Disentanglement in Bipartite Continuous-Variable Systems
ABSTRACT Entanglement in bipartite continuous-variable systems is investigated in the
presence of partial losses, such as those introduced by a realistic quantum
communication channel, e.g. by propagation in an optical fiber. We find that
entanglement can vanish completely for partial losses, in a situa- tion
reminiscent of so-called entanglement sudden death. Even states with extreme
squeezing may become separable after propagation in lossy channels. Having in
mind the potential applications of such entangled light beams to optical
communications, we investigate the conditions under which entanglement can
survive for all partial losses. Different loss scenarios are examined and we
derive criteria to test the robustness of entangled states. These criteria are
necessary and sufficient for Gaussian states. Our study provides a framework to
investigate the robustness of continuous-variable entanglement in more complex
multipartite systems.
- [Show abstract] [Hide abstract]
ABSTRACT: A system of two initially entangled qubits interacting with a bosonic environment is considered. The interaction induces a loss of the initial entanglement of the two qubits, and for specific initial states it causes entanglement sudden death. An investigation of the modifications on the entanglement dynamics by a single pulse control field, performed in the two qubit system, shows that the control field can not only protect entangled states against sudden death but also induce a revival of entanglement in the two qubit system.Physica A: Statistical Mechanics and its Applications 05/2013; 392(10):2615–2622. · 1.68 Impact Factor - SourceAvailable from: Gaetano Nocerino[Show abstract] [Hide abstract]
ABSTRACT: Quantum properties are soon subject to decoherence once the quantum system interacts with the classical environment. In this paper we experimentally test how propagation losses, in a Gaussian channel, affect the bipartite Gaussian entangled state generated by a subthreshold type-II optical parametric oscillator. Experimental results are discussed in terms of different quantum markers, as teleportation fidelity, quantum discord, and mutual information, and continuous-variable (CV) entanglement criteria. To analyze state properties we have retrieved the composite system covariance matrix by a single homodyne detector. We experimentally found that, even in the presence of a strong decoherence, the generated state never disentangles and keeps breaking the quantum limit for the discord. This result proves that the class of CV entangled states discussed in this paper would allow, in principle, to realize quantum teleportation over an infinitely long Gaussian channel.Physical Review A 10/2012; 86(4). · 3.04 Impact Factor - [Show abstract] [Hide abstract]
ABSTRACT: An entanglement witness approach to quantum coherent state key distribution and a system for its practical implementation are described. In this approach, eavesdropping can be detected by a change in sign of either of two witness functions, an entanglement witness S or an eavesdropping witness W. The effects of loss and eavesdropping on system operation are evaluated as a function of distance. Although the eavesdropping witness W does not directly witness entanglement for the system, its behavior remains related to that of the true entanglement witness S. Furthermore, W is easier to implement experimentally than S. W crosses the axis at a finite distance, in a manner reminiscent of entanglement sudden death. The distance at which this occurs changes measurably when an eavesdropper is present. The distance dependance of the two witnesses due to amplitude reduction and due to increased variance resulting from both ordinary propagation losses and possible eavesdropping activity is provided. Finally, the information content and secure key rate of a continuous variable protocol using this witness approach are given.05/2013;
Page 1
Early Stage Disentanglement in Bipartite Continuous-Variable Systems
F. A. S. Barbosa1, A. S. Coelho1, A. J. de Faria1, K. N. Cassemiro2, A. S. Villar2,3, and M. Martinelli1
1Instituto de F´ ısica, Univ. de S˜ ao Paulo, Caixa Postal 66318, 05315-970 S˜ ao Paulo, SP, Brazil.
2Max Planck Inst. for the Science of Light, G¨ unther-Scharowsky-str. 1 / Bau 24, 91058 Erlangen, Germany.
3University of Erlangen-Nuremberg, Staudtstr. 7/B2, 91058 Erlangen, Germany.∗
(Dated: September 23, 2010)
Entanglement may vanish even for a limited amount of loss imposed on the quantum state, a
phenomenon known as entanglement sudden death in discrete systems, or, as proposed, early stage
disentanglement (ESD) in continuous variables. The resilience to ESD, or entanglement robustness,
is an essential property of entangled systems intended to convey quantum information through
realistic quantum channels. We investigate the robustness of bipartite entanglement in continuous-
variables systems, which we classify according to their robustness in different loss scenarios. The
sufficient conditions specifying each robustness class are determined. These conditions turn out to
be necessary and sufficient for Gaussian states. We find that in general states subject to ESD lie
close to the border between entangled and separable states.
PACS numbers: 03.67.Mn 03.67.Hk 03.65.Ud 42.50.Dv
I.INTRODUCTION
Entanglement is one of the fundamental aspects of
quantum mechanics and is considered a primary resource
for applications of quantum information [1, 2]. In order
to be considered a useful resource, entanglement must
be resilient to losses on the process of information distri-
bution between parties through quantum channels. It is
known, however, that certain entangled quantum states
cannot survive even a limited amount of loss to the envi-
ronment, a process known as entanglement sudden death.
The evolution of an entangled system into a separa-
ble one due to decoherence is one of the subtle prop-
erties of entangled systems, and it may become a limi-
tation for applications in quantum information process-
ing. For instance, entangled two-level systems (qubits)
are among the candidates for quantum computation. For
this kind of system it has been predicted and experimen-
tally demonstrated that entanglement can be completely
lost even for partial loss of coherence to the environ-
ment [3, 4].
Another class of conceptually simple systems are
continuous-valued observables
statistics. Coherent and squeezed states of the harmonic
oscillator belong to this class. Light fields are commonly
employed to produce such states in the laboratory, since
lasers and non-linear crystals are readily available. It
has been recently observed that these states may also
undergo early stage disentanglement (ESD) for finite
losses to the environment [5, 6]. This phenomenon is
a finite-loss parallel to the finite-time disentanglement
observed in discrete systems.
In this paper, we theoretically analyze the conditions
leading to continuous-variables early stage disentangle-
ment in the simplest case of bipartite systems. We inves-
possessingGaussian
∗Electronic address: mmartine@if.usp.br
tigate the role that a lossy channel play on bipartite en-
tanglement as it would be transmitted between two par-
ties. The property of resilience to losses will be referred
to as ‘robustness’. Robust entanglement will thus corre-
spond to the actual desired resource for the distribution
of entanglement among parties in a quantum communi-
cation network. We show that entanglement of bipartite
Gaussian entangled states can be assessed by the entan-
glement criteria currently existent in the literature. We
arrive at sufficient conditions for the robustness of bipar-
tite systems. For Gaussian states, the conditions become
also necessary.
