Separability Properties of Three-mode Gaussian States

Abstract

We derive a necessary and sufficient condition for the separability of tripartite three mode Gaussian states, that is easy to check for any such state. We give a classification of the separability properties of those systems and show how to determine for any state to which class it belongs. We show that there exist genuinely tripartite bound entangled states and point out how to construct and prepare such states. Comment: 11 pages, 2 figures (.eps); results extended to 1x1xn modes; references updated; version to appear in Phys. Rev. A

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arXiv:quant-ph/0103137v2 29 Aug 2001
Separability Properties of Three-mode Gaussian States
G. Giedke(1), B. Kraus(1), M. Lewenstein(2), and J. I. Cirac(1)
(1) Institut f¨ ur Theoretische Physik, Universit¨ at Innsbruck, A-6020 Innsbruck, Austria
(2) Institut f¨ ur Theoretische Physik, Universit¨ at Hannover, 30163 Hannover, Germany
We derive a necessary and sufficient condition for the separability of tripartite three mode Gaus-
sian states, that is easy to check for any such state. We give a classification of the separability
properties of those systems and show how to determine for any state to which class it belongs. We
show that there exist genuinely tripartite bound entangled states and point out how to construct
and prepare such states.
PACS numbers: PACS numbers: 03.67.Hk, 03.65.Ta
I.INTRODUCTION
Entanglement of composite quantum systems is central
to both the peculiarities and promises of quantum infor-
mation. Consequently, the study of entanglement of bi-
and multipartite systems has been the focus of research in
quantum information theory. While pure state entangle-
ment is fairly well understood, there are still many open
questions related to the general case of mixed states. The
furthest progress has been made in the study of systems
of two qubits: it has been shown that a state of two
qubits is separable if and only if its partial transpose is
positive (PPT-property) [1] and a closed expression for
the entanglement of formation was derived [2]. Moreover,
it was shown [3] that all entangled states of two qubits
can be distilled into maximally entangled pure states by
local operations. This property of distillability is of great
practical importance, since only the distillable states are
useful for certain applications such as long-distance quan-
tum communication, quantum teleportation or cryptog-
raphy [4].
In higher dimensions much less is known: the PPT-
property is no longer sufficient for separability as proved
by the existence of PPT entangled states in
systems [5]. These states were later shown to be bound
entangled [6]: even if two parties (Alice and Bob) share
an arbitrarily large supply of such states, they cannot
transform (“distill”) it into even a single pure entangled
state by local quantum operations and classical commu-
nication. Meanwhile, a number of additional necessary
or sufficient conditions for inseparability have been found
for finite dimensional bipartite systems, which use prop-
erties of the range and kernel of the density matrix ρ and
its partial transpose ρTAto establish separability ([7] and
references therein).
When going from two to more parties, current knowl-
edge is even more limited. Pure multipartite entangle-
ment was first considered in [8]. A classification of N-
partite mixed states according to their separability prop-
erties has been given [9]. But even for three qubits there
is currently no general way to decide to which of these
classes a given state belongs [10].
entanglement [11] and entanglement distillation [12] for
multi-party systems have been obtained.
?2⊗
?4
Results on bound
Recently increasing attention was paid to infinite di-
mensional systems, the so-called continuous quantum
variables (CV), in particular since the experimental re-
alization of CV quantum teleportation [13, 14]. Quan-
tum information with CV in general is mainly con-
cerned with the family of Gaussian states, since these
comprise essentially all the experimentally realizable CV
states.A practical advantage of CV systems is the
relative ease with which entangled states can be gen-
erated in the lab [14, 15].
ity and distillability of Gaussian states were reported in
[16, 17, 18, 19, 20, 21, 22]. One finds striking similari-
ties between the situations of two qubits and two one–
mode CV systems in a Gaussian state: PPT is necessary
and sufficient for separability [17, 18], and all inseparable
states are distillable [19]. Generalizing the methods re-
viewed in [7] it was shown that for more than two modes
at either side PPT entangled states exist [20]. In [21] a
computable measure of entanglement for bipartite Gaus-
sian states was derived.
First results on separabil-
The study of CV multipartite entanglement was initi-
ated in [23, 24], where a scheme was suggested to create
pure CV N-party entanglement using squeezed light and
N−1 beamsplitters. In fact, this discussion indicates that
tripartite entanglement has already been created (though
not investigated or detected) in the CV quantum telepor-
tation experiment [14].
In this paper we provide a complete classification of
tri-mode entanglement (according to the scheme [9]) and
obtain – in contrast to the finite dimensional case – a sim-
ple, directly computable criterion that allows to deter-
mine to which class a given state belongs. We show that
none of these classes are empty and in particular provide
examples of genuine tripartite bound entangled states,
i.e. states of three modes A, B, and C that are separable
whenever two parties are grouped together but cannot be
written as a mixture of tripartite product states.
Before we can derive our results we need to introduce
some notation and collect a number of useful facts about
our main object of study: Gaussian states.

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II.GAUSSIAN STATES
In quantum optics and in other scenarios described by
continuous quantum variables, not all states on the infi-
nite dimensional Hilbert space are equally accessible in
current experiments. In fact, the set of Gaussian states
comprises essentially all genuinely CV states that can
currently be prepared in the lab. This, and the mathe-
matical simplicity of these states are the reasons why CV
quantum information has so far considered almost exclu-
sively Gaussian states, as will the present paper. This
section summarizes results on Gaussian states that we
need in the following and introduces some notation.
We consider systems composed of n distinguishable in-
finite dimensional subsystems, each with Hilbert space
?0 − ?
which plays an important role in the following calcula-
tions [25].
For such systems, it is convenient to describe the state
ρ by its characteristic function
H = L2( ?). These could be implemented quantum op-
tically by different modes of the electromagnetic field,
hence each of these subsystems will be referred to as a
“mode”. To each mode belong the two canonical ob-
servables Xk,Pk,k = 1,...,n with commutation relation
[Xk,Pk] = i. Defining Rk= Xk,Rn+k= Pk these rela-
tion are summarized as [Rk,Rl] = −iJkl, using the anti-
symmetric 2n × 2n matrix
J =
?
0
?
.(1)
χ(x) = tr[ρD(x)].(2)
Here x = (q,p), q,p ∈
?nis a real vector, and
D(x) = e−i?
k(qkXk+pkPk).(3)
The characteristic function contains all the information
about the state of the system, that is, one can construct
ρ knowing χ. Gaussian states are exactly those for which
χ is a Gaussian function of the phase space coordinates
x [26],
χ(x) = e−1
4xTγx−idTx,(4)
where γ is a real, symmetric, strictly positive matrix, the
correlation matrix (CM), and d ∈
the displacement. Note that both γ and d are directly
measurable quantities, their elements γkland dkare re-
lated to the expectation values and variances of the oper-
ators Rk. A Gaussian state is completely determined by
γ and d. Note that the displacement of a (known) state
can always be adjusted to d = 0 by a sequence of unitaries
applied to individual modes. This implies that d is irrel-
evant for the study of nonlocal properties. Therefore we
will occasionally say, e.g., that “a CM is separable” when
the Gaussian state with this CM is separable. Also, from
now on in this paper “state” will always mean “Gaussian
state” (unless stated otherwise).
?2nis a real vector,
Not all real, symmetric, positive matrices γ correspond
to the CM of a physical state. There are a number of
equivalent ways to characterize physical CMs, which will
all be useful in the following. We collect them in
Lemma 1 (Correlation Matrices)
For a real, symmetric 2n×2n matrix γ > 0 the following
statements are equivalent:
γ is the CM of a physical state, (5a)
γ + Jγ−1J ≥ 0,(5b)
γ − iJ ≥ 0,(5c)
γ = ST(D ⊕ D)S,(5d)
for S symplectic [27] and D ≥
? diagonal [28].
Proof:
(5a) ⇔ (5d) see [29, Prop. 4.22].
(5a) ⇔ (5b) see [26]; (5a) ⇔ (5c) see [20];
A CM corresponds to a pure state if and only if (iff)
D =
(5d) that for pure states Ineq. (5b) becomes an equality
and dim[ker(γ − iJ)] = n. It is clear from Eq. (5d) that
for every CM γ there exists a pure CM γ0 such that
γ0 ≤ γ. This will allow us to restrict many proofs to
pure CMs only. Note that for a pure 2n × 2n CM γ it
holds that trγ ≥ 2n.
A very important transformation for the study of en-
tanglement is partial transposition [1]. Transposition is
an example of a positive but not completely positive map
and therefore may reveal entanglement when applied to
part of an entangled system. On phase space transposi-
tion corresponds to the transformation that changes the
sign of all the p coordinates (q,p) ?→ Λ(q,p) = (q,−p)
[18] and leaves the q’s unchanged. For γ and d this means
(γ,d) ?→ (ΛγΛ,Λd). Using this, the NPT-criterion for in-
separability [1] translates very nicely to Gaussian states.
Consider a bipartite system consisting of m modes on
Alice’s side and n modes on Bob’s (m × n-system in the
following). Let γ be the CM of a Gaussian m × n-state
and denote by ΛA= Λ ⊕
A’s system only. Then we have the following criterion for
inseparability:
?, i.e. iff detγ = 1 (e.g. [26]). It is easy to see from
? the partial transposition in
Theorem 1 (NPT criterion)
Let γ be the CM of a 1×n system, then γ corresponds to
a inseparable state if and only if ΛAγΛAis not a physical
CM, i.e. if and only if
ΛAγΛA?≥ iJ.(6)
We say that γ “is NPT” if (6) holds.

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Proof:
case.
See [18] for N = 1 and [20] for the general
Occasionally it is convenient to apply the orthogonal
operation ΛAto the right hand side of Ineq. (6) and write
˜JA≡ ΛAJΛA.
For states of at least two modes at both sides Condi-
tion (6) is still sufficient for inseparability, but no longer
necessary as shown by Werner and Wolf, who have con-
sidered a family of 2 × 2 entangled states with positive
partial transpose [20]. In the same paper, it was shown
that
Theorem 2 (Separability of Gaussian States)
A state with CM γ is separable iff there exist CMs γA,γB
such that
γ ≥ γA⊕ γB.(7)
It is observed in [20] that if Ineq. (7) can be fulfilled,
then the state with CM γ can be obtained by local op-
erations and classical communication from the product
state with CM γp = γA⊕ γB, namely by mixing the
states (γp,d) with the d’s distributed according to the
Gaussian distribution ∝ exp?−dT(γ − γp)−1d?.
cient condition for separability, it is not a practical crite-
rion, since to use it, we have to prove the existence or non-
existence of CMs γA,γB. Instead, a criterion would allow
to directly calculate from γ whether the corresponding
state is separable or not. Theorem 2 and its extension
to the 3-party situation are the starting point for the
derivation of such a criterion for the case of three-mode
three-party states in the following main section of this
paper.
Note that while Theorem 2 gives a necessary and suffi-
III. TRI-MODE ENTANGLEMENT
When systems that are composed of N > 2 parties are
considered, there are many “types” of entanglement due
to the many ways in which the different subsystems may
be entangled with each other. We will use the scheme in-
troduced in [9], to classify three-mode tripartite Gaussian
states. The important point is that from the extension of
Theorem 2 we can derive a simple criterion that allows to
determine which class a given state belongs to. This is in
contrast to the situation for three qubits, where up until
now no such criterion is known. In particular, we show
that none of these classes are empty and we provide an
example of a genuine tripartite bound entangled state,
i.e. a state of three modes A, B, and C that is separa-
ble whenever two parties are grouped together but can-
not be written as a mixture of tripartite product states
and therefore cannot be prepared by local operations and
classical communication of three separate parties.
A.Classification
The scheme of [9] considers all possible ways to group
the N parties into m ≤ N subsets, which are then them-
selves considered each as a single party. Now, it has to
be determined whether the resulting m-party state can
be written as a mixture of m-party product states. The
complete record of the m-separability of all these states
then characterizes the entanglement of the N-party state.
For tripartite systems, we need to consider four
cases, namely the three bipartite cases in which AB,
AC, or BC are grouped together, respectively, and the
tripartite case in which all A, B, and C are separate.
We formulate a simple extension to Theorem 2 to
characterize mixtures of tripartite product states
Theorem 2’ (Three-party Separability)
A Gaussian three-party state with CM γ can be written
as a mixture of tripartite product states iff there exist
one-mode correlation matrices γA,γB,γC such that
γ − γA⊕ γB⊕ γC≥ 0.(8)
Such a state will be called fully separable.
Proof:
The proof is in complete analogy with that of
Theorem 7 in [20] and is therefore omitted here.
A state for which there are a one-mode CM γA and
a two-mode CM γBC such that γ − γA⊕ γBC ≥ 0 is
called A−BC biseparable (and similarly for the two other
bipartite groupings). In total, we have the following five
different entanglement classes:
Class 1 Fully inseparable states are those which are not
separable for any grouping of the parties.
Class 2 1-mode biseparable states are those which are
separable if two of the parties are grouped together,
but inseparable with respect to the other groupings.
Class 3 2-mode biseparable states are separable with re-
spect to two of the three bipartite splits but insep-
arable with respect to the third.
Class 4 3-mode biseparable states separable with respect
to all three bipartite splits but cannot be written
as a mixture of tripartite product states.
Class 5 The fully separable states can be written as a
mixture of tripartite product states.
Examples for Class 1 (the GHZ-like states of [24]),
Class 2 (two-mode squeezed vacuum in the first two and
the vacuum in the third mode), and Class 5 (vacuum
state in all three modes) are readily given; we will pro-
vide examples for Classes 3 and 4 in Subsection IV below.
How can we determine to which Class a given state
with CM γ belongs? States belonging to Classes 1, 2,
or 3 can be readily identified using the NPT-criterion
(Theorem 1). Denoting the partially transposed CM by
˜ γx= ΛxγΛx,x = A,B,C, we have the following equiva-
lences:

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Lemma 2 (Classification)
˜ γA?≥ iJ, ˜ γB?≥ iJ, ˜ γC?≥ iJ ⇔ Class 1
(∗)˜ γA?≥ iJ, ˜ γB?≥ iJ, ˜ γC≥ iJ ⇔ Class 2
(∗)˜ γA?≥ iJ, ˜ γB≥ iJ, ˜ γC≥ iJ ⇔ Class 3
˜ γA≥ iJ, ˜ γB≥ iJ, ˜ γC≥ iJ ⇔ Class 4 or 5, (12)
where the (∗) reminds us to consider all permutations of
the indices A, B, and C.
(9)
(10)
(11)
The proof follows directly from the definitions of the dif-
ferent classes and Theorem 1.
What is still missing is an easy way to distinguish be-
tween Class 4 and Class 5. Thus to complete the classifi-
cation we now provide a criterion to determine whether a
CM γ satisfying Ineqs. (12) is fully separable or 3-mode
biseparable, that is we have to decide whether there exist
one-mode CMs γA,γB,γC such that (8) holds, in which
case γ is fully separable. In the next subsection we will
describe a small set consisting of no more than nine CMs
among which γAis necessarily found if the state is sepa-
rable.
B.Criterion for Full Separability
This subsection contains the main result of the pa-
per: a separability criterion for PPT 1 × 1 × 1 Gaussian
states, i.e. states whose CM fulfills Ineqs. (12). We start
from Theorem 2’ and obtain in several steps a simple,
directly computable necessary and sufficient condition.
The reader mainly interested in this result may go di-
rectly to Theorem 3, from where she will be guided to
the necessary definitions and Lemmas.
Since the separability condition in Theorem 2’ is for-
mulated in terms of the positivity of certain matrices the
following lemma will be very useful throughout the pa-
per. We consider a self-adjoint (n+m)×(n+m) matrix
M that we write in block form as
?
where A,B,C are n × n,m × m, and n × m matrices,
respectively.
M =
A C
C†B
?
,(13)
Lemma 3 (Positivity of self-adjoint matrices)
A self-adjoint matrix M as in (13) with A ≥ 0,B ≥ 0 is
positive if and only if for all ǫ > 0
A − C
1
B + ǫ ?C†≥ 0, (14)
or, equivalently, if and only if
kerB ⊆ kerC(15a)
and
A − C1
BC†≥ 0, (15b)
where B−1is understood in the sense of a pseudoinverse
(inversion on the range).
