↓{a
0
}) = ↓ f
−1
(a
0
).
Proof. For the left-hand side we have g
0
(↓{a
0
}) =
{b ∈ P | f(b) ∈ ↓{a
0
}} = {b ∈ P | f(b) a
0
}
by applying (70a), while for the right-hand side we
have ↓ f
−1
(a
0
) = ↓{a ∈ P | f(a) = a
0
} = {b ∈
P | ∃a ∈ P s.t. b a, f(a) = a
0
} = {b ∈ P | ∃a ∈
f
−1
(a
0
) s.t. b a}, by definitions. So we need that
for all a
0
∈ P
0
and b ∈ P (∃a ∈ f
−1
(a
0
) s.t. b
a) ⇔ f(b)
0
a
0
.
To see the “if” direction, we have that if f(b)
0
a
0
then for b ∈ f
−1
(f(b)) there exists a ∈ f
−1
(a
0
) such
that b a by (63b).
To see the “only if” direction, we have that if ∃a ∈
f
−1
(a
0
) such that b a, then f (b)
0
f(a) by
the monotonicity (64) of f, then f(b)
0
a
0
, since
f(a) = a
0
by the assumption a ∈ f
−1
(a
0
).
Lemma 13. In the above setting (and assuming
(63)), for all a
0
, b
0
∈ P
0
, we have b
0
0
a
0
⇔
g
0
(↓{b
0
}) v g
0
(↓{a
0
}).
Proof. First, we have b
0
0
a
0
⇔ ↓{b
0
} v
0
↓{a
0
} obvi-
ously. Second, we have ↓{b
0
} v
0
↓{a
0
} ⇔ g
0
(↓{b
0
}) v
g
0
(↓{a
0
}), which is a special case of Lemma 10, for
principal ideals in Q
0
.
Lemma 14. In the above setting (and assuming
(63)), for all a
0
∈ Q
0
, we have
g
0
(a
0
) =
_
a
0
∈a
0
↓ f
−1
(a
0
). (71)
Proof. We have
W
a
0
∈a
0
↓ f
−1
(a
0
) =
W
a
0
∈a
0
g
0
(↓{a
0
}) =
W
a
0
∈a
0
S
b
0
∈↓{a
0
}
f
−1
(b
0
) =
S
a
0
∈a
0
,b
0
a
0
f
−1
(b
0
) =
S
a
0
∈a
0
f
−1
(b
0
) = g
0
(a
0
), by using Lemma 12, (70b),
exploiting that a
0
is a down-set (65a), then (70b)
again, respectively.
Although it will not be used, the following lemma
shows that the whole construction in this section
works also if up-sets are used instead of down-sets.
Lemma 15. Lemmas 8, 9, 10, 11, 12, 13 and 14
hold also if up-sets are used instead of down-sets in
the construction (67).
Proof. All the proofs can be repeated with slight mod-
ifications.
A.4 Application to the set and integer parti-
tions
Now we turn to the case of the main text, for which
the machinery developed in the previous sections is
applied. That is, for the roles of (P, ) and (P
0
,
0
)
we have the poset (P
I
, ) of set partitions (2b) with
the refinement relation (2c), and the poset (
ˆ
P
I
, ) of
integer partitions (18b) with the refinement relation
(18c).
First, we have the σ ∈ S(L) permutations of the
elementary subsystems L, acting naturally on the
partitions ξ = {X
1
, X
2
, . . . , X
|ξ|
} ∈ P
I
as σ(ξ) =
{σ(i) | i ∈ X}
X ∈ ξ
∈ P
I
. We will make
use of the obvious observations that all the permu-
tations of a partition ξ ∈ P
I
are of the same type,
s(σ(ξ)) = s(ξ) ∈
ˆ
P
I
; and all partitions of a given
type
ˆ
ξ ∈
ˆ
P
I
can be transformed into one another by
suitable permutations, that is, for all ξ
1
, ξ
2
∈ s
−1
(
ˆ
ξ),
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there exists at least one σ ∈ S(L) by which ξ
2
= σ(ξ
1
).
Also, for all σ ∈ S(L) permutations, υ ξ if and only
if σ(υ) σ(ξ). Using these, we can show that (63)-
like conditions hold for this case.
Lemma 16. In the setting of the main text, the fol-
lowing conditions hold
∀
ˆ
ξ, ˆυ ∈
ˆ
P
I
, if ˆυ
ˆ
ξ then
∀ξ ∈ s
−1
(
ˆ
ξ), ∃υ ∈ s
−1
(ˆυ), s.t. υ ξ;
(72a)
∀
ˆ
ξ, ˆυ ∈
ˆ
P
I
, if ˆυ
ˆ
ξ then
∀υ ∈ s
−1
(ˆυ), ∃ξ ∈ s
−1
(
ˆ
ξ), s.t. υ ξ;
(72b)
∀ξ, υ, ζ ∈ P
I
, if ζ υ ξ
and s(ζ) = s(ξ), then s(υ) = s(ξ).