A thorough investigation reveals the possibility of dis-
tinct entanglement dynamics as losses are imposed on
both subsystems. A formal classification based on that
is introduced, consisting of four robustness classes. On
the one extreme, the entanglement of fully robust states
vanishes only for total attenuation of both beams. On
the opposite extreme, ESD states become separable for
partial attenuations on either beam or a combination of
them and are therefore strongly subject to ESD. Two in-
termediate classes of partially robust states show either
robustness or ESD depending on the way losses are in-
troduced.
This paper is organized as follows. In Section II we
establish notation and the basic reservoir model respon-
sible for the losses on the quantum state. In Section III
a sufficient criterion to determine the robustness of the
entangled state is demonstrated. In Section IV we ex-
tend the robustness criterion, resulting in a necessary and
sufficient robustness condition for all Gaussian bipartite
states. The different classes of entanglement robustness
against losses in each channel are defined in Section V. In
Section VI particular quantum states commonly treated
in the literature are used as examples for a better intu-
itive understanding of the factors leading to ESD. Sec-
tion VII focuses on the main physical results and implica-
tions of our findings. We present our concluding remarks
in Section VIII.
arXiv:1009.4255v1 [quant-ph] 22 Sep 2010
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2
II. ENTANGLEMENT AND ESD IN A LOSSY
GAUSSIAN CHANNEL
The quantum properties of Gaussian states are com-
pletely characterized by the second order moments of the
appropriate observables. The choice of observables de-
pends on the system under consideration. In the case
of the electromagnetic field, we will consider the am-
plitude and phase quadratures, respectively written as
ˆ pj= i(ˆ a†
annihilation ˆ aj and creation ˆ a†
j = 1,2 stand for the two field modes we will consider.
The quadrature operators obey the commutation rela-
tion [ˆ qj, ˆ pj] = 2i, from which follows the unity value for
the standard quantum level (SQL), representing the noise
power present in the quadrature fluctuations of a coher-
ent state.
It is useful to organize the operator moments in the
form of a 4 × 4 covariance matrix V . Its entries are the
averages of the symmetric products of quadrature fluctu-
ation operators
?
whereˆξ = (ˆ q1, ˆ p1, ˆ q2, ˆ p2)Tis the column vector of quadra-
ture operators, and δˆξ =ˆξ−?ˆξ? are the fluctuation oper-
ators with zero average. Similar notation will be valid for
the individual quadratures, e.g. δˆ p1. The noise power is
proportional to the variance of the fluctuation, denoted
for a given quadrature by e.g. ∆2ˆ p1 = ?(δˆ p1)2?. The
Heisenberg uncertainty relation can be expressed as [7, 8]
j− ˆ aj) and ˆ qj= (ˆ a†
j+ ˆ aj) in terms of the field
joperators. The indices
V =1
2
δˆξδˆξT+ (δˆξδˆξT)T?
, (1)
V + iΩ
?J 0
≥
0, (2)
where Ω =
0 J
?
, and J =
?
01
−1 0
?
.
The covariance matrix can be divided in three 2×2 sub-
matrices, from which two (Aj) represent the reduced co-
variance matrices of the individual subsystems and one
(C) expresses the correlations between the subsystems
?A1
The correlations originate from both classical and quan-
tum backgrounds, and cannot be directly associated to
entanglement without considering the properties of each
subsystem. A particular trivial case of separable subsys-
tems, however, occurs when C is the null matrix. For
entangled systems, an additional uncorrelated classical
noise will be connected to the occurrence of ESD, as we
will present.
For bipartite Gaussian states, the covariance matrix
allows a perfect detection of entanglement between the
subsystems [8, 9]. In other words, the applicable entan-
glement criteria admit a necessary and sufficient form, as
will be made clear in the next section. In the remaining
of this paper, this fact will be crucial to classify these
V =
C
CTA2
?
. (3)
states according to their resilience to interactions with
the environment.
First we need to adopt a model for the quantum chan-
nel. Here we consider the realistic case of a lossy bosonic
channel, equivalent to the attenuation of light by random
scattering. Losses are modeled by independent beam
splitters placed on the beam paths. Each beam splitter
transformation combines one field mode with the vacuum
field. It can be associated to a reservoir at zero temper-
ature. Another realistic reservoir model is discussed in
Section VII.
The Gaussian attenuation channel transforms the field
operators according to [10, 11]
j=?Tjˆ aj+?1 − Tjˆ a(E)
where Tj is the beam splitter transmittance and ˆ a(E)
the annihilation operator from the environment. The re-
sulting transformation acting on the two modes of inter-
est (tracing out the reservoir) is not symplectic. It acts
on the covariance matrix as
ˆ aj−→ ˆ a?
j
, (4)
j
is
V?= L(V ) = L(V − I)L + I,
where L = diag(√T1,√T1,√T2,√T2) is the loss matrix
and I is the 4 × 4 identity matrix.
Starting from the situation where V represents an en-
tangled state, the question we want to answer regards
the behavior of entanglement as the covariance matrix
undergoes the transformation of Eq. (5).
(5)
III.
CRITERION AND ROBUSTNESS
THE DUAN ENTANGLEMENT
We will direct our attention, in a first moment, to the
entanglement criterion presented in Ref. [9], here referred
to as the Duan criterion. According to them, a sufficient
condition for the existence of entanglement is offered by
the corresponding witness
WD= ∆2ˆ u + ∆2ˆ v −
?
a2+1
a2
?
< 0, (6)
where
ˆ u =
1
√2
?
|a|ˆ p1+1
aˆ p2
?
and ˆ v =
1
√2
?
|a|ˆ q1−1
aˆ q2
?
(7)
are EPR-like collective operators, and a is an arbitrary,
real, nonzero number.
We will reserve the symbol ‘W’ for witnesses in general.
They will assume negative values if a given property is
present. Since the Duan witness is only a sufficient wit-
ness of entanglement, a state satisfying WD ≥ 0 could
be either separable or entangled.
attractive because it does not require full knowledge of
the covariance matrix, simplifying the detection of en-
tanglement in experiments. The downside is its limited
detection ability.
The witness WD is
Page 3
3
For a = 1, ˆ u and ˆ v become the original EPR opera-
tors [12, 13]. In this case, the process of entanglement de-
tection is equivalent to the balanced beam splitter trans-
formation of the input fields and the measurement of
squeezing in the two output fields [14]. Alternatively,
one can measure the quadrature noises ∆2ˆ piand ∆2ˆ qiof
each field and the cross correlations cp= ?δˆ p1δˆ p2? and
cq= ?δˆ q1δˆ q2?. The optimum choice for the parameter a
which minimizes WDis a2=?σ2/σ1, where σiexpresses
σj= ∆2ˆ pj+ ∆2ˆ qj− 2 = trAj− 2.
The sign indeterminacy in a is solved by taking into ac-
count the signs of the quadrature correlations.
these considerations, one arrives at the minimized form
of the Duan criterion
the excess noise in each subsystem,
(8)
With
WM= σ1σ2− (cp− cq)2< 0.
Eq. (9) provides the first insight on the robustness of
bipartite Gaussian states. The crucial fact to be observed
is that the sign of WM is conserved by attenuations. In
fact, using Eq. (5), the correlations transform as c?
√T1T2cpand c?
noises become σ?
factorizes in the entanglement witness,
(9)
p=
q=√T1T2cq, whilst the individual excess
j= Tjσj. The attenuation operation
W?
M= T1T2WM. (10)
Therefore, an initially entangled state satisfying Eq. (9)
will not disentagle under partial losses.
Entangled states satisfying the Duan criterion do not
disentangle for partial losses imposed on any mode: they
are fully robust. Among them lie the two-mode entangled
states, a large class of states for which both EPR-like
observables are squeezed [14–16].
Since WM is only a sufficient witness, the existence of
robust states for which WM≥ 0 cannot be excluded. In
what follows, we will demonstrate a necessary and suf-
ficient criterion for robustness in Gaussian states, effec-
tively determining the boundary between robust states
and those subject to ESD.
IV.ENTANGLEMENT ROBUSTNESS:
GENERAL CONDITIONS
In order to obtain clear-cut conditions for the robust-
ness of entanglement, we must employ a necessary and
sufficient entanglement criterion. By analyzing whether
the subsystems remain entangled or become separable
during attenuation, we will classify all bipartite Gaus-
sian states.
A.The PPT Criterion
We find a convenient separability criterion in the
requirement of positivity under partial transposition
(PPT) of the density matrix for separable states [17, 18].
An entangled state, on the other hand, will necessarily
lead to a negative partially transposed density matrix,
which is non-physical.
The partial transposition (PT) of the density operator
is equivalent to the operation of time-reversal applied to a
single subsystem. On the covariance matrix level, a time-
reversal is obtained by changing the sign of the momen-
tum (for harmonic oscillators), or the sign of the phase
quadrature of one mode (for electromagnetic fields), in
this manner affecting the sign of its correlations [8].
Physical validity is assessed using Eq. (2). The uncer-
tainty relation can be recast into a more explicit form by
expressing it in terms of the determinants of the covari-
ance matrix and its submatrices as
1 + detV − 2detC −
?
i=1,2
detAj≥ 0.(11)
The PT operation modifies the sign of detC, resulting in
the following condition for entanglement [8]
Wppt= 1 + detV + 2detC −
?
i=1,2
detAj< 0. (12)
In contraposition, all separable states fulfill Wppt ≥ 0.
Therefore, Wpptis a necessary entanglement witness, and
the equation Wppt = 0 traces a clear boundary in the
space of bipartite Gaussian states, discriminating the
subspaces of separable and entangled states from one an-
other.
It is convenient to recall here that the purities of Gaus-
sian states are directly related to the determinant of the
covariance matrices [19]
µ = (detV )−1
µj = (detAj)−1
2,(13)
2, (14)
so that the entanglement witness of Eq. (12) involves the
total purity of the systems, the purity of each subsystem,
and the shared correlations.
B.Covariance Matrix under Attenuation
Applying the witness of Eq. (12) to the attenuated
covariance matrix of Eq. (5), one obtains
W?
ppt(T1,T2) = 1+detV?+2detC?−
?
j=1,2
det(A?
i), (15)
from which W?
it follows that the individual submatrices transform as
C?=√T1T2C and A?
tions. These forms mean that the bilinear dependence of
Eq. (9) on T1and T2leading to a constant sign of the wit-
ness is not expected on Eq (15). Therefore, robustness
is not guaranteed for states satisfying Wppt < 0. The
change of the witness sign is the signature of ESD.
ppt(T1= 1,T2= 1) = Wppt. From Eq. (5),
j= Tj(Aj− I) + I under attenua-
Page 4
4
In Appendix A, we derive the explicit transmittance-
dependent form of W?
and detV as combinations of quantities for which attenu-
ation scales polynomially, obtaining local rotation invari-
ant terms. Disregarding the trivial solutions T1,2 = 0,
corresponding to separability by total attenuation of ei-
ther mode, the term T1T2factors out. The sufficient and
necessary entanglement witness of Eq. (15) assumes the
form
ppt(T1,T2). We write detAj, detC,
W?
ppt(T1,T2) = T1T2WE(T1,T2). (16)
The entanglement witness WEpreserves the sign of Wppt
with the advantage of maintaining only the relevant de-
pendence on T1and T2for the detection of robustness.
They are thus equivalent robustness witnesses. It reads
WE(T1,T2) = T1T2Γ22+ T2Γ12+ T1Γ21+ Γ11. (17)
The expressions for the coefficients Γij in terms of the
covariance matrix entries are given in Appendix A.
The different dynamics of entanglement under losses
appear in the witnesses W?
four entangled states (three of them presenting ESD)
plus a separable state under attenuation. The plots show
W?
pptand WE. Fig. 1 depicts
ppt(T1,T2) based on the covariance matrix
V =
∆2q1
0
cq
0
0cq
0
0
cp
0
∆2p1
0
cp
∆2q2
0∆2p2
,
(18)
constructed from diagonal submatrices. This simple form
of V suffices to span all types of entanglement dynamics
on Gaussian states.
The curves of Fig. 1a–d were specifically obtained from
V =
2.55
0
cq
0
0cq
0
0
1.80
0
−1.26
−1.26
0
1.80
2.55
0
.
(19)
The matrix is even simpler in this case of symmetric
modes, since both present the same quantum statistics.
By varying the parameter cqthree different types of en-
tanglement dynamics are observed.
1.275), a state violating the Duan criterion maintains the
witness negative for all values of partial attenuations, re-
sulting in a fully robust state. Disentanglement does not
occurs for finite losses in any of the fields. In Fig. 1b, the
choice cq= 0.893 characterizes a state for which ESD oc-
curs for partial attenuations on a single mode or on both
modes. This represents the class of states most suscepti-
ble to ESD. Fig. 1c (cq= 0.3825) illustrates a separable
state, which naturally remains separable throughout the
whole region of attenuations.
A more subtle entanglement dynamics appears in
Fig. 1d (cq= 1.033). The state is robust against any sin-
gle mode attenuation but may become separable if both
In Fig. 1a (cq =
modes are attenuated simultaneously. Finally, for asym-
metric modes, the system may be robust against losses
on one mode, but not on the other. This is observed
in Fig. 1e, where W?
matrix
This particular covariance matrix is obtained from
Eq. (19) by imposing the attenuation T2 = 0.40. The
state remains robust against losses on mode 2, but the
partial attenuation on this mode favors the disentangle-
ment with respect to losses on mode 1. In Section VII we
discuss on the possibility that the entanglement dynam-
ics change by attenuation. Figs. 1b, c, e are examples of
partially robust states, for which both ESD and robust-
ness may take place depending on the particular sequence
of losses.
pptis calculated for the covariance
V =
2.55
0
0.653
0
00.653
0
1.62
0
0
1.80
0
−0.797
−0.797
0
1.32
.
(20)
C. Full Robustness
We would like to find a witness capable of identifying
fully robust states directly from the covariance matrix.
To obtain the necessary condition, we note from Eq. (17)
that the entanglement dynamics close to complete at-
tenuation is dominated by Γ11. Therefore, an initially
entangled state WE(T1= 1,T2= 1) < 0 with Γ11> 0,
must become separable for sufficiently high attenuation,
from which we derive the witness
Wfull= Γ11= σ1σ2− tr(CTC) + 2detC.
Wfull≤ 0 supplies a simple, direct, and general condi-
tion for testing the entanglement robustness of bipartite
Gaussian states.
Eq. (9) obtained from the Duan criterion is a partic-
ular case of Eq. (21) when the correlation submatrix is
diagonal. Wfullis manifestly invariant under local rota-
tions in phase space and scales linearly with attenuations
on both fields. Using local rotations to diagonalize the
correlation matrix C, we obtain
(21)
W(D)
full= σ1σ2− (cp− cq)2≤ 0, (22)
which coincides with WM. Therefore, for Gaussian states
given by covariance matrices with diagonal correlation
submatrix, WM is a necessary and sufficient robust en-
tanglement witness, but only sufficient otherwise. It is
a particular case of the more general robustness witness
Wfull.
Wfull is a necessary and sufficient witness of robust
entanglement, just as Wpptis a necessary and sufficient
witness of entanglement. However, the symmetry proper-
ties of Wfulldiffer from those of Wppt, since its invariance
holds only under local rotations, but not under general lo-
cal symplectic transformations. Rotational invariance is
Page 5
5
Figure 1: Possible behaviors of the PPT entanglement witness W?
transmitance T1 and T2. (a) Fully robust entanglement. (b) ESD for any combination of beam attenuations. (c) Separable
state. (d) Two-sided partial robustness: the state is robust for any individual attenuation, but not for a combination of
attenuations, such as equal attenuations. (e) One-sided partial robustness, i.e., the state is robust with regard to attenuations
on one mode but presents ESD for attenuations on the other mode.
pptunder attenuation, as a function of the beam splitter
indeed expected and corresponds to the arbitrary choice
of measurement basis for the quadratures. In the basis
that diagonalizes A1and A2, the determinants appear-
ing in Wpptare given by products of the quadrature vari-
ances. In the case of Wfull, the dependence on the traces
of Aj implicit in σj of Eq. (8) shows that it is a wit-
ness based on the sum of variances. A criterion based on
the product of variances will be less restrictive than one
based on the sum of variances [21]. Wfull is naturally
more restrictive than the entanglement criterion, since
it reveals the particular property of robustness against
losses of certain entangled states.
D. Partial Robustness
The previous discussion based on the general assump-
tion that both subsystems may suffer attenuation is now
particularized to situations where a single subsystem is
exposed to a lossy channel. We will define witnesses ca-
pable of identifying partial robustness, in addition to sys-
tems presenting full robustness. These conditions will be
less restrictive than Eq. (21), but still more restrictive
than Eq. (12).
Let us consider the case T2= 1 for definiteness. The
attenuated witness of Eq. (17) becomes
WE(T1,T2= 1) = (Wppt− W1)T1+ W1, (23)
where
W1= Wfull+ Γ21
(24)
(see Appendix A for the expression of Γ21). The analysis
of W1 follows the same lines used in the case of fully
robust states, with the simplification that the witness
depends linearly on the attenuation. Thus, there is only
one possible path cutting the plane WE(T1,T2= 1) = 0.
The beam splitter transmission where ESD occurs is
Tc
1=
W1
W1− Wppt. (25)
From Wppt< 0, it follows that 0 < W1< W1− Wppt, to
assure that Tc
1exists as a meaningful physical quantity
(0 < Tc
1< 1) whenever W1> 0.
Therefore, an entangled state satisfying W1≤ 0 is ro-
bust against losses in channel 1, and W1is the witness
for this type of robustness. The corresponding analy-
sis regarding attenuations on the subsystem 2 yields the
Page 6
6
witness
W2= Wfull+ Γ12,(26)
with the same properties of W1. A relation analogous to
Eq. (25) holds for Tc
2. Both witnesses are invariant under
local rotations, as expected.
V.ROBUSTNESS CLASSES
Based on the different dynamics of entanglement of
Fig. (1), we develop a classification of bipartite entan-
gled states according to their resilience to losses. We
take guidance on the sign of the entanglement witness
WE(T1,T2), which is a hyperbolic paraboloid surface.
The contour defined by the condition WE(T1,T2) = 0
provides a complete description of the entanglement dy-
namics in terms of Γij. As depicted in Fig. 1, four special
dynamical behaviors are recognized by a detailed analy-
sis. Bipartite entangled Gaussian states can be assigned
to four different classes:
(i). Fully robust states remain entangled for any partial
attenuation: WE(T1,T2) < 0,∀T1,2.
(ii). Two-sided robust states remain entangled against
losses on any single mode, but may become disen-
tangled for some combinations of partial attenua-
tions on both modes: WE(T1,T2 = 1) < 0,∀T1,
and WE(T1= 1,T2) < 0,∀T2.
(iii). One-sided robust states remain entangled against
losses on one specific mode, but may disentangle
for partial losses on the other mode: WE(T1,T2=
1) < 0,∀T1, or WE(T1= 1,T2) < 0,∀T2.
(iv). ESD states disentangle for partial attenuation on
any mode or combinations of partial attenuations
on both modes.
In addition, a fifth class can be introduced to account for
the separable states, in this manner extending our classi-
fication to all possible bipartite Gaussian states. Classes
(i)–(iii) show some robustness to losses, although only
class (i) is robust in all situations. In the same manner,
classes (ii)–(iv) are susceptible to some form of ESD, but
only class (iv) presents ESD in any attenuation scenario.
