Separability for mixed states with operator Schmidt rank two
Gemma De las Cuevas
1
, Tom Drescher
2
, and Tim Netzer
2
1
Institute for Theoretical Physics, Technikerstr. 21a, A-6020 Innsbruck, Austria
2
Department of Mathematics, Technikerstr. 13, A-6020 Innsbruck, Austria
The operator Schmidt rank is the minimum
number of terms required to express a state as
a sum of elementary tensor factors. Here we
provide a new proof of the fact that any bipar-
tite mixed state with operator Schmidt rank
two is separable, and can be written as a sum
of two positive semidefinite matrices per site.
Our proof uses results from the theory of free
spectrahedra and operator systems, and illus-
trates the use of a connection between decom-
positions of quantum states and decomposi-
tions of nonnegative matrices. In the multipar-
tite case, we prove that any Hermitian Matrix
Product Density Operator (MPDO) of bond
dimension two is separable, and can be writ-
ten as a sum of at most four positive semidefi-
nite matrices per site. This implies that these
states can only contain classical correlations,
and very few of them, as measured by the en-
tanglement of purification. In contrast, MP-
DOs of bond dimension three can contain an
unbounded amount of classical correlations.
1 Introduction
Entanglement is an essential ingredient in many appli-
cations in quantum information processing and quan-
tum computation [NC00, HHHH09]. Mixed states
which are not entangled are called separable, and they
may contain only classical correlations—in contrast
to entangled states, which contain quantum correla-
tions. As many other problems in theoretical physics
and elsewhere, the problem of determining whether a
state is entangled or not is NP-hard [Gur03, Gha10].
This does not prevent the existence of multiple sepa-
rability criteria [HHHH09]. One example are criteria
based on the rank of a bipartite mixed state 0 6 ρ ∈
M
d
1
⊗ M
d
2
(where M
d
denotes the set of complex
matrices of size d × d, and A > 0 denotes that A is
positive semidefinite) [KCKL00, HLVC00]. Another
example is the realignment criterion [KW03, Rud03],
which is based on the operator Schmidt decomposi-
tion of the state.
Here, we focus on the operator Schmidt rank of ρ,
which is the minimum p so that
ρ =
p
X
α=1
A
α
⊗ B
α
, (1)
where A
α
∈ M
d
1
and B
α
∈ M
d
2
, i.e. these matrices
need not fulfill any conditions of Hermiticitiy or pos-
itivity. Clearly, a product state (i.e. a state without
classical or quantum correlations) has p = 1, but it is
generally not clear what kind of correlations a state
with p > 1 has.
The first main result of this paper is that, if a state
has operator Schmidt rank two, then it is separable.
Moreover, it can be written as a sum of only two pos-
itive semidefinite matrices at each site. Our proof is
constructive and gives a method to obtain the posi-
tive semidefinite matrices, as we will explicitly show.
To state our result formally, denote by PSD
d
the set
of d × d positive semidefinite matrices.
Theorem 1 (Bipartite case). Let ρ be a positive
semidefinite matrix 0 6 ρ ∈ M
d
1
⊗M
d
2
where d
1
, d
2
are arbitrary, such that
ρ =
2
X
α=1
A
α
⊗ B
α
where A
α
∈ M
d
1
and B
α
∈ M
d
2
. Then ρ is separable
and can be written as
ρ =
2
X
α=1
σ
α
⊗ τ
α
where σ
α
∈ PSD
d
1
and τ
α
∈ PSD
d
2
.
To the best of our knowledge, this result was first
proven by Cariello in [Car14, Lemma 4.5] (see also
[Car17, Theorem 3.44] and this blog post [Joh]), and
extended to the case that one side is infinite di-
mensional in [Gie18, Theorem 3.3]. (See also the
related inequalities [Car14, Theorem 5.2, Theorem
5.3].) Here we provide a new proof for this result,
using other techniques and a new context. The tech-
niques are based on free spectrahedra, which form
a relatively young field of study within convex alge-
braic geometry, and whose connections with decom-
positions of quantum states are explored in [Net19].
In particular, the core result of this paper is a rela-
tion between separable states and minimal operator
systems (Theorem 11), and its consequences (Corol-
lary 13), which we leverage to prove Theorem 1. (See
Ref. [BN18] for other recent connections between free
spectrahedra and quantum information.) The context
of this work is a connection between decompositions
of quantum states and decompositions of nonnegative
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arXiv:1903.05373v4 [quant-ph] 11 Nov 2019
matrices presented in [DN19]. Theorem 1 is inspired
by an analogous result for nonnegative matrices, and
can be seen as a generalisation thereof, as we will show
in Section 4.3.
What happens in the multipartite case? For pure
states, multipartite entanglement features many novel
properties with respect to the bipartite case, see e.g.
[SWGK18]. For a mixed state that describes the state
of a spin chain in one spatial dimension, 0 6 ρ ∈
M
d
1
⊗ M
d
2
⊗ ··· ⊗ M
d
n
, the multipartite analogue
of Eq. (1) is the Matrix Product Density Operator
(MPDO) form [VGRC04, ZV04],
ρ =
D
X
α
1
,...,α
n−1
=1
A
[1]
α
1
⊗ A
[2]
α
1
,α
2
⊗ ··· ⊗ A
[n]
α
n−1
,
where A
[1]
α
∈ M
d
1
, A
[2]
α,β
∈ M
d
2
, . . . , A
[n]
α
∈ M
d
n
.
The minimal such D is also called the operator
Schmidt rank of ρ. If, additionally, we require that
each of the local matrices is Hermitian, i.e. A
[1]
α
∈
Her
d
1
, A
[2]
α,β
∈ Her
d
2
, . . . , A
[n]
α
∈ Her
d
n
, where Her
d
denotes the set of d × d Hermitian matrices, then
the minimal such D is called the Hermitian operator
Schmidt rank of ρ [DN19].
One of the problems of the MPDO form is that
the amount of quantum or classical correlations of
the state (as measured by the entanglement of pu-
rification [THLD02]) is not upper bounded by the
operator Schmidt rank [DSPGC13, DN19]. Instead,
the amount of correlations is upper bounded by the
purification rank of ρ, which is the smallest op-
erator Schmidt rank of a matrix L that satisfies
ρ = LL
†
, i.e. that makes explicit the positivity of
ρ [DSPGC13, DN19]. Since the purification rank
can be arbitrarily larger than the operator Schmidt
rank [DSPGC13], the latter generally does not tell
us anything about the correlations of the state. In
particular, there are families of states with operator
Schmidt rank three whose purification rank diverges
[DSPGC13]. A similar thing happens for the Hermi-
tian operator Schmidt rank: while it is lower bounded
by the operator Schmidt rank, it does not allow us to
upper bound the amount of correlations of the state.
The exception to this rule is again the trivial case of
product states, in which case the operator Schmidt
rank, its Hermitian counterpart and the purification
rank are all one.