(72c)
Proof. Recall that ˆυ
ˆ
ξ by definition if there exist
υ
1
∈ s
−1
(ˆυ) and ξ
1
∈ s
−1
(
ˆ
ξ), such that υ
1
ξ
1
,
see (18c), in accordance with (62). Then, for all ξ
2
∈
s
−1
(
ˆ
ξ) we can construct υ
2
which is υ
2
ξ
2
, simply as
υ
2
:= σ(υ
1
) by any of the permutations implementing
ξ
2
= σ(ξ
1
), leading to (72a).
(72b) can be proven analogously.
(72c) can be proven by noting that for all ξ, υ ∈ P
we have that if υ ξ then |υ| ≤ |ξ|, by (2c); and
|ξ| = |s(ξ)|, by (17). Then ζ υ ξ and s(ζ) = s(ξ)
lead to |ζ| ≤ |υ| ≤ |ξ| with |ζ| = |ξ|, form which
we have |υ| = |ξ|. Now we need that |υ| = |ξ| and
υ ξ lead to υ = ξ (then s(υ) = s(ξ)), which holds,
because (2c) must be fulfilled with the same number
of parts: let us number them accordingly as Y
i
⊆ X
i
for ξ = {X
1
, X
2
, . . . , X
k
} and υ = {Y
1
, Y
2
, . . . , Y
k
},
with the constraints of being disjoint and
S
k
i=1
X
i
=
S
k
i=1
Y
i
= L, which leads to Y
i
= X
i
.
Recall that Lemma 1 states that if a poset (P, )
is transformed by a function f as given in (62), then
(P
0
,
0
) = (f(P ), f ()) is a poset if the conditions
(63) hold. Now Lemma 16 shows that for the poset
(P
I
, ), if transformed by s as given in (18b) and (18c)
in the main text, the corresponding conditions (72)
hold, so (
ˆ
P
I
, ) = (s(P ), s()) is a poset.
Recall also that if we form the down-set lattices
(66), then Lemma 4 and Lemma 5 together state that
(67) commutes, Lemma 6 states that the conditions
in (63) are inherited. Then with Lemma 7, we have
that in the main text, the three-level construction is
well-defined as follows.
Corollary 17. The diagram (14) in the main text
commutes.
With Lemma 12 and (71), we also have that for the
first two levels of the construction in the main text,
the following identities concerning the principal ideals
hold.
Corollary 18. In the setting of the main text, for all
ˆ
ξ ∈
ˆ
P
I
and
ˆ
ξ ∈
ˆ
P
II
, we have
∨s
−1
(↓{
ˆ
ξ}) = ↓ s
−1
(
ˆ
ξ), (73a)
∨s
−1
(
ˆ
ξ) =
_
ˆ
ξ∈
ˆ
ξ
↓ s
−1
(
ˆ
ξ). (73b)
With Lemma 13 and Lemma 10, we also have that
for the first two levels of the construction in the main
text, the following order isomorphisms hold.
Corollary 19. In the setting of the main text, for all
ˆ
ξ, ˆυ ∈
ˆ
P
I
and
ˆ
ξ,
ˆ
υ ∈
ˆ
P
II
, we have
ˆυ
ˆ
ξ ⇐⇒ ∨s
−1
(↓{ˆυ}) ∨s
−1
(↓{
ˆ
ξ}), (74a)
ˆ
υ
ˆ
ξ ⇐⇒ ∨s
−1
(
ˆ
υ) ∨s
−1
(
ˆ
ξ). (74b)
B Miscellaneous proofs
B.1 Monotonicity of correlation measures
Here we show the ξ-LO monotonicity of the ξ-
correlation (5a), from which the LO monotonicity of
the ξ-correlation (10a) follows, because the latter one
is a minimum of some of the former ones. A quantum
channel (completely positive trace preserving map
[6, 7]) Φ is ξ-LO, if it can be written as Φ =
N
X∈ξ
Φ
X
with quantum channels Φ
X
: D
X
→ D
X
for all X ∈ ξ.
With this,
C
ξ
Φ(%)
(5a)
= min
σ∈D
ξ-unc
D
Φ(%)
σ
(75)
≤ min
σ
0
∈Φ(D
ξ-unc
)
D
Φ(%)
σ
0
= min
σ∈D
ξ-unc
D
Φ(%)
Φ(σ)
(76)
≤ min
σ∈D
ξ-unc
D
%
σ
(5a)
= C
ξ
(%),
where the first inequality comes from
Φ(D
ξ-unc
) ⊆ D
ξ-unc
, (75)
which holds for ξ-LOs; and the second inequality
comes from the monotonicity of the relative entropy
(1b),
D
Φ(%)
Φ(σ)
≤ D(%kσ), (76)
which holds for all quantum channels [6, 7].
Note that the monotonicity shown here is a particu-
lar case of a much more general property of monotone
distance based geometric measures [65] in resource
theories [75]: the monotonicity with respect to free
maps, by which the set of free states is mapped onto
itself.
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