Proof: The only difficulty in the proof arises if kerB ?=
0. Therefore we consider the matrices Mǫ, where B in
(13) is replaced by Bǫ= B +ǫ ? (ǫ > 0), which avoid this
problem and which are positive ∀ǫ > 0 iff M ≥ 0. In a
second simplifying step we note that Mǫ≥ 0 ∀ǫ > 0 iff
M′
ǫǫ
Now direct calculation shows the claim: we can write a
general f ⊕g as f ⊕
thogonal to the range of (B−1/2
ǫ
g) = f†(A−CB−1
which is clearly positive, if (14) holds. With the choice
h⊥= 0 and h = −f it is seen that (14) is also necessary.
That the second condition is equivalent is seen as fol-
lows: If Ineq. (14) holds ∀ǫ > 0 there cannot be vec-
tor ξ ∈ kerB and ξ ?∈ kerC since for such a ξ we have
ξT?
and if (15a) holds then (14) converges to (15b). Con-
versely, if (15a) holds, then CB−1C†is well-defined and
Ineq. (15b) implies it ∀ǫ > 0.
As mentioned above, in this section we exclusively con-
sider three-mode CMs γ that satisfy Ineqs. (12).
write γ in the form of Eq. (13) as
ǫ= ( ? ⊕ B−1/2
)M( ? ⊕ B−1/2
?
) ≥ 0.
(B−1/2
ǫ
C†)h + h⊥
?
, where h⊥is or-
C†). Then (f⊕g)†M′
ǫ C†(f+h)+h†
ǫ(f⊕
⊥h⊥,
ǫ C†)f+(f+h)†CB−1
A − C
1
B+ǫ ?C†?
ξ < 0 for sufficiently small ǫ > 0,
We
γ =
?
A
CTB
C
?
, (16)
where A is a 2 × 2 matrix, whereas B is a 4 × 4 matrix.
We observe that Ineqs. (12) impose some conditions on
γ that will be useful later on:
Observation 1 Let γ satisfy Ineqs. (12), then
γ ≥


σAiJ
0
0
00
0σBiJ
0σCiJ

, (17)
where σx∈ {0,±1},∀x = A,B,C.
Proof: Ineqs. (12) say that γ ±iJ ≥ 0 and γ ±i˜Jx≥ 0
∀x. By adding these positive matrices all combinations
of σxcan be obtained.
From this it follows:
Observation 2 For a PPT CM γ as in Eq. (16)
ker(B + iJ),ker(B + i˜J) ⊆ kerC,
where˜J = J ⊕(−J) is the partially transposed J for two
modes.
(18)
Proof:
ate consequence of Lemma 3 applied to the matrices
γ − 0 ⊕ iJ ⊕ (±iJ), which are positive by Obs. 1.
Then the matrices
Cond. (18) on the kernels is an immedi-
˜ N ≡ A − C
1
B − i˜JCT,
1
B − iJCT
(19a)
N ≡ A − C
(19b)
are well-defined and

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Observation 3 It holds that both
trN,tr˜ N > 0, (20)
Proof: Cond. (20) is true since, again by Lemma 3 and
Obs. 1, both N and˜ N are positive and N±iJ,˜N±iJ ≥ 0.
This implies that N,˜ N cannot be zero, which is the only
positive matrix with vanishing trace. Therefore trN,tr˜ N
are strictly positive.
The remainder of this section leads in several steps to
the separability criterion. First, we simplify the condition
(8) by reducing it to a condition which involves only one
one-mode CM γA.
Lemma 4 A PPT 3-mode CM γ is fully separable if and
only if there exists a one-mode CM γAsuch that
˜ N ≥ γA,
N ≥ γA,
(21a)
(21b)
where N,˜ N were defined in Eqs. (19).
of generality we require γA to be a pure state CM, i.e.
detγA= 1.
Without loss
Proof:
alent to the existence of one-mode CMs γA,γB,γC≥ iJ
such that γ−γA⊕γB⊕γC≥ 0. Let γxstand for γA,B,C.
