Experimentally friendly bounds on non-Gaussian entanglement from second moments

0.14 MB·English

by  Gerardo Adesso

Experimentally friendly bounds onnon-Gaussianentanglement from second moments
Gerardo Adesso
SchoolofMathematicalSciences,UniversityofNottingham,UniversityPark,NottinghamNG72RD,UK.and
DipartimentodiMatematicaeInformatica,Universita´ degliStudidiSalerno,ViaPonteDonMelillo,84084Fisciano(SA),Italy
(Dated:January15,2009)
Wedemonstratethattheentanglement inaclassof two-modenon-Gaussian statesobtainedbysubtracting
photonsfromGaussiantwinbeamscanbeboundedfromaboveandfrombelowbyfunctionalsofthesecond
momentsonly. Knowledgeofthecovariancematrixthussufficesforanentanglement quantificationwithap-
preciableprecision. Theabsoluteerrorintheentanglementestimationscaleswiththenon-Gaussianityofthe
consideredstates.
PACSnumbers:03.67.Mn,03.65.Ud,42.50.Dv.
9
0
0
I. INTRODUCTION their preparation. While for Gaussian states all the informa-
2
tionisencodedinthesecondmomentsofthecanonicaloper-
n
ators(collectedin thecovariancematrix),foranyotherstate
a Interfacesbetweenlightandmatterarekeybuildingblocks
aninfinitehierarchyofmomentsisinprincipleneededforan
J ofafuturequantumweb,aglobalsecurecommunicationnet-
exactentanglementquantification[13]. Thistranslates,inex-
2 workwherethemanipulationandtransmissionofinformation
perimentalterms,intothedemandforacompletestatetomog-
2 areregulatedbythequantumlaws[1]. Thetransferofquan-
tum states and the distribution of correlations across the in- raphy[14],aprocesswhichistime-andresource-consuming

] especiallyfortwoormoremodes.
terfaces are enabled by the common mathematical language
h
p of canonically conjugate observables with continuous spec- Inthispaperweprovideanadvanceinthecharacterization
- trum, suchas the quadraturesof lightandthe collectivespin ofnon-Gaussianentanglementwhichreducesthecomplexity
nt componentsof atomic ensembles. It is thusvery fascinating ofitsexperimentaldeterminationexactlytothesamelevelof
a to witness how the second generation of quantum informa- Gaussianstates. Thiscannotbepossibleforanygenericnon-
u tionresearchisfocusingmoreandmore,fromboththeoretical Gaussianstate: henceherewefocusontheimportantclassof
q andexperimentalviewpoints,onthe characterizationofcon- photon-subtractedstates(PSS)underquiterealisticconditions
[
tinuous variable (CV) entanglement and its applications for [10,11],whichrepresentthepreferredresourcesformostcur-
2 communication,computationandmetrology[2]. rentandfutureapplicationsrequiringnon-Gaussianity. Com-
v bining results from the extremality of Gaussian states [15]
Outoftheinfinite-dimensionalHilbertspaceofCVstates,
0 with an analysis of quadrature correlations, we derive ana-
aspecialclassofstateshasplayedaprominentroleinrecent
9 lyticallowerandupperboundswhichindividuatetheentropy
years: Gaussian states. Their mathematical treatment is ad-
1
ofentanglementofpuretwo-modePSS(obtainedfromGaus-
3 vantageousthankstoacompactformalismbasedonsymplec-
sian twin beams by conditionalsubtractionof k photonsper
. tic analysis, anda veryaccuratedegreeofcontrolisreached
1 beam)within a verynarrowband, with relativeerrorvanish-
in their experimental realizations with light and matter [3].
0
ing with increasing entanglement. The bandwidth is linked
Therearehowevermanytaskswhichareimpossiblebyusing
9
to the degree of non-Gaussianity [16] of the analyzed PSS.
0 only the Gaussian states and operationstoolbox(e.g., entan-
Theresultsareextendedtoboundtheentanglementofforma-
: glement distillation [4] and universal quantum computation
v tion of a class of mixed PSS obtained by means of realistic
[5]), and many other tasks which can be sharply improved
i
X bysuitablyresortingtosomenon-Gaussianity(e.g. CV tele- “on/off”typephotodetectors. Crucially, allthese boundsare
only functions of the second moments of the non-Gaussian
r portation [6] and loss estimation [7]). These premises have
a states. A novel method for the fast and reliable reconstruc-
spurredastonishingprogressesin the experimentalengineer-
tion of the complete covariance matrix of optical two-mode
ing of non-Gaussian states, such as Fock states and states
CVstateshasbeenrecentlydemonstrated[17,18]: ourresult
obtained by deGaussifying Gaussian resources via addition
provesthatitcanbeappliedtorealdeGaussifiedPSSaswell
and/or subtraction of single photons [8]. It has been in par-
anditisenoughtoquantitativelyestimateentanglementwith
ticularverifiedexperimentallythataphotonsubtractionfrom
surprisinglyhighaccuracy.
atwo-modesqueezedentangledGaussianstateleadstoanen-
hancementof the entanglement(at fixed squeezing) [9], and
itisknownthatsuchresourcemightbeusedfora moreper-
formant quantum teleportation of classical and nonclassical
II. PRELIMINARIES
states(aprimitiveofthequantuminternet)[6,10,11]andfor
yet-to-be-achieveddemonstrationsofloophole-freeBelltests
WedealwithaCVsystemofN =2bosonicmodes,asso-
ofnonlocