a r X i v : q u a n t  p h / 0 5 1 0 0 5 2 v 2 6 M a r 2 0 0 7
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CHAPTER 1Bipartite and Multipartite Entanglement of Gaussian States
Gerardo Adesso and Fabrizio Illuminati
Dipartimento di Fisica “E. R. Caianiello”, Universit`a di Salerno;CNRCoherentia, Gruppo di Salerno; and INFN Sezione di NapoliGruppoCollegato di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy Email: gerardo@sa.infn.it, illuminati@sa.infn.it
In this chapter we review the characterization of entanglement in Gaussian states of continuous variable systems. For twomode Gaussian states,we discuss how their bipartite entanglement can be accurately quantiﬁed in terms of the global and local amounts of mixedness, and efﬁciently estimated by direct measurements of the associated purities.For multimode Gaussian states endowed with local symmetry with respect to a given bipartition, we show how the multimode block entanglement can be completely and reversibly localized onto a single pairof modes by local, unitary operations. We then analyze the distributionof entanglement among multiple parties in multimode Gaussian states.We introduce the continuousvariable tangle to quantify entanglementsharing in Gaussian states and we prove that it satisﬁes the CoﬀmanKunduWootters monogamy inequality. Nevertheless, we show that pure,symmetric three–mode Gaussian states, at variance with their discretevariable counterparts, allow a promiscuous sharing of quantum correlations, exhibiting both maximum tripartite residual entanglement andmaximum couplewise entanglement between any pair of modes. Finally,we investigate the connection between multipartite entanglement and theoptimal ﬁdelity in a continuousvariable quantum teleportation network.We show how the ﬁdelity can be maximized in terms of the best preparation of the shared entangled resources and, viceversa, that this optimalﬁdelity provides a clearcut operational interpretation of several measuresof bipartite and multipartite entanglement, including the entanglementof formation, the localizable entanglement, and the continuousvariabletangle.
1
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G. Adesso and F. Illuminati
1. Introduction
One of the main challenges in fundamental quantum theory as well asin quantum information and computation sciences lies in the characterization and quantiﬁcation of bipartite entanglement for mixed states, and inthe deﬁnition and interpretation of multipartite entanglement both for purestates and in the presence of mixedness. While important insights have beengained on these issues in the context of qubit systems, a less satisfactoryunderstanding has been achieved until recent times on higherdimensionalsystems, as the structure of entangled states in Hilbert spaces of high dimensionality exhibits a formidable degree of complexity. However, and quiteremarkably, in inﬁnitedimensional Hilbert spaces of continuousvariablesystems, ongoing and coordinated eﬀorts by diﬀerent research groups haveled to important progresses in the understanding of the entanglement properties of a restricted class of states, the socalled Gaussian states. Thesestates, besides being of great importance both from a fundamental pointof view and in practical applications, share peculiar features that maketheir structural properties amenable to accurate and detailed theoreticalanalysis. It is the aim of this chapter to review some of the most recentresults on the characterization and quantiﬁcation of bipartite and multipartite entanglement in Gaussian states of continuous variable systems,their relationships with standard measures of purity and mixedness, andtheir operational interpretations in practical applications such as quantumcommunication, information transfer, and quantum teleportation.
2. Gaussian States of Continuous Variable Systems
We consider a continuous variable (CV) system consisting of
N
canonical bosonic modes, associated to an inﬁnitedimensional Hilbert space
H
and described by the vector ˆ
X
=
{
ˆ
x
1
,
ˆ
p
1
,...,
ˆ
x
N
,
ˆ
p
N
}
of the ﬁeld quadrature (“position” and “momentum”) operators. The quadrature phase operators are connected to the annihilation ˆ
a
i
and creation ˆ
a
†
i
operators of each mode, by the relations ˆ
x
i
= (ˆ
a
i
+ ˆ
a
†
i
) and ˆ
p
i
= (ˆ
a
i
−
ˆ
a
†
i
)
/i
. Thecanonical commutation relations for the ˆ
X
i
’s can be expressed in matrix form: [ ˆ
X
i
,
ˆ
X
j
] = 2
i
Ω
ij
, with the symplectic form Ω =
⊕
ni
=1
ω
and
ω
=
δ
ij
−
1
−
δ
ij
+1
, i,j
= 1
,
2.Quantum states of paramount importance in CV systems are the socalled Gaussian states,
i.e.
states with Gaussian characteristic functionsand quasi–probability distributions1. The interest in this special class of states (important examples include vacua, coherent, squeezed, thermal, andsqueezedthermal states of the electromagnetic ﬁeld) stems from the feasi
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Bipartite and Multipartite Entanglement of Gaussian States
3
bility to produce and control them with linear optical elements, and fromthe increasing number of eﬃcient proposals and successful experimental implementations of CV quantum information and communication processesinvolving multimode Gaussian states (see Ref. 2 for recent reviews). Bydeﬁnition, a Gaussian state is completely characterized by ﬁrst and second moments of the canonical operators. When addressing physical properties invariant under local unitary transformations, such as mixedness andentanglement, one can neglect ﬁrst moments and completely characterizeGaussian states by the 2
N
×
2
N
real covariance matrix (CM)
σ
, whose entries are
σ
ij
= 1
/
2
{
ˆ
X
i
,
ˆ
X
j
}−
ˆ
X
i
ˆ
X
j
. Throughout this chapter,
σ
willbe used indiﬀerently to indicate the CM of a Gaussian state or the stateitself. A real, symmetric matrix
σ
must fulﬁll the RobertsonSchr¨odingeruncertainty relation3
σ
+
i
Ω
≥
0
,
(1)to be a
bona ﬁde
CM of a physical state. Symplectic operations (
i.e.
belonging to the group
Sp
(2
N,
R
)
=
{
S
∈
SL
(2
N,
R
) :
S
T
Ω
S
= Ω
}
) actingby congruence on CMs in phase space, amount to unitary operations ondensity matrices in Hilbert space. In phase space, any
N
mode Gaussianstate can be transformed by symplectic operations in its Williamson diagonal form4
ν
, such that
σ
=
S
T
ν
S
, with
ν
= diag
{
ν
1
,ν
1
,...ν
N
,ν
N
}
.The set Σ =
{
ν
i
}
of the positivedeﬁned eigenvalues of

i
Ω
σ

constitutesthe symplectic spectrum of
σ
and its elements, the socalled symplecticeigenvalues, must fulﬁll the conditions
ν
i
≥
1, following from Eq. (1) andensuring positivity of the density matrix associated to
σ
. We remark thatthe full saturation of the uncertainty principle can only be achieved bypure
N
mode Gaussian states, for which
ν
i
= 1
∀
i
= 1
,...,N
. Instead,those mixed states such that
ν
i
≤
k
= 1 and
ν
i>k
>
1, with 1
≤
k
≤
N
,partially saturate the uncertainty principle, with partial saturation becoming weaker with decreasing
k
. The symplectic eigenvalues
ν
i
are determinedby
N
symplectic invariants associated to the characteristic polynomial of the matrix

i
Ω
σ

. Global invariants include the determinant Det
σ
=
i
ν
2
i
and the quantity ∆ =
i
ν
2
i
, which is the sum of the determinants of allthe 2
×
2 submatrices of
σ
related to each mode5.The degree of information about the preparation of a quantum state
̺
can be characterized by its
purity
µ
≡
Tr
̺
2
, ranging from 0 (completelymixed states) to 1 (pure states). For a Gaussian state with CM
σ
one has6
µ
= 1
/
√
Det
σ
.
(2)As for the entanglement, we recall that positivity of the CM’s partialtranspose (PPT)7is a necessary and suﬃcient condition of separability
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G. Adesso and F. Illuminati
for (
M
+
N
)mode bisymmetric Gaussian states (see Sec. 4) with respectto the
M

N
bipartition of the modes8, as well as for (
M
+
N
)mode Gaussian states with fully degenerate symplectic spectrum9. In the special, butimportant case
M
= 1, PPT is a necessary and suﬃcient condition for separability of all Gaussian states10
,
11. For a general Gaussian state of any
M