For this reason, we will refer to classes (ii)–(iii) as par-
tially robust states.
With the witnesses previously defined from the value
of WE(T1,T2) on the vertices of the physically achievable
values of attenuations, we can define the necessary crite-
ria to classify the Gaussian states according to their ro-
bustness of entanglement. Therefore, an entangled state
is one-sided robust with respect to subsystem 1, if
W1≤ 0 (27)
and similarly, it is one-sided robust with respect to sub-
system 2 if
W2≤ 0. (28)
These witnesses offers a necessary condition for one-sided
robustness (class (iii)). A two-sided robust state, belong-
ing therefore to class (ii), must satisfy both conditions
simultaneously,
W1≤ 0 and W2≤ 0. (29)
According to our definition, the set of two-sided robust
states are also one-sided robust. There is a practical rea-
son for choosing this inclusive definition of robustness
classes. Several quantum communication protocols in
continuous variables can be realized by one of the parties
(Alice) locally producing the entangled state and sending
one mode to a remote location while keeping the remain-
ing one. The other party (Bob) then performs unitary
operations on the state according to instructions send by
Alice through a classical channel. The success of such
communication schemes strongly depend on the losses
that the subsystem of Bob may undergo to the environ-
ment, or to an eavesdropper (Eve). In this situation,
Alice must produce an entangled state that is at least
one-sided robust in order to avoid problems with sig-
nal degradation. Two-sided robust or fully robust states,
by fulfilling a more restrictive condition, can clearly be
employed in those protocols, replacing one-sided robust
states. Therefore, one-sided robustness might suffice to
establish a quantum communication channel in some ap-
plications.
Moreover, since the witnesses are sufficient for any
continuous-variable state, a non-Gaussian state satisfying
the condition for one-sided robustness but not the one for
two-sided robustness may actually remain entangled due
to higher order correlations, but its one-sided robustness
can be assured. Based on the same arguments used be-
fore, a fully robust state could also replace any partially
robust state in quantum communication protocol.
VI.PARTICULAR CASES
In this section we analyze particular Gaussian states of
interest. For instance, we have already seen from Eq. (18)
that the simplified Gaussian state with ?ˆ pjˆ qj?? = 0 is
sufficient to span all robustness classes. The intent here
is to offer a deeper understanding of common physical
scenarios leading to entanglement robustness or ESD.
A.Symmetric Modes and Quadratures - Fully
Robust States
Let us first consider systems with certain symmetries,
such that their covariance matrices are restricted to sim-
ple forms. A case of physical interest is the completely
symmetric state, for which ∆2ˆ p1 = ∆2ˆ q1 = ∆2ˆ p2 =
∆2ˆ q2= s and ?δˆ p1δˆ p2? = ?δˆ q1δˆ q2? = c, and ?δˆ pjδˆ qj?? = 0.
Page 7
7
The covariance matrix has the form
V =
s
0
c
0 −c 0
0
s
0
c
0 −c
s
0
0
s
.
(30)
Such states are generated, for instance, by the interfer-
ence of (symmetric) squeezed states on a balanced beam
splitter (entangled squeezed states) [14–16]. In this case
one has s = ν cosh2r and c = ν sinh2r, where r is the
squeezing parameter and ν ≥ 1 accounts for an even-
tual thermal mixedness, representing a correlated classi-
cal noise between the systems.
The highly symmetric covariance matrix of Eq. (30)
results in the witnesses
Wppt= (s2− c2+ 1)2− 4s2
(31)
and
Wfull= 4[(s − 1)2− c2] = 4(s2− c2+ 1 − 2s),
from which one directly sees that Wppt< 0 and Wfull< 0
lead to the same condition (s − 1 − |c| < 0). Therefore,
states with symmetry between the two modes and the
two quadratures are fully robust.
(32)
12345
?1.0
?0.5
0.0
0.5
1.0
n
C
Figure 2: State space of a fully symmetric system is presented
in terms of variance s and normalized correlation ¯ c. Separa-
ble state lies in the yellow region. States in red region are
robustly entangled. The white region contains non-physical
states violating µ ≤ 1. We can see that the ESD does not
happen in these systems.
In this situation, the state space can be written in
terms of two parameters, the quadrature variance s and
the normalized correlation ¯ c = c/s. Fig. 2 depicts the re-
gions of separable states and robust entanglement. The
lack of ESD in these systems indicates that strong sym-
metries lead to entanglement robustness, even when clas-
sical noise is present, as long as it is correlated.
Generalizing the highly symmetric states treated here,
all covariance matrices in the standard form II of Ref. [9]
can be shown to be fully robuts. The reason is that in
this case the Duan criterion, which we have shown is
sufficient condition for fully robustness, coincides with
the PPT criterion. For these states, the parameter a
of Eq. (6) results in EPR-like operators coinciding with
the optimized choice of collective operators, so that the
state also satisfies the minimized Duan witness WM of
Eq. (9). We note that in this case the covariance matrix
is not necessarily as symmetric as in Eq. (30). Therefore,
the symmetry of the covariance matrix is not a necessary
condition for robustness, but a sufficient one, since such
states are naturally in the standard form II.
B. Symmetric Modes and Asymmetric
Quadratures
Following the final discussion, we may relax the sym-
metries of the state to investigate entanglement robust-
ness. States which are symmetric on both modes but
asymmetric on the quantum statistics of the quadratures
have been recently observed to present ESD [6]. The sys-
tem under investigation consisted of the twin light beams
produced by an optical parametric oscillator, described
by a covariance matrix of the form
V =
∆2q
0
cq
0
0cq
0
0
cp
0
∆2p
0
cp
∆2q
0∆2p
.
(33)
The entanglement and robustness witnesses read
?(∆2p)2− c2
Wppt=
p
??(∆2q)2− c2
−2∆2p∆2q + 2cpcq+ 1
q
?
(34)
and
Wfull= (∆2p + ∆2q − 2)2− (cq− cp)2. (35)
In this situation, the subsystems have equal purities
(µS = 1/?∆2p∆2q).
(∆2q)2− c2
tions ¯ cp= cp/∆2p and ¯ cq= cq/∆2q for simplicity. They
are bounded by −1 ≤ ¯ cj≤ 1.
In Fig. 3 the robustness condition is mapped in terms
of the correlations for a fixed purity µS= 0.626, showing
the regions corresponding to different robustness classes.
The inset includes the location of the states in Fig. 1a–
d. Robust entangled state (a) falls within the red region
in Fig. 3, while the separable state (c) is located in the
yellow region. Within the blue region, two different types
The quadrature variances and
correlations are constrained by (∆2p)2− c2
q≥ 0. We introduce the normalized correla-
p≥ 0 and
Page 8
8
?1.0
?0.50.00.51.0
?1.0
?0.5
0.0
0.5
1.0
Cq
Cp
?a??a?