The second main result of this paper is that if a mul-
tipartite state has Hermitian operator Schmidt rank
two, then it is separable. Moreover, it admits a sep-
arable decomposition of bond dimension two, which
means that it can be written as a sum of at most four
positive semidefinite matrices per site. Formally:
Theorem 2 (Multipartite case). Let ρ be a positive
semidefinite matrix, 0 6 ρ ∈ M
d
1
⊗M
d
2
⊗···⊗M
d
n
where d
1
, d
2
, . . . , d
n
are arbitrary, such that
ρ =
2
X
α
1
,...,α
n−1
=1
A
[1]
α
1
⊗ A
[2]
α
1
,α
2
⊗ ··· ⊗ A
[n]
α
n−1
where A
[1]
α
∈ Her
d
1
, A
[2]
α,β
∈ Her
d
2
, . . . , A
[n]
α
∈ Her
d
n
.
Then ρ is separable and can be written as
ρ =
2
X
α
1
,...,α
n−1
=1
σ
[1]
α
1
⊗ σ
[2]
α
1
,α
2
⊗ ··· ⊗ σ
[n]
α
n−1
where σ
[1]
α
∈ PSD
d
1
, σ
[2]
α,β
∈ PSD
d
2
, . . ., σ
[n]
α
∈
PSD
d
n
.
This situation is summarised in Table 1.
States Product Separable
hosr 1 2 3
puri-rank 1 2 unbounded
sep-rank 1 2 unbounded
Table 1: Multipartite mixed states ρ with hermitian operator
Schmidt rank (cf. Definition 3) hosr(ρ) = 1 are product
states, and the separable rank sep-rank (Definition 5) and
purification rank puri-rank (Definition 6) need also be 1. If
hosr(ρ) = 2 then ρ must be separable and the other two
ranks also need be 2 (Theorem 2). If hosr(ρ) = 3 and ρ
is separable, then the other two ranks may diverge with the
system size [DSPGC13].
We remark that [Car14, Corollary 4.8] showed that
if a state ρ has tensor rank 2, then it is separable.
Recall that the tensor rank, tsr(ρ), is the minimal D
required to express ρ as
ρ =
D
X
α=1
A
[1]
α
⊗ A
[2]
α
⊗ . . . A
[n]
α
.
Theorem 2, in contrast, shows that if the Hermitian
operator Schmidt rank of a state ρ is 2, then ρ is
separable and its separable rank is 2 (the latter will
be defined in Definition 5). Since both the hermi-
tian operator Schmidt rank (hosr) and the tensor rank
(tsr) are lower bounded by the operator Schmidt rank
(osr),
osr ≤ hosr, osr ≤ tsr,
our Theorem 2 does not follow from [Car14, Corollary
4.8] (For more relations between the ranks of several
types of tensor decompositions, see [DHN19]). Note
also that Theorem 2 does not only show that ρ is
separable but also that the separable rank (see Defi-
nition 5) is 2.
This paper is structured as follows. In Section 2 we
present the preliminary material needed for this work.
In Section 3 we prove the core result, Theorem 11. In
Section 4 we focus on the bipartite case: we prove
Theorem 1, provide a method to obtain the separa-
ble decomposition, and explore several implications of
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this result. In Section 5 we focus on the multipartite
case: we prove Theorem 2, and explore its implica-
tions. Finally, in Section 6 we conclude and present
an outlook.
2 Preliminaries
Here we present the preliminary material needed to
prove the results of this work. First we will define the
relevant decompositions of mixed states (Section 2.1),
and finally define free spectrahedra and minimal op-
erator systems (Section 2.2).
We use the following notation. The set of posi-
tive semidefinite matrices, complex Hermitian matri-
ces, and complex matrices of size r × r is denoted
PSD
r
, Her
r
and M
r
, respectively. We denote that a
matrix A is positive semidefinite (psd) by A > 0. The
identity matrix is denoted by I. We will also often
ignore normalizations, as they do not modify any of
the ranks [DN19]. For this reason, we often refer to
positive semidefinite matrices as states.
2.1 Decompositions of mixed states
Throughout this paper we consider a multipartite pos-
itive semidefinite matrix 0 6 ρ ∈ M
d
1
⊗ M
d
2
⊗
··· ⊗ M
d
n
. We now review the MPDO, the Hermi-
tian MPDO form, the separable decomposition and
the local purification of ρ. They are all discussed and
characterized in Ref. [DN19].
Definition 3. The Matrix Product Density Operator
(MPDO) form [VGRC04, ZV04] of ρ is given by
ρ =
D
X
α
1
,...,α
n−1
=1
A
[1]
α
1
⊗ A
[2]
α
1
,α
2
⊗ ··· ⊗ A
[n]
α
n−1
, (2)
where A
[l]
α
∈ M
d
l
for l = 1, n, and A
[l]
α,β
∈ M
d
l
for
1 < l < n. The minimum such D is called the opera-
tor Schmidt rank of ρ, denoted osr(ρ).
The MPDO form can also be defined for matrices
which are not Hermitian or positive semidefinite. In
this case, it is simply called the Matrix Product Oper-
ator form (since ρ is no longer a density operator), and
the corresponding minimum dimension is also called
the operator Schmidt rank.
We also need to consider a modified version of the
MPDO form, in which the local matrices are Hermi-
tian.
Definition 4. The Hermitian Matrix Product Den-
sity Operator (Hermitian MPDO) form [DN19] of ρ
is given by
ρ =
D
X
α
1
,...,α
n−1
=1
A
[1]
α
1
⊗ A
[2]
α
1
,α
2
⊗ ··· ⊗ A
[n]
α
n−1
, (3)
where A
[l]
α
∈ Her
d
l
for l = 1, n, and A
[l]
α,α
0
∈ Her
d
l
for
1 < l < n. The minimum such D is called the Her-
mitian operator Schmidt rank of ρ, denoted hosr(ρ).
In this paper, we say that ρ is separable if it admits
a separable decomposition. That this is a meaningful
definition is shown in [DN19].
Definition 5. A separable decomposition of ρ is a
form where
ρ =
D
X
α
1
,...,α
n−1
=1
σ
[1]
α
1
⊗ σ
[2]
α
1
,α
2
⊗ ··· ⊗ σ
[n]
α
n−1
(4)
where each of these matrices is psd, i.e. σ
[1]
α
∈ PSD
d
1
,
σ
[n]
α
∈ PSD
d
n
, and σ
[j]
α,β
∈ PSD
d
j
for 1 < j < n.
The minimal such D is called the separable rank of ρ,
denoted sep-rank(ρ).
We now review the local purification form.
Definition 6. The local purification form of ρ is an
expression ρ = LL
†
where L is in Matrix Product Op-
erator form. The minimal operator Schmidt rank of
such an L is called the purification rank of ρ, denoted
puri-rank(ρ):
puri-rank(ρ) = min{osr(L)|ρ = LL
†
}.
Finally we review some results about these ranks.
Proposition 7 ([DN19]). Let
0 6 ρ ∈ M
d
1
⊗ M
d
2
⊗ ··· ⊗ M
d
n
.
Then
(i) osr(ρ) = 1 ⇐⇒ puri-rank(ρ) = 1 ⇐⇒
sep-rank(ρ) = 1.