N
bipartition, the PPT criterion is replaced by another necessary andsuﬃcient condition stating that a CM
σ
corresponds to a separable stateif and only if there exists a pair of CMs
σ
A
and
σ
B
, relative to the subsystems
A
and
B
respectively, such that the following inequality holds11:
σ
≥
σ
A
⊕
σ
B
. This criterion is not very useful in practice. Alternatively,one can introduce an operational criterion based on a nonlinear map, thatis independent of (and strictly stronger than) the PPT condition12.In phase space, partial transposition amounts to a mirror reﬂection of one quadrature in the reduced CM of one of the parties. If
{
˜
ν
i
}
is thesymplectic spectrum of the partially transposed CM ˜
σ
, then a (1+
N
)mode(or bisymmetric (
M
+
N
)mode) Gaussian state with CM
σ
is separableif and only if ˜
ν
i
≥
1
∀
i
. A proper measure of CV entanglement is the
logarithmic negativity
13
E
N
≡
log
˜
̺
1
, where
·
1
denotes the tracenorm, which constitutes an upper bound to the
distillable entanglement
of the state
̺
. It can be computed in terms of the symplectic spectrum ˜
ν
i
of ˜
σ
:
E
N
= max
0
,
−
i
:˜
ν
i
<
1
log ˜
ν
i
.
(3)
E
N
quantiﬁes the extent to which the PPT condition ˜
ν
i
≥
1 is violated.
3. Two–Mode Gaussian States: Entanglement andMixedness
Two–mode Gaussian states represent the prototypical quantum statesof CV systems, and constitute an ideal testground for the theoretical andexperimental investigation of CV entanglement14. Their CM can be writtenis the following block form
σ
≡
α γ γ
T
β
,
(4)where the three 2
×
2 matrices
α
,
β
,
γ
are, respectively, the CMs of the tworeduced modes and the correlation matrix between them. It is well known10that for any two–mode CM
σ
there exists a local symplectic operation
S
l
=
S
1
⊕
S
2
which takes
σ
to its standard form
σ
sf
, characterized by
α
= diag
{
a, a
}
,
β
= diag
{
b, b
}
,
γ
= diag
{
c
+
, c
−
}
.
(5)
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Bipartite and Multipartite Entanglement of Gaussian States
5
States whose standard form fulﬁlls
a
=
b
are said to be symmetric. Anypure state is symmetric and fulﬁlls
c
+
=
−
c
−
=
√
a
2
−
1. The uncertaintyprinciple Ineq. (1) can be recast as a constraint on the
Sp
(4
,
R
)
invariantsDet
σ
and ∆(
σ
) = Det
α
+ Det
β
+2Det
γ
, yielding ∆(
σ
)
≤
1+ Det
σ
. Thestandard form covariances
a
,
b
,
c
+
, and
c
−
can be determined in terms of the two local symplectic invariants
µ
1
= (Det
α
)
−
1
/
2
= 1
/a, µ
2
= (Det
β
)
−
1
/
2
= 1
/b,
(6)which are the marginal purities of the reduced single–mode states, and of the two global symplectic invariants
µ
= (Det
σ
)
−
1
/
2
= [(
ab
−
c
2+
)(
ab
−
c
2
−
)]
−
1
/
2
,
∆ =
a
2
+
b
2
+ 2
c
+
c
−
,
(7)where
µ
is the global purity of the state. Eqs. (67) can be inverted to
provide the following physical parametrization of two–mode states in termsof the four independent parameters
µ
1
, µ
2
, µ
, and ∆15:
a
= 1
µ
1
, b
= 1
µ
2
, c
±
=
√
µ
1
µ
2
4
ǫ
−
±
ǫ
+
,
(8)with
ǫ
∓
≡
[∆
−
(
µ
1
∓
µ
2
)
2
/
(
µ
21
µ
22
)]
2
−
4
/µ
2
. The uncertainty principleand the existence of the radicals appearing in Eq. (8) impose the followingconstraints on the four invariants in order to describe a physical state
µ
1
µ
2
≤
µ
≤
µ
1
µ
2
µ
1
µ
2
+

µ
1
−
µ
2

,
(9)2
µ
+ (
µ
1
−
µ
2
)
2
µ
21
µ
22
≤
∆
≤
1 + 1
µ
2
.
(10)The physical meaning of these constraints, and the role of the extremalstates (
i.e.
states whose invariants saturate the upper or lower bounds of Eqs. (910)) in relation to the entanglement, will be investigated soon.
In terms of symplectic invariants, partial transposition corresponds toﬂipping the sign of Det
γ
, so that ∆ turns into ˜∆ = ∆
−
4Det
γ
=
−
∆ +2
/µ
21
+ 2
/µ
22
. The symplectic eigenvalues of the CM
σ
and of its partialtranspose ˜
σ
are promptly determined in terms of symplectic invariants2
ν
2
∓
= ∆
∓
∆
2
−
4
/µ
2
,
2˜
ν
2
∓
= ˜∆
∓
˜∆
2
−
4
/µ
2
,
(11)where in our naming convention
ν
−
≤
ν
+
in general, and similarly for the˜
ν
∓
. The PPT criterion yields a state
σ
separable if and only if ˜
ν
−
≥
1.Since ˜
ν
+
>
1 for all two–mode Gaussian states, the quantity ˜
ν
−
alsocompletely quantiﬁes the entanglement, in fact the logarithmic negativity Eq. (3) is a monotonically decreasing and convex function of ˜
ν
−
,
E
N
= max
{
0
,
−
log ˜
ν
−
}
. In the special instance of symmetric Gaussian