?b??b?
?c??c?
?d??d?
?0.75 ?0.7 ?0.65
0.1
0.25
0.4
0.55
Figure 3: The space of symmetric two-mode states is plotted
as a function of the normalized correlations ¯ cp and ¯ cq. Sep-
arable states lie in the yellow region; robust entangled states
are comprised within the red region; two-sided robust states
are in the dark-blue region, and one-sided robust states are in
the blue region. The white region contains unphysical states.
The boundaries of physical states are given by global unitary
purity and the uncertainty principle. Here we use ∆2p = 1.80
and ∆2q = 2.55, normalized by the SQL.
of states subject to ESD are present. State (d) is two-
sided robust, lying close to the boundary of robust states.
State (b) shows ESD for partial losses in general, lying
on the boundary to separable states.
Alternatively, following the treatment briefly described
in Ref. [6], the covariance matrix of Eq. (33) can be
parametrized in terms of the uncorrelated subspaces, de-
fined in terms of combinations of the two subsystems
through the EPR operators,
ˆ p±=
1
√2(ˆ p1± ˆ p2)(36)
and
ˆ q±=
1
√2(ˆ q1± ˆ q2).(37)
The covariance matrix is diagonal on this basis,
Entanglement can be directly observed from the product
of squeezed variances of the proper pair of EPR opera-
tors, (ˆ p+, ˆ q−) or (ˆ p−, ˆ q+). Additionally, the entanglement
V+−=
∆2ˆ p+
0
0
0
00
0
0
0
0
∆2ˆ q+
0
0
∆2ˆ p−
0∆2ˆ q−
.
(38)
and the ESD criteria of symmetric two-mode systems of
Eqs. (34)–(35) can be written in the simpler forms,
Wppt = WprodWprod,
Wfull = WsumWsum,
(39)
(40)
where
Wsum = ∆2ˆ p++ ∆2ˆ q−− 2,
Wsum = ∆2ˆ p−+ ∆2ˆ q+− 2,
Wprod = ∆2ˆ p+∆2ˆ q−− 1,
Wprod = ∆2ˆ p−∆2ˆ q+− 1.
The distinction between robust and partially ro-
bust entanglement is clearly illustrated with symmetric
modes. Considering attenuation solely on mode 1, the
condition for partial robustness of Eq. (27) yields
W1= WsumWprod+ WprodWsum.(41)
The condition W1= 0 defines the border between par-
tial robustness and ESD, being less stringent than the
full robustness condition of Eq. (40) and more restrictive
than the entanglement condition of Eq. (39). The fact
that Wppt< 0 is a necessary and sufficient entanglement
condition and Wfull< 0 is only sufficient can be stated
as
Wfull< 0 =⇒ Wppt< 0.(42)
Since ˆ p±and ˆ q±are conjugated observables, Wprodand
Wprod (or Wsum and Wsum) cannot be simultaneously
negative. The uncertainty relation implies
Wsum< 0 =⇒ Wsum> 0,
Wprod< 0 =⇒ Wprod> 0.
(43)
(44)
In this context, the condition of Eq. (42) can be restated
as
Wsum< 0 =⇒ Wprod< 0.(45)
For W1= 0,
WsumWprod= −WprodWsum.
This equation holds only if Wprod < 0 and Wsum > 0
(or Wprod< 0 and Wsum> 0). The other possibilities
are forbidden by Eqs. (43)–(45). This demonstrates that
W1= 0 lies between the curves Wppt= 0 and Wfull= 0.
Fig. 4 shows the regions of entanglement robustness,
ESD and separability for a typical scenario.
sumes fixed values for the partial purities of the matrix
V+−, µ+= 1/?∆2ˆ p+∆2ˆ q+and µ−= 1/?∆2ˆ p−∆2ˆ q−,
terms of ∆2ˆ p− and ∆2ˆ q+. The entanglement criterion
of Eq. (39) sets the border between the separable states
(yellow) and the entangled states (blue regions and red
region). Eq. (40) defines the boundary between full (red)
(46)
It as-
to write the entanglement and robustness conditions in
Page 9
9
0.00.51.0 1.52.02.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
?2q?
?2p?
Figure 4: The space of symmetric two-mode states is plotted
as a function of the EPR variances ∆2ˆ q+ and ∆2ˆ p−. Separa-
ble states lie in the yellow region; robust entangled states are
comprised within the red region; states which undergo ESD
are in the lighter and darker blue region. Darker blue region
confines the partially robust entangled states. The partial
purities are µ− = 0.7267 and µ+ = 0.4529.
and partial robustness (dark blue). The border between
partial robustness (dark blue) and ESD (light blue) is
given by the condition W1 = 0.
ESD reported in Ref. [6] was obtained for partially ro-
bust states lying in the region delimited by the conditions
Wsum> 0 and W1< 0.
The observation of
C.System in Standard Form I
The last case we consider is a covariance matrix in the
standard form I [8, 9], produced by a suitable combina-
tion of squeezed states and beams splitters. It represents
two different modes with symmetric quadratures,
The entanglement and full robustness witnesses read
V =
s
0
cq 0
0 cp 0
0 cq 0
s0 cp
t0
t
.
(47)
Wppt= (st − c2
and
q)(st − c2
p) − s2− t2+ 2cqcp+ 1 (48)
Wfull= 4(s − 1)(t − 1) − (cq− cp)2.
The subsystems have purities µ1 = s−1and µ2 = t−1.
We define the normalized correlation ¯ cj = cj/√st =
(49)
?1.0
?0.50.0 0.51.0
?1.0
?0.5
0.0
0.5
1.0
Cq
Cp
Figure 5: The space of states represented by covariance ma-
trices in standard form I is plotted as a function of the nor-
malized correlations ¯ cq and ¯ cp. Separable states lie in the
yellow region; robust entangled states are comprised within
the red region; states which undergo ESD are located in the
blue region. The boundaries of physical states are given by
the maximum global purity. The local purities are µ1 = 0.5
and µ2 = 0.4.
cj√µ1µ2 as before, and plot the conditions above in
terms of them.
A covariance matrix in standard form I may also
present ESD for certain parameters, as shown in Fig. 5.
Differently from the case of symmetric modes, ESD in
such a system does not occur for symmetric correlations,
¯ cq= −¯ cp, independently of the purities µ1and µ2. This
case of standard form I is also a particular case of stan-
dard form II, for which symmetry properties guarantee
the full robustness of entanglement.
VII.DISCUSSION
In the recent history of entanglement detection in bi-
partite continuous-variable systems and specifically in
Gaussian states, the excessively restrictive (sufficient)
conditions initially considered were slowly substituted
by clear-cut (necessary and sufficient) conditions which
could trace a border between entanglement and separa-
bility. On the first extreme lies the EPR criterion [24],
based on a hypothetical violation of the uncertainty prin-
ciple by inferred variances, employed in the first demon-
stration of entanglement between the faint fields pro-
duced by an OPO below threshold [25]. This criterion
was followed by the simple form of the Duan criterion
of Eq. (6), based on the sum of variances of EPR-like
Page 10
10
operators, and is still extensively used in experiments for
its simplicity [14–16, 26, 27]. Finally, the necessary and
sufficient entanglement criterion of Eq. (12) here referred
to as the PPT criterion [8, 9], used in situations when the
detection of entanglement requires deeper knowledge of
the state [5], can be shown to be equivalent to a product of
variances of EPR-like operators [21]. For non-Gaussian
states, more extensive tests must be performed to detect
entanglement beyond the covariance matrix [28].
The two criteria based on the sum (Duan) and on the
product (PPT) of variances were evoked in this paper
in the context of bipartite entanglement robustness and
ESD under losses. The sum criterion assumes in this case
the role of a necessary and sufficient witness for entan-
glement robustness. For the general attenuation scenario
of losses imposed on both modes, it inequivocally distin-
guishes fully robust states from those subject to some
form of ESD. This fact could indeed be expected, since
equal losses imposed on both beams will linearly lead
both EPR-like operators to the shot noise level, in this
manner preserving the violation of the Duan criterion and
thus the entanglement. Unequal losses can be accounted
for by simply including an additional loss to balance the
losses between the modes. In this manner, a state which
would eventually stop violating the Duan criterion after
unequal losses could be brought back to violation by the
last operation. Since the latter is Gaussian and as such
cannot create entanglement, one is forced to conclude
that entanglement was already present before, and as a
consequence was not lost. In this manner, states violat-
ing the Duan inequality can be understood to be robust
by a straightforward reasoning.
Among the entangled states violating the Duan crite-
rion lie the two-mode entangled states, for which both
EPR-like observables present squeezing. As mentioned
above, those are the states commonly produced in experi-
ments involving parametric downconversion. Thus, those
sources of entangled modes could be safely used in lossy
channels for quantum communications. A second pos-
sibility of entanglement robustness is offered by symme-
try. Entangled states symmetric on both the quadratures
and modes violate the Duan criterion even if an arbitrary
amount of correlated classical noise is present on the sys-
tem, decreasing its purity to an unlimited extent.
fact, correlated classical noise is largely irrelevant to the
presence of ESD even for asymmetric states. Conversely,
uncorrelated classical noise seems to play an important
role in causing ESD. An appealing example is given by
the OPO operating above threshold. This system should
produce pure states, therefore deprived of any classical
noise. In reality, the non-linear crystal includes thermal
noise on the two modes, making ESD possible [6, 29].
At this point it is important to consider that the
above physical picture is only valid for the simplest case
of quantum channels attenuating the modes by random
scattering. This model is certainly accurate to describe
detection losses and communication schemes relying on
the exchange of modes through short distances. A more
In
general model would have to consider the inclusion of
classical noise on the system by the channel, and specif-
ically uncorrelated noise. In other words, the channel
noise contribution would not be limited to the vacuum
noise. The addition of noise is likely to occur over larger
distances, for instance by atmospheric turbulence on
open-air quantum communication schemes or by phonon
coupling on long-distance optical trasmission through op-
tical fibers. Those models are likely to impose an upper
limit on the amount of uncorrelated classical noise to
which a given entangled state is resilient. We presume
that bipartite systems presenting a higher degree of en-
tanglement (as given by common entanglement measures
such as the logarithmic negativity [30, 31]) will show re-
silience to higher levels of classical noise, as hinted by the
fact the ESD states lie close to the border with separable
states (Fig. 3). In any case, the simple channel model we
consider imposes the weakest restriction on the robust-
ness of bipartite entanglement, i.e. the limit of acceptable
losses in the channel before entanglement is destroyed. A
state which fails to show robustness under these circun-
stances will not stand the influence of more destructive
channels.
The less stringent notion of partial robustness, despite
seeming a less important particularization, touches a very
fundamental issue. Bipartite Gaussian states are the sim-
plest possible states in continuous-variable systems, and
the attenuation on one mode is the simplest dissipative
process. Thus, violating the condition of Eq. (27) im-
plies that ESD may occur even in the simplest scenario.
This reasoning consolidates ESD as a fundamental phe-
nomenon, discarding it as a consequence of complexity.
Partially robust states can transit between robustness
classes by attenuations, as revealed by Fig. 1e. A two-
sided robust state can become a one-sided robust state,
and subsequently an ESD state. The opposite however
is not possible due to the fact that entanglement cannot
be distilled by Gaussian operations [22, 23]. The proof is
stated in Appendix B.
VIII.CONCLUSION
We have shown that the subtle properties of entangle-
ment are present even in the simplest quantum systems.
Sudden death of entanglement, demonstrated in the dis-
crete variable domain, is also present in their counterpart
in the domain of continuous variables. Disentanglement
of Gaussian bipartite states is a quite surprising feature
since the correlations between the modes, which are re-
sponsible for entanglement, only vanish for complete at-
tenuation. In this sense, we could say that the corre-
lations between the subsystems are transiting from the
quantum to the classical regime by independent degra-
dation on the subsystems.
For Gaussian bipartite system, we have demonstrated
that more than having a necessary and sufficient crite-
rion for entanglement, we can also develop a necessary
Page 11
11
and sufficient criterion for robustness of the state against
interaction with a lossy environment. The sum criterion
of Ref. [9] has been shown to discriminate among the en-
tangled states many of those which are robust against
losses.
The robustness conditions presented in this paper rep-
resent necessary and sufficient conditions for the classi-
fication of the Gaussian state according to its entangle-
ment dynamics under losses. For non-Gaussian states,
those same conditions are still valid as sufficient condi-
tions, but the situation in which higher order correlations
might keep the modes entangled even beyond a certain
level of losses cannot be discarded.
We hope that our contribution can provide further in-
sight into the dynamics of entanglement in realistic quan-
tum channels.
Acknowledgments
This work was supported by the Conselho Nacional
de Desenvolvimento Cient´ ıfico e Tecnol´ ogico (CNPq) and
the Funda¸ c˜ ao de Amparo ` a Pesquisa do Estado S˜ ao Paulo
(FAPESP). KNC and ASV acknowledge support from
the AvH Foundation. We kindly acknowledge P. Nussen-
zveig for fruitful discussions.
Appendix A: Attenuated Witness
We would like to obtain an explicit expression for
W?
bipartite system (noises and correlations).
that the procedure cannot be directly realized by first
bringing V?(or V ) to a standard form and then ap-
plying the attenuation, since local symplectic operations
S ∈ Sp(2,?)⊕Sp(2,?) do not commute with the attenu-
ation operation, L(SV ST) ?= SL(V )ST[11, 20]. Conse-
quently, invariant quantities under global and local sym-
plectic transformations are not necessarily conserved by
attenuations, such as the global and local purities. On
the other hand, SL(V )ST= L(SV ST) is satisfied only
if SST= I, i.e. S must be a local phase space rota-
tion, S ∈ SO(2,?) ⊕ SO(2,?). Therefore, a criterion for
entanglement robustness should depend solely on local
rotational invariants.
We derive the explicit behavior of the witness W?
with attenuation. Writing the PPT separability criterion
in terms of the symplectic invariants [8], we obtain
ppt(T1,T2) in terms of the physical parameters of the
We note
ppt
Wppt = 1 + detV + 2detC −
?
j=1,2
detAj, (A1)
detV = detA1detA2+ detC2− Λ4,
Λ4 = tr(A1JCJA2JCTJ).
(A2)
(A3)
After attenuation the matrices A1, A2, and C become
?
C?=
A?
i= Ti(Ai− I) + I,
T1T2C,(A4)
(A5)
To derive Eq. (17), we express the symplectic invari-
ants in terms of quantities presenting similar behav-
ior. Two such quantities are obtained from Eq. (5) and
Eq. (A4),
det(V?− I) = T2
detC?= T1T2detC.
1T2
2det(V − I),(A6)
(A7)
Since for any 2 × 2 matrix M the following expressions
are valid,
det(M − I) = detM − trM + 1,
tr(M − I) = tr(M) − 2,
(A8)
(A9)
one obtains
??
j− σ?
j= T2
σ?
j= Tjσj,
j(?j− σj),(A10)
(A11)
where σi= trAi−2, and ?i= detAi−1 is the deviation
from a pure state (impurity), which is zero for a pure
state and positive for any mixed state.
Applying Eq. (A8) for det(V − I), we find quantities
which scale polynomially on the beam attenuations,
detV = det(V − I) + η,
η = σ1(?2− σ2) + σ2(?1− σ1) + σ1σ2
+ det(A1) + det(A2) + Λ1+ Λ2− ΛC− 1
Λ1 = tr(CTJ(A1− I)JC),
Λ2 = tr(CJ(A2− I)JCT),
ΛC = tr(CTC)
(A12)
(A13)
(A14)
where the last three quantities scale as
Λ?
1= T2
1T2Λ1
,Λ?
2= T1T2
2Λ2,(A15)
(A16)Λ?
C= T1T2ΛC.
Substituting Eq. (A12) in Eq. (A1) and applying the
attenuation operation, we arrive at
W?
ppt(T1,T2) =
?
i,j=1,2
Ti
1Tj
2Γij, with(A17)
Γ22 = det(V − I) = det(V ) − η,
Γ12 = σ1(?2− σ2) + Λ2,
Γ21 = σ2(?1− σ1) + Λ1,
Γ11 = σ1σ2− ΛC+ 2det(C),
The function W?
Gaussian states when subject to losses.
pptdescribes the dynamics of all bipartite
Page 12
12
Appendix B: Geometry of WE
The entanglement witness under attenuations,
WE(T1,T2) = T1T2Γ22+ T2Γ12+ T1Γ21+ Γ11, (B1)
forms a surface known as a hyperbolic paraboloid. Such
surface have the property that projections of WE on
planes determined by constant T1and T2are linear func-
tions. Projections on planes with constant linear com-
binations of T1 and T2 are parabolas.
lines with WE = const. are hyperbolas whose asymp-
totes are straight lines defined by T1 = const. and
T2 = const. These hyperbolas represent the boundary
between separable and entangled states for the particu-
lar case WE(T1,T2) = 0.
Given the hyperbola defined by WE= 0, we can inter-
pret T2as a function of T1, denoting it as F = T2(T1),
and calculate ∂F/∂T1, to obtain
The contour
∂F
∂T1
=Γ11Γ22− Γ12Γ21
(Γ12+ Γ22T1)2. (B2)
The denominator is always positive. To calculate the nu-
merator, we consider a covariance matrix transformed by
local rotations, so that the matrix C is diagonalized [9].
This transformation maintains the Γij invariant. Hence
the numerator in terms of transformed mean values is
Γ11Γ22− Γ12Γ21= −[k1(γcq+ ζcp) + k2(αcq+ βcp)]2
−[αγcq− βζcp− (cpcq− k1k2)(cp− cq)]2,
where
α = ∆2ˆ p1− 1 , β = ∆2ˆ q1− 1,
γ = ∆2ˆ p2− 1 , ζ = ∆2ˆ q2− 1,
cp=1
2?{δˆ p1,δˆ p2}? , cq=1
k1=1
2?{δˆ q1,δˆ q2}?,
2?{δˆ p1,δˆ q1}? , k2=1
2?{δˆ p2,δˆ q2}?.
The numerator is clearly non-positive. Then the hyper-
bola F is a piecewise monotonically decreasing function.
Using G = T1(T2) instead of F, we arrive at the same
result. This negative signal implies that the hyperbolas
branches have only one possible orientation.
Moreover, considering the sign of WE on the vertices
of the square defined by the physical range of attenua-
tion values, only three possibilities remain for WE(T1=
0,T2 = 0) = Wfull and WE(T1 = 1,T2 = 1) = Wppt:
they are either both positive or both negative, or Wfullis
positive and Wpptis negative. These constraints over WE
are obtained noting that, for all T1and T2, WE(T1,T2) is
a necessary and sufficient entanglement witness of the
Gaussian system after the respective attenuation.
Wfull is a particular form of a sufficient entanglement
criterion, the Duan criterion, then we can state that
Wppt≥ 0 ⇒ Wfull≥ 0 or Wfull< 0 ⇒ WE(T1,T2) < 0
for all T1,T2> 0. The last scenario to analyze is when
Wppt≥ 0 and Wfull≥ 0, but there are T1and T2, such
that WE< 0. This situation would imply the possibility
of entanglement distillation by Gaussian operations (real-
ized by attenuations in the situation considered), contra-
dicting a well established result of quantum information
science [22, 23].
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