(ii) osr(ρ) ≤ puri-rank(ρ)
2
, and this bound is tight
for pure states.
(iii) If ρ is separable, then osr(ρ) ≤ sep-rank(ρ), and
puri-rank(ρ) ≤ sep-rank(ρ).
It is also known that there is a separation between
osr(ρ) and puri-rank(ρ), and between puri-rank(ρ)
and sep-rank(ρ) [DN19].
2.2 Free spectrahedra
Here we review the definitions of (free) spectrahedra,
which are discussed more thoroughly in [Net19], as
well as the definition of minimal operator system.
Definition 8. A spectrahedron is a set of the follow-
ing form
S(A
1
, . . . , A
m
) = {(b
1
, . . . , b
m
) ∈ R
m
|
m
X
i=1
b
i
A
i
> 0}.
where A
1
, . . . A
m
∈ Her
s
are Hermitian matrices.
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The free spectrahedron is its non-commutative gen-
eralization:
Definition 9. For A
1
, . . . , A
m
∈ Her
s
and r ∈ N we
define
FS
r
(A
1
, . . . , A
m
) :={(B
1
, . . . , B
m
) ∈ Her
m
r
|
m
X
i=1
A
i
⊗ B
i
> 0} (5)
and call it the r-level of the free spectrahedron defined
by A
1
, . . . , A
m
. The collection
FS(A
1
, . . . , A
m
) := (FS
r
(A
1
, . . . , A
m
))
r≥1
is called the free spectrahedron defined by A
1
, . . . , A
m
.
Note that the free spectrahedron at level 1 coincides
with the spectrahedron.
Finally we review the definition of minimal opera-
tor system (see, for example, Ref. [FNT16] for more
details).
Definition 10. Let C ⊆ R
m
be a closed salient con-
vex cone (i.e. a closed cone that does not contain a
nontrivial subspace) with nonempty interior. Define
C
min
r
:=
(
n
X
i=1
c
i
⊗ P
i
| n ∈ N, c
i
∈ C, P
i
∈ PSD
r
)
Note that since the c
i
are from R
m
, elements from
C
min
r
can be understood as m-tuples of Hermitian ma-
trices of size r (exactly as elements from the r-level
of a free spectrahedron).
The minimal operator system is defined as
C
min
=
C
min
r
r≥1
.
3 Separable states and minimal oper-
ator systems
In this section we prove the core result of this paper,
namely Theorem 11, which is a relation between sep-
arable states and minimal operator systems. We will
extract several consequences thereof in Corollary 13,
which will be exploited later on to prove Theorem 1
and Theorem 2.
Theorem 11. Let P
1
, . . . , P
m
∈ Her
s
be R-linearly
independent, and Q
1
, . . . , Q
m
∈ Her
t
with
ρ =
m
X
i=1
P
i
⊗ Q
i
> 0. (6)
Define
C := FS
1
(P
1
, . . . , P
m
) ⊆ R
m
and
V := span
R
{P
1
, . . . , P
m
} ⊆ Her
s
.
Then the following are equivalent:
(i). (Q
1
, . . . , Q
m
) ∈ C
min
t
(ii). ρ admits a separable decomposition
ρ =
n
X
j=1
˜
P
j
⊗
˜
Q
j
(7)
with
˜
P
j
∈ V for all j.
Proof. (i)⇒(ii): By definition of C
min
there are
v
1
, . . . , v
n
∈ C and H
1
, . . . , H
m
∈ Her
t
with all H
i
> 0
and
Q
i
=
n
X
j=1
v
ji
H
j
for all i. Substituting in the expression of ρ (Eq. (6)),
we obtain
ρ =
n
X
j=1
m
X
i=1
v
ji
P
i
!
⊗ H
j
.
From v
j
∈ C = FS
1
(P
1
, . . . , P
m
) it follows that
P
i
v
ji
P
i
> 0 for all j, which proves (ii).
(ii) ⇒ (i): Assume that ρ is of form (7) with all
˜
P
j
,
˜
Q
j
> 0 and
˜
P
j
=
m
X
i=1
v
ji
P
i
for some v
j
= (v
j1
, . . . , v
jm
) ∈ R
m
. Substituting in
the expression of ρ [Eq. (7)] we obtain that
ρ =
m
X
i=1
P
i
⊗
n
X
j=1
v
ji
˜
Q
j
.
Comparing this with expression (6) and using that
P
1
, . . . , P
m
are linearly independent, we obtain
Q
i
=
n
X
j=1
v
ji
˜
Q
j
for all i, i.e.
(Q
1
, . . . , Q
m
) =
n
X
j=1
v
j
⊗
˜
Q
j
∈ C
min
t
.
Remark 12. In Theorem 11, “(ii) ⇒(i)” fails without
the assumption
˜
P
j
∈ V for all j. To see this, consider
the following example taken from [HKM13]. Consider
P
1
=
1 0 0
0 1 0
0 0 1
, P
2
=
0 1 0
1 0 0
0 0 0
, P
3
=
0 0 1
0 0 0
1 0 0
Q
1
=
1 0
0 1
, Q
2
=
1
2
0
0 0
, Q
3
=
0
3
4
3
4
0
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and
ρ :=
3
X
i=1
P
i
⊗ Q
i
One easily checks that ρ > 0, and since all appearing
matrices are symmetric, ρ passes the separability test
of partial transposition [HHHH09]. This test is how-
ever sufficient for separability in the 3 × 2 case, so ρ
is separable.
Now consider the Pauli matrices
σ
z
=
1 0
0 −1
, σ
x
=
0 1
1 0
,
and note that
C = FS
1
(P
1
, P
2
, P
3
) = FS
1
(I
2
, σ
z
, σ
x
) ⊆ R
3
is the circular cone, i.e. the cone defined by the real
numbers a, b, c such that b
2
+ c
2
≤ a
2
, and a ≥ 0.
However, on the other hand,
I
2
⊗ Q
1
+ σ
z
⊗ Q
2
+ σ
3
⊗ Q
3
0,
which implies that (Q
1
, Q
2
, Q
3
) /∈ C
min
2
.
Corollary 13. Let P
1
, . . . , P
m
∈ Her
s
be R-linearly
independent, and set
C := FS
1
(P
1
, . . . , P
m
) ⊆ R
m
.
(i). If FS(P
1
, . . . , P
m
) = C
min
then for any choice of
Q
1
, . . . , Q
m
∈ Her
t
with
ρ =
m
X
i=1
P
i
⊗ Q
i
> 0,
the state ρ is separable.
(ii). If C is a simplex cone, then for any choice of
Q
1
, . . . , Q
m
∈ Her
t
with
ρ =
m
X
i=1
P
i
⊗ Q
i
> 0,
the state ρ is separable. Furthermore, it admits
a separable decomposition with at most m terms,
that is, sep-rank(ρ) ≤ m.
(iii). If m = 2, then for any choice of Q
1
, Q
2
∈ Her
t
with
ρ = P
1
⊗ Q
1
+ P
2
⊗ Q
2
> 0,
the state ρ is separable and admits a separa-
ble decomposition with at most 2 terms, i.e.
sep-rank(ρ) ≤ 2.
Proof. (i): The condition ρ > 0 just means
(Q
1
, . . . , Q
m
) ∈ FS
t
(P
1
, . . . , P
m
) = C
min
t
.
So the result follows from Theorem 11.
(ii): From [FNT16] we know that if C is a simplex
cone, then
C
min
= FS(P
1
, . . . , P
m
)
is automatically true. So the result follows from (i).
Note that the number of terms in the constructed
separable decomposition is m, since in the proof of
Theorem 11 “(i) ⇒ (ii)” we can choose the v
i
as the
extreme rays of C, of which there are m many.
(iii): For C = FS
1
(P
1
, P
2
) ⊆ R
2
there are three
possibilities. First, C might be a simplex cone, so the
claim follows from (ii). Second, C might be a single
ray. In this case one easily checks that ρ is in fact a
product state, so sep-rank(ρ) = 1. Finally C = {0}
implies ρ = 0, so everything is trivial.
4 Bipartite states with operator
Schmidt rank two
In this section we focus on bipartite states which have
operator Schmidt rank two. We will first prove The-
orem 1 (Section 4.1), and then provide a method to
obtain the separable decomposition of such bipartite
states (Section 4.2). The method closely follows the
proof and thereby illustrates our construction. Then
we will put Theorem 1 in the broader context of
decompositions of nonnegative matrices, and derive
some implications for the ranks (Section 4.3). Fi-
nally we will derive further implications for quantum
channels and the entanglement of purification (Sec-
tion 4.4).
4.1 Proof of Theorem 1
In this section we prove Theorem 1. The key miss-
ing element is Lemma 14, which tells us that in the
bipartite case we can force the local matrices to be
Hermititian without increasing the number of terms.
Lemma 14 (Hermiticity in the bipartite case). Let
P
1
, . . . , P
r
∈ M
s
and Q
1
, . . . , Q
r
∈ M
t
be two sets of
linearly independent matrices such that
ρ =
r
X
α=1
P
α
⊗ Q
α
is Hermitian. Then ρ can be expressed as
ρ =
r
X
α=1
A
α
⊗ B
α
where A
α
∈ Her
s
and B
α
∈ Her
t
.
Proof. First note that a Hermitian decomposition
ρ =
P
r
0
α=1
A
α
⊗ B
α
(with each A
α
and B
α
Her-
mitian, and possibly r
0
≥ r) can always be found,
since Her
s
⊗Her
t
= Her
st
. This holds simply because
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Her
s
⊗ Her
t
⊆ Her
st
and the two vector spaces have
the same dimension, namely s
2
t
2
.
To see that r
0
= r, assume that ρ =
P
r
0
α=1
A
α
⊗
B
α
is a minimal Hermitian decomposition, i.e. where
{A
α
}
α
are linearly independent over R, and so are
{B
α
}
α
. But this implies that they are also linearly
independent over C, since consider λ
α
∈ C and the
equation
X
α
λ
α
A
α
= 0.
Summing and subtracting its complex conjugate
transpose of this equation, we see that the real and
imaginary part of each λ
α
needs to vanish. Thus
r = r
0
.
Now we are ready to prove Theorem 1. Let us first
state the theorem again.
Theorem (Theorem 1). Let ρ be a bipartite state 0 6
ρ ∈ M
d
1
⊗ M
d
2
where d
1
, d
2
are arbitrary, such that
ρ =
2
X
α=1
A
α
⊗ B
α
where A
α
∈ M
d
1
and B
α
∈ M
d
2
. Then ρ is separable
and can be written as
ρ =
2
X
α=1
σ
α
⊗ τ
α
where σ
α
∈ PSD
d
1
and τ
α
∈ PSD
d
2
.
Proof. By Lemma 14, ρ can be written with Hermi-
tian local matrices, namely as
ρ =
2
X
α=1
P
α
⊗ Q
α
where P
α
∈ Her
d
1
and Q
α
∈ Her
d
2
. By Corollary 13
(iii), there is a separable decomposition with only two
terms,
ρ =
2
X
α=1
σ
α
⊗ τ
α
.
4.2 Method to obtain the separable decompo-
sition
We now provide an explicit procedure to obtain the
separable decomposition of a bipartite state with op-
erator Schmidt rank two. We remark that another
such procedure was provided in [Car17, Theorem
3.44]. The procedure follows the proof of Theorem 1,
as the latter is constructive. In the following we focus
on presenting the method, rather than on justifying
why it works. At the end of this section, we will apply
this method to an example.
First, the input and output to this method are as
follows:
Input A decomposition of a bipartite positive
semidefinite matrix
ρ =
2
X
α=1
P
α
⊗ Q
α
> 0.
where the P
α
’s and the Q
α
’s are linearly inde-
pendent.
1
Output σ
α
> 0 and τ
α
> 0 such that
ρ =
2
X
α=1
σ
α
⊗ τ
α
. (8)
The method works as follows:
Step 1 Making the matrices Hermitian.
1. Express P
α
= P
0
α
+ iP
1
α
, where P
0
α
and iP
1
α
are the Hermitian and antihermitian part of
P
α
, respectively
2
, and similarly for Q
α
.
2. Find two matrices of the set
{P
0
1
, P
1
1
, P
0
2
, P
1
2
} which are R-linearly
independent, i.e. that are not a multiple of
each other.
3. Express the other two matrices as a real lin-
ear combination of the two linearly indepen-
dent matrices. Substitute this in ρ and ob-
tain an expression of the form
ρ = A ⊗ C + B ⊗ D > 0 (9)
where each matrix is Hermitian.
Step 2 Finding the extreme rays v, u ∈ R
2
:
S(A, B) = {(x, y) ∈ R
2
|xA + yB > 0}
= convex-cone{v, u} ⊆ R
2
.
Denote the ith diagonal element of C, D by C
ii
,
D
ii
, respectively. Note that C
ii
A + D
ii
B > 0 by
construction. There are two cases:
Case 1: There is an i such that C
ii
A + D
ii
B > 0.
Since this is positive definite, there is an
invertible matrix P such that P
†
(C
ii
A +
D
ii
B)P = I. Set
˜
A = P
†
AP,
˜
B = P
†
BP
and let λ
min
, λ
max
be the smallest and
largest eigenvalue of
C
ii
˜
B − D
ii
˜
A.
The extreme rays are
u = −(λ
min
C
ii
+ D
ii
, λ
min
D
ii
− C
ii
),
v = (λ
max
C
ii
+ D
ii
, λ
max
D
ii
− C
ii
).
1
This could be, for example, the operator Schmidt decom-
position of ρ.
2
Namely P
0
α
= (P
α
+ P
†
α
)/2 and P
1
α
= −i(P
α
− P
†
α
)/2.
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Case 2: For all i, C
ii
A+D
ii
B ≯ 0. Then (C
ii
, D
ii
) is
an extremal ray for any i. If there are two i’s
whose corresponding vectors (C
ii
, D
ii
) are
linearly independent, then we have already
found the two extremal rays. If all (C
ii
, D
ii
)
are linearly dependent, then we only have
one extremal ray. In this case, consider
(C
0
ii
, D
0
ii
) = (C
ii
, D
ii
) ± ε(−D
ii
, C
ii
).