By Lemma 3 this is equivalent to ∃γxsuch that Xǫ≡
B − CT
But iff there exist such γxthen (Lemma 3) the inequality
also holds for ǫ = 0 and the kernels fulfill (15a). This
is true iff the matrix X ≡ X′
separable state, i.e. (Theorem 1) iff X′≥ i˜J,iJ. Using
B ≥ i˜J,iJ [which holds since γ fulfills Ineqs. (12)] we
obtain that γ is separable iff there exists γA≥ iJ such
that
By Theorem 2’ full separability of γ is equiv-
1
Aǫ−γAC ≥ γB⊕ γC, ∀ǫ > 0, where Aǫ≡ A + ǫ ?.
0is a CM belonging to a
?A − γA C
CT
B′
k
?
2= B − i˜J. Since Condition
≥ 0, k = 1,2,(22)
where B′
(15a) holds, this is (Lemma 3) equivalent to Ineqs. (21).
That we can always choose detγA = 1 follows directly
from Eq. (5d) and the remark after Lemma 1.
While we can always find a γA fulfilling Ineq. (21b),
since γ belongs to a PPT state (and there exists a two-
mode CM γBC≥ iJ such that γA⊕ γBCis smaller than
γ), it may well happen that Ineq. (21a) cannot be satis-
fied at all, or that it is impossible to have both Ineqs. (21)
fulfilled for one γAsimultaneously. Note that due to In-
eqs. (12), N and˜ N as above are always positive. From
Ineqs. (21) we observe that
1= B − iJ and B′
Observation 4 for the CM γ of a separable state it is
necessary to have
trN,tr˜ N ≥ 2,
detN,det˜ N > 0,
(23a)
(23b)
where γ as in Eq. (16) and N,˜ N as in Eqs. (19).
Proof: A self-adjoint 2×2 matrix is positive iff its trace
and determinant are positive. Since the trace of the RHS
of both Ineqs. (21) is ≥ 2 [remark after Lemma 1] the
same is necessary for the LHS. Also, since detγA = 1,
which implies that γA has full rank, any matrix ≥ γA
must also have full rank [31] and thus a strictly positive
determinant.
For a self-adjoint positive 2 × 2 matrix
?
we show
R =
a b
b∗c
?
,(24)
Lemma 5 There exists a CM γA≤ R if and only if there
exist (y,z) ∈
?2such that
trR ≥ 2
?y
?
1 + y2+ z2, (25a)
detR + 1 + LT
z
?
≥ trR
?
1 + y2+ z2, (25b)
where
L = (a − c,2Reb).(26)
Proof: As noted in Lemma 4 we need only look for γA
with detγA= 1. We parameterize
γA=
?x + yz
zx − y
?
, (27)
with real parameters x,y,z and x2= 1 + y2+ z2for
purity. This is a CM iff γA− iJ ≥ 0 (Lemma 1), that
is iff trγA= 2x ≥ 0 (where we use that positivity of the
a 2 × 2 matrix is equivalent to the positivity of its trace
and determinant and det(γA−iJ) = 0 by construction).
By the same argument, R − γA ≥ 0 leads to the two
conditions (25).
The inequalities (25) have a simple geometrical inter-
pretation that will be useful for the proof of the promised
criterion: Ineq. (25a) restricts (y,z) to a circular disk
C′of radius
Ineq. (25b) describes a (potentially degenerate) ellipse
E (see Fig. 2), whose elements are calculated below, and
the existence of a joint solution to Ineqs. (25) is therefore
equivalent to a nonempty intersection of C′and E.
Applying this now to the matrices (19) we find that
in order to simultaneously satisfy both conditions in
Lemma 4, the intersection between the two ellipses E,˜E
and the smaller of the two concentric circles C′,˜C′(which
we denote in the following by C) must be nonempty. This
condition leads to three inequalities in the coefficients
of the matrices˜ N,N which can be satisfied simultane-
ously if and only if the PPT trimode state is separable.
Thus we can reformulate the condition for separability
(Lemma 4) as follows
?(trR)2/4 − 1 around the origin, while
Lemma 6 (Reformulated Separability Condition)
A three-mode state with CM γ satisfying Ineqs. (12) is