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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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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

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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-

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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

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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

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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

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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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