For small ε, the new point will be in the in-
terior of S(A, B). The kernel of C
0
ii
A+D
0
ii
B
is then precisely the joint kernel of A and B.
After splitting it off, the new matrices A
0
, B
0
will define the same spectrahedral cone and
C
0
ii
A
0
+ D
0
ii
B
0
> 0. Hence we are in Case 1.
Note that it is very unlikely that Case 2 of Step 2
happens, as the set of all (C, D) ∈ R
2
satisfying
C · A + D · B > 0 but C · A + D · B ≯ 0 is a set
of measure zero. In most cases there will be an i
which provides an interior point.
Step 3 Finding the positive semidefinite matrices
H
1
, H
2
such that A, B
C = u
1
H
1
+ v
1
H
2
,
D = u
2
H
1
+ v
2
H
2
,
where u = (u
1
, u
2
) and v = (v
1
, v
2
). Since u, v
are linearly independent, this system of equations
has a unique solution.
Substituting in (9) we obtain that the positive
semidefinite matrices of (8) are
σ
1
= u
1
A + u
2
B, σ
2
= v
1
A + v
2
B
τ
1
= H
1
, τ
2
= H
2
.
We now illustrate this procedure with a simple ex-
ample, namely a two-qubit state. In this case the neg-
ativity of the partial transpose is a sufficient criterion
for separability. We use the standard conventions for
qubits, where I = |0ih0| + |1ih1|, σ
z
= |0ih0| − |1ih1|,
and σ
x
= |0ih1| + |0ih1|.
Example 15 (Two qubits correlated in σ
x
basis).
Consider the two-qubit state
ρ =
1
2
(I ⊗ I + σ
x
⊗ σ
x
).
This has osr(ρ) = 2 and is thus separable. In this case
the separable decomposition can be found by inspec-
tion, namely
ρ =
1
2
(|+, +ih+, +| + |−, −ih−, −|), (10)
where |±i = (|0i ± |1i)/
√
2, which indeed has
sep-rank(ρ) = 2.
We now apply the method above to this state to
obain the separable decomposition. From now on we
ignore normalization, i.e. we consider 2ρ.
Step 1 The matrices are already Hermitian.
Step 2 The spectrahedron defined by I, σ
x
is
S(I, σ
x
) = {(a, b)|aI + bσ
x
> 0}
The extreme rays are u = (1, −1) and v = (1, 1).
Step 3 The positive semidefinite matrices H
1
, H
2
that satisfy
I = H
1
+ H
2
, σ
x
= −H
1
+ H
2
are H
1
= |−ih−| and H
2
= |+ih+|. Thus
σ
1
= |−ih−|, σ
2
= |+ih+|,
τ
1
= I − σ
x
, τ
2
= I + σ
x
which is indeed the decomposition of (10).
4.3 Context and implications of Theorem 1
In this section we put Theorem 1 in the context of
Ref. [DN19], which presents a relation between de-
compositions of quantum states and decompositions
of nonnegative matrices. Namely, Ref. [DN19] and
[DHN19] show that for bipartite positive semidefinite
matrices which are diagonal in the computational ba-
sis,
ρ =
X
i,j
M
i,j
|i, jihi, j|, (11)
certain decompositions of ρ correspond to decompo-
sitions of the nonnegative matrix
3
M =
X
i,j
M
i,j
|iihj|. (12)
In particular, the operator Schmidt decomposition,
the separable decomposition, and the local purifica-
tion form of ρ correspond to the singular value de-
composition, the nonnegative factorisation [Yan91],
and the positive semidefinite factorisation [FMP
+
12]
of M, respectively. (There are also correspondences
between the translationally invariant (t.i.) versions,
by which the t.i. separable decomposition and the t.i.
local purification form of ρ correspond to the com-
pletely positive factorisation [BDSM15] and the com-
pletely positive semidefinite transposed factorisation
of M [LP15], respectively). Moreover, each factori-
sation has an associated rank, and these correspon-
dences imply that the operator Schmidt rank of ρ
equals the rank of M , the separable rank of ρ equals
the nonnegative rank of M, and the purification rank
of ρ equals the positive semidefinite rank of M. In a
3
That is, entrywise nonnegative. A Hermitian matrix with
nonnegative eigenvalues is called positive semidefinite.
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formula, if ρ is of the form (11) and M of the form
(12), then
osr(ρ) = rank(M)
sep-rank(ρ) = rank
+
(M) (13)
puri-rank(ρ) = psd-rank(M)
Now, it is known that:
Proposition 16 ([FGP
+
15]). Let M be a nonnega-
tive matrix. Then rank(M) = 2 =⇒ rank
+
(M) =
psd-rank(M) = 2.
Theorem 1 can be seen as a generalization of Propo-
sition 16 to the case that ρ is not diagonal in the com-
putational basis. And, as we will see in Section 5.3,
Theorem 2 can be seen as a generalization of Proposi-
tion 16 to the multipartite case. The precise implica-
tions of Theorem 1 for the ranks of bipartite positive
semidefinite matrices are the following.
Corollary 17 (of Theorem 1). Let ρ be a bipartite
positive semidefinite matrix, 0 6 ρ ∈ M
d
1
⊗ M
d
2
where d
1
, d
2
are arbitrary. Then the following are
equivalent:
(i) osr(ρ) = 2.
(ii) hosr(ρ) = 2.
(iii) sep-rank(ρ) = 2.
Moreover, they imply that:
(iv) puri-rank(ρ) = 2 and ρ is separable.
Proof. (i) implies (ii): is shown in Lemma 14.
(ii) implies (i): since osr(ρ) ≤ hosr(ρ) we only have
to see that osr(ρ) = 1 cannot hold. If osr(ρ) = 1
then ρ = A ⊗ B for some matrices A, B, but since
ρ > 0, A and B must both be positive semidefinite
or both negative semidefinite. In either case they are
both Hermitian, which shows that hosr(ρ) = 1, and
contradicts the assumption hosr(ρ) = 2.
(i) implies (iii): is shown in Theorem 1.
(iii) implies (i): From Proposition 7 (iii) we have
that osr(ρ) ≤ sep-rank(ρ). But osr(ρ) = 1 cannot
hold, since this would imply that sep-rank(ρ) = 1 by
Proposition 7 (i), which contradicts the assumption
that sep-rank(ρ) = 2.
(iii) implies (iv): From Proposition 7 (iii) we have
that puri-rank(ρ) ≤ sep-rank(ρ), thus we only have to
see that puri-rank(ρ) = 1 cannot be true. By Propo-
sition 7 (i) this implies that sep-rank(ρ) = 1, which
contradicts the assumption that sep-rank(ρ) = 2.
This leaves the following situation for bipartite pos-
itive semidefinite matrices. Note that we only need to
analyse osr, since osr(ρ) = hosr(ρ) by Lemma 14.
• If osr(ρ) = 1, then ρ is a product state, and
hosr(ρ) = sep-rank(ρ) = puri-rank(ρ) = 1 by
Proposition 7 (i).
• If osr(ρ) = 2 then ρ is a separable state, and
hosr(ρ) = sep-rank(ρ) = puri-rank(ρ) = 2 by
Corollary 17.
• Ref. [GPT13] provides a family of nonnegative
matrices {M
t
∈ R
t×t
≥0
}
t≥3
, for which rank(M
t
) =
3 for all t but psd-rank(M
t
) diverges with t.
Defining the bipartite state
ρ
t
=
t
X
i,j=1
(M
t
)
i,j
|i, jihi, j|,
and using relations (13), this immediately implies
that there is a family {ρ
t
} for which osr(ρ
t
) = 3
for all t but puri-rank(ρ
t
) diverges. By Propo-
sition 7 (iii), this implies that sep-rank(ρ
t
) also
diverges with t.
This is summarised in Table 2, and this situation
is to be compared with the multipartite case, sum-
marised in Table 3.
osr(ρ) = hosr(ρ) Bipartite state ρ
1 Product state
2 Separable state, and separable rank and purification rank are both 2
3 If separable, purification rank can be unbounded,
and thus separable rank can be unbounded too
Table 2: Overview of the properties of bipartite positive semidefinite matrices ρ with small operator Schmidt rank (osr). Recall
that the operator Schmidt rank equals the hermitian operator Schmidt rank (hosr) by Lemma 14.
4.4 Further implications of Theorem 1
We now derive some further implications of Theo-
rem 1, first concerning quantum channels, and then
the entanglement of purification.
First we focus on quantum channels, which are com-
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pletely positive and trace preserving maps. Recall
that every completely positive map E : M
d
2
→ M
d
1
is dual to a positive semidefinite matrix 0 6 ρ ∈
M
d
1
⊗M
d
2
(sometimes called the Choi matrix of E)
via the following relation [Jam72]
ρ =
1
√
d
1
d
2
(E ⊗id)(
d
2
X
j,k=1
|j, jihk, k|) (14)
where id is the identity map. Moreover, E is trace
preserving if and only if the Choi matrix satisfies that
tr
1
(ρ) = I/d
2
, where tr
1
indicates the trace over the
first subsystem, and I is the identity matrix of size
d
2
. A quantum channel E : M
d
2
→ M
d
1
is called
entanglement breaking if (E ⊗ I)(σ) is separable for
all 0 6 σ ∈ M
d
1
⊗ M
d
2
[HSR03]. It is shown in Ref.
[HSR03] that E is entanglement breaking if and only if
its Choi matrix ρ is separable. Note that determining
whether a quantum channel is entanglement breaking
is NP-hard [Gha10].
Corollary 18 (of Theorem 1). Let d
1
, d
2
≥ 2 be arbi-
trary and E : M
d
2
→ M
d
1
be a quantum channel such
that the rank of E (i.e. the dimension of the image) is
2. Then E is entanglement breaking.
Proof. If the dimension of the image of E is 2, then E
can be written as E(X) =
P
2
α=1
P
α
tr(Q
t
α
X), where
t denotes transposition, and P
1
, P
2
are linearly inde-
pendent, as are Q
1
, Q
2
. From (14) it is immediate to
see that the Choi matrix is
ρ =
1
√
d
1
d
2
2
X
α=1
P
α
⊗ Q
α
,
i.e. it has operator Schmidt rank 2. By Corollary 13
(ii), ρ is separable, and thus E is entanglement break-
ing.
Note that the statement of Corollary 18 also holds
if E is not trace preserving, since in that case the Choi
matrix also has operator Schmidt rank 2.
Now we turn to the implications of Theorem 1 for
the correlations in the system. We consider again a
bipartite state, which we now denote by ρ
AB
. The en-
tanglement of purification E
p
(ρ
AB
) [THLD02] is de-
fined as
E
p
(ρ
AB
) = min
ψ
{E(|ψi
A,A
0
,B,B
0
))|tr
A
0
B
0
|ψihψ| = ρ}
where the entropy of entanglement E(|ψi
A,A
0
,B,B
0
) is
defined as
E(|ψi
A,A
0
,B,B
0
) = S
1
(ρ
AA
0
),
where ρ
AA
0
= tr
BB
0
|ψi
AA
0
BB
0
hψ| and S
1
is the von
Neumann entropy, S
1
(ρ) = −tr(ρ log
d
ρ), where ρ has
size d ×d. The entanglement of purification is a mea-
sure of the amount of classical and quantum correla-
tions of the system [THLD02]. It follows from Theo-
rem 1 that if a state has operator Schmidt rank two,
then its entanglement of purification is very small:
Corollary 19 (of Theorem 1). Let 0 6 ρ ∈ M
d
1
⊗
M
d
2
be such that osr(ρ) = 2. Then its entanglement
of purification satisfies E
p
(ρ) ≤ log
d
1
2.
Proof. From Corollary 17 we have that osr(ρ) = 2
implies puri-rank(ρ) = 2. From [DN19] we know that
E
p
(ρ) ≤ log
d
1
(puri-rank(ρ)), from which the result
follows.
5 Multipartite states with Hermitian
operator Schmidt rank two
In this section we focus on multipartite states with
Hermitian operator Schmidt rank two. First we prove
Theorem 2 (Section 5.1), then analyse how to impose
Hermiticity in the multipartite case (Section 5.2), and
finally discuss the context and derive several implica-
tions of this result (Section 5.3).
5.1 Proof of Theorem 2
In this section we prove Theorem 2. The main differ-
ence between the bipartite and the multipartite case
(i.e. Theorem 1 and Theorem 2) is that in the bipartite
case we need not assume that the matrices are Her-
mitian. This is because Lemma 14 allows to enforce
the Hermiticity of the matrices without increasing the
number of terms, i.e. it shows that osr(ρ) = hosr(ρ).
Yet, this need not be true in the multipartite case,
where we can only prove the looser relations between
osr and hosr presented in Section 5.2. In particu-
lar, they do not guarantee that if osr(ρ) = 2 then
hosr(ρ) = 2. Note that, ultimately, we need Hermi-
tian matrices in order to use Corollary 13 (iii).
Let us now state and prove Theorem 2.
Theorem (Theorem 2). Let ρ be a positive semidef-
inite matrix, 0 6 ρ ∈ M
d
1
⊗M
d
2
⊗···⊗ M
d
n
where
d
1
, d
2
, . . . , d
n
are arbitrary, such that
ρ =
2
X
α
1
,...,α
n−1
=1
A
[1]
α
1
⊗ A
[2]
α
1
,α
2
⊗ ··· ⊗ A
[n]
α
n−1
where A
[1]
α
∈ Her
d
1
, A
[2]
α,β
∈ Her
d
2
, . . . , A
[n]
α
∈ Her
d
n
.
Then ρ is separable and can be written as
ρ =
2
X
α
1
,...,α
n−1
=1
σ
[1]
α
1
⊗ σ
[2]
α
1
,α
2
⊗ ··· ⊗ σ
[n]
α
n−1
where σ
[1]
α
∈ PSD
d
1
, σ
[2]
α,β
∈ PSD
d
2
, . . ., σ
[n]
α
∈
PSD
d
n
.
Note that we are assuming that σ
[1]
1
and σ
[1]
2
are
linearly independent, and similarly for the other sites,
since otherwise the sum over the corresponding index
would only need to take one value.
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Proof. We prove the statement by induction on n. In
the bipartite case we have
ρ =
2
X
α=1
A
[1]
α
⊗ A
[2]
α
,
where A
[1]
α
and A
[1]
α
are Hermitian. This is a particular
case of Theorem 1, and thus it follows that
ρ =
2
X
α=1
σ
[1]
α
⊗ σ
[2]
α
,
where σ
[1]
α
and σ
[2]
α
are positive semidefinite. For the
induction step we also need the following property: if
another state admits a representation with the same
A
[1]
α
but different A
[2]
α
, then its separable decomposi-
tion can be chosen with the same σ
[1]
α
, i.e. only σ
[2]
α
is modified. This follows directly from the proof of
Theorem 1.
Now, for the induction step, assume that the state-
ment holds for n sites. We first write the positive
semidefinite matrix ρ on n + 1 sites as
ρ =
2
X
α
n
=1
B
[1...n]
α
n
⊗ A
[n+1]
α
n
(15)
where
B
[1...n]
α
n
=
2
X
α
1
,...,α
n−1
=1
A
[1]
α
1
⊗ A
[2]
α
1
,α
2
⊗ ··· ⊗ A
[n]
α
n−1
,α
n
.
Note that A
[n+1]
1
, A
[n+1]
2
are Hermitian, and so are
B
[1...n]
1
and B
[1...n]
2
. Thus, applying Corollary 13 (iii)
to ρ in Eq. (15), we find that it is separable and can
be written as
ρ =
2
X
α
n
=1
σ
[1...n]
α
n
⊗ σ
[n+1]
α
n
(16)
where σ
[1...n]
α
n
∈ PSD
d
1
·d
2
···d
n
and σ
[n+1]
α
n
∈ PSD
d
n+1
.
Moreover, σ
[1...n]
α
n
is a real linear combination of B
[1...n]
1
and B
[1...n]
2
,
σ
[1...n]
α
n
=
2
X
j=1
λ
α
n
j
B
[1...n]
j
=
2
X
α
1
,...,α
n−1
A
[1]
α
1
⊗ ··· ⊗
2
X
j=1
λ
α
n
j
A
[n]
α
n−1
,j
(17)
Thus, both σ
[1...,n]
1
and σ
[1...,n]
2
are positive semidef-
inite, defined on n sites, and of Hermitian operator
Schmidt rank 2, as Eq. (17) shows. Note also that
σ
[1...,n]
1
and σ
[1...,n]
2
only differ at the n-th site. By
the induction hypothesis, both σ
[1...,n]
α
n
have separa-
ble rank 2 representations, using the same positive
semidefinite matrices at the first n − 1 sites. Explic-
itly,
σ
[1...n]
α
n
=
2
X
α
1
,...,α
n−1
=1
σ
[1]
α
1
⊗ σ
[2]
α
1
,α
2
⊗ ··· ⊗ σ
[n]
α
n−1
,α
n
,
where σ
[1]
α
∈ PSD
d
1
, σ
[2]
α,β
∈ PSD
d
2
. . . Inserting this
expression in ρ of Eq. (16) we obtain a separable de-
composition of ρ with separable rank 2.
Finally, note that if only the A
[n+1]
α
n
in the ini-
tial representation of ρ are changed (Eq. (15)), then
the separable decomposition that we have just con-
structed will only differ at the last site.
5.2 Imposing Hermiticity in the multipartite
case
Here we analyse how to impose Hermiticity of the lo-
cal matrices for multipartite states, and then prove
a modified version of Theorem 2 in which the local
matrices need not be Hermitian but linearly indepen-
dent.
As we have seen in Lemma 14, in the bipartite case
we can impose Hermiticity of the local matrices with-
out increasing the number of terms, that is, the op-
erator Schmidt rank and its Hermitian counterpart
coincide, hosr(ρ) = osr(ρ). In the multipartite case
we can only give the following bound, which we do
not know if it is tight.
Proposition 20 (Hermiticity in the multipartite
case). Let ρ be a Hermitian matrix in M
d
1
⊗ ··· ⊗
M
d
n
, where d
1
, . . . , d
n
are arbitrary. Then
osr(ρ) ≤ hosr(ρ) ≤ 2
n−1
osr(ρ),
except if n = 2, in which case hosr(ρ) = osr(ρ).
Proof. The statement for n = 2 corresponds to
Lemma 14. For general n, we consider the MPDO
form of ρ [Eq. (2)], and express each matrix as a sum
of its Hermitian and anti-Hermitian part, namely as
A
[l]
α
=
1
X
k=0
i
k
A
[l]k
α
for l = 1, n, and
A
[l]
α,β
=
1
X
k=0
i
k
A
[l]k
α,β
for 1 < l < n. Substituting in the MPDO form of ρ
we obtain
ρ =
1
X
k
1
,...,k
n
=0
i
k
D
X
α
1
,...,α
n−1
=1
A
[1]k
1
α
1
⊗A
[2]k
2
α
1
,α
2
⊗···⊗A
[n]k
n
α
n−1
,
where k = k
1
+ ··· + k
n
. Since ρ is Hermitian, all
terms where k is odd must vanish. Thus the above
expression contains 2
n−1
terms, all of which are are
Hermitian. Thus hosr(ρ) ≤ 2
n−1
osr(ρ).
Accepted in Quantum 2019-10-31, click title to verify. Published under CC-BY 4.0. 10
With the help of an additional condition we can
prove an analogous result to the bipartite case.
4
The
following result is presented for finitely many matrices
ρ
i
, because this is needed in the induction step of the
proof. Afterwards, we will prove a corollary for a
single ρ.
Proposition 21. Let
ρ
1
, . . . , ρ
r
∈ M
d
1
⊗ M
d
2
⊗ ··· ⊗ M
d
n
be Hermitian matrices such that
ρ
i
=
k
X
α
1
,...,α
n−1
=1
P
[1]
α
1
⊗ P
[2]
α
1
,α
2
⊗ ··· ⊗ P
[n−1]
α
n−2
,α
n−1
⊗ P
[n]
α
n−1
,i
where at each site the local matrices of ρ
i
are linearly
independent. Then ρ
1
, . . . , ρ
r
can be written as
ρ
i
=
k
X
α
1
,...,α
n−1
=1
A
[1]
α
1
⊗A
[2]
α
1
,α
2
⊗···⊗A
[n−1]
α
n−2
,α
n−1
⊗A
[n]
α
n−1
,i
where all local matrices are Hermitian.
Proof. The proof is by induction on n. The case n = 2
is precisely Lemma 14, where the fact that the Her-
mitian matrices at site one can be chosen the same
for all ρ
i
follows from choosing a Hermitian basis of
span{P
[1]
1
, . . . , P
[1]
k
}.
For a general n, we proceed as in the proof of Theo-
rem 2. It is easy to see that linear independence of the
local matrices at each site is maintained throughout
the construction.
Corollary 22. Let 0 6 ρ ∈ M
d
1
⊗M
d
2
⊗···⊗M
d
n
be positive semidefinite with
ρ =
2
X
α
1
,...,α
n−1
=1
P
[1]
α
1
⊗ P
[2]
α
1
,α
2
⊗ ··· ⊗ P
[n−1]
α
n−2
,α
n−1
⊗ P
[n]
α
n−1
where at each site the local matrices are linearly inde-
pendent. Then ρ is separable with sep-rank(ρ) = 2.
Proof. Clear from Proposition 21 and Theorem 2.
5.3 Context and implications of Theorem 2
As mentioned in Section 4.3, Theorem 2 can be seen
as a generalization of Proposition 16 to the case that
ρ is a general quantum state (i.e. not diagonal in the
computational basis), and it is multipartite. In partic-
ular, Theorem 2 implies the following relations among
ranks for the multipartite case.
Corollary 23 (of Theorem 2). Let ρ be a multipartite
positive semidefinite matrix,
0 6 ρ ∈ M
d
1
⊗ M
d
2
⊗ ··· ⊗ M
d
n
,
where d
1
, d
2
, . . . , d
n
are arbitrary. Then the following
are equivalent:
4
We thank an anonymous referee for suggesting this argu-
ment.
(i) hosr(ρ) = 2.
(ii) sep-rank(ρ) = 2.
Moreover, they imply the following:
(iii) puri-rank(ρ) = 2 and ρ is separable.
Proof. (i) implies (ii): is shown in Theorem 2.
(ii) implies (i): From Proposition 7 we have that
hosr(ρ) ≤ sep-rank(ρ), and hosr(ρ) = 1 implies
sep-rank(ρ) = 1, which contradicts the assumption.
(ii) implies (iii): From Proposition 7 (iii) we have
that puri-rank(ρ) ≤ sep-rank(ρ), and from Proposi-
tion 7 (i) puri-rank(ρ) = 1 implies that sep-rank(ρ) =
1, which contradicts the assumption.
This leaves the following situation for multipar-
tite positive semidefinite matrices of small operator
Schmidt rank (summarised in Table 3):
• If osr(ρ) = 1, then ρ is a product state, and
hosr(ρ) = sep-rank(ρ) = puri-rank(ρ) = 1 by
Proposition 7 (i).
• If osr(ρ) = 2 and hosr(ρ) = 2, then ρ is a
separable state, and osr(ρ) = sep-rank(ρ) =
puri-rank(ρ) = 2 by Corollary 23.
• If osr(ρ) = 2 and hosr(ρ) > 2, we cannot con-
clude anything about the state. (Note that from
Proposition 20 we only know that, if osr(ρ) = 2,
then 2 ≤ hosr(ρ) ≤ 2
n
, where n is the number of
sites, i.e. ρ ∈ M
d
1
⊗ ··· ⊗ M
d
n
).
• Ref. [DSPGC13] provides a family of states
{ρ
n
∈ M
2
⊗ ··· ⊗ M
2
(n times)}
n≥1
which are diagonal in the computational basis
(thus, separable) for which osr(ρ
n
) = 3 for all
n, and puri-rank(ρ
n
) diverges with n. By Propo-
sition 7 (iii), this implies that sep-rank(ρ
n
) also
diverges with n.
In words, the case hosr(ρ) = 2 behaves very simi-
larly to the trivial case hosr(ρ) = 1, as one can deter-
mine the kind of state ρ is, and one can upper bound
the amount correlations. On the other hand, the sep-
aration between osr(ρ) and puri-rank(ρ) appears for
a very small value of osr, namely osr(ρ) = 3.
Accepted in Quantum 2019-10-31, click title to verify. Published under CC-BY 4.0. 11
osr(ρ) hosr(ρ) Multipartite state ρ
1 1 Product state
2 2 Separable state, and separable rank and purification rank are both 2
2 > 2 ?
3 ≥ 3 If separable, purification rank can be unbounded,
and thus separable rank can be unbounded too
Table 3: Overview of the properties of multipartite positive semidefinite matrices ρ with small operator Schmidt rank (osr)
and hermitian operator Schmidt rank (hosr).
6 Conclusions & Outlook
We have presented two main results. First, we have
shown that any bipartite positive semidefinite matrix
of operator Schmidt rank two is separable, and admits
a separable decomposition with two positive semidef-
inite matrices per site (Theorem 1). We have also
provided a step-by-step method to obtain this sep-
arable decomposition (Section 4.2), and have drawn
some consequences of this result concerning the re-
lation among the ranks (Corollary 17, see also the
summary of Table 2), for quantum channels (Corol-
lary 18), and for the amount of correlations of the
state (Corollary 19). While this result was already
known [Car14], we have provided a new proof thereof.
In particular, we have leveraged a relation between
minimal operator systems and separable states (The-
orem 11) and its implications (Corollary 13).
Second, we have shown that any multipartite
positive semidefinite matrix of Hermitian operator
Schmidt rank two is separable, and it has separable
rank two (Theorem 2). We have drawn consequences
of this result for the ranks (Corollary 23, see also the
summary of Table 3).
This work leaves a number of interesting open ques-
tions. A first question concerns the generalisation of
these results to the case of operator Schmidt rank
three. Bipartite states ρ ∈ M
d
⊗ M
d
0
with opera-
tor Schmidt rank three must be separable if d = 2
or d
0
= 2 [Car14], but this is no longer true in the
case that d > 2 and d
0
> 2. Further results for these
states are shown in [CMHW18, Corollary III.5]. It
would be interesting to analyse the consequences of
these results for the multipartite case, and compare
them to Table 3.
A second question is to clarify the relation between
the operator Schmidt rank and the hermitian oper-
ator Schmidt rank in the multipartite case. In par-
ticular, it would be interesting to find out whether
Proposition 20 is tight or can be improved, and to
find direct relations between the hosr and the separa-
ble rank and/or the purification rank.
On a broader perspective, it would be worth study-
ing which of these results extends to the translation-
ally invariant (t.i.) case. Ref. [DN19] presents the t.i.
versions of the MPDO form, the separable form, and
the local purification form. For bipartite states which
are diagonal in the computational basis, the ranks as-
sociated to each of these decompositions correspond
to well-studied ranks of nonnegative matrices (such
as the cp rank and the cpsd rank), as mentioned in
Section 4.3. Exploiting these relations may allow us
to generalize results about ranks in that case, too,
and gain further insight into the decompositions of
t.i. quantum states.
More generally, this paper illustrates how fruitful
the connection between quantum states and (free)
spectrahedra can be. The latter provide a novel set of
techniques that can shed further light into the decom-
positions of quantum states. This is being explored
in [Net19].
Acknowledgements. We thank an anonymous ref-
eree for suggesting Proposition 21 and bringing Refs.
[Car17, Gie18] to our attention. We thank A. M¨uller–
Hermes, M. Studi´nski and N. Johnston for bringing
Refs. [Car14, Car15] to our attention. T. D. and T.
N. are supported by the Austrian Science Fund FWF
through project P 29496-N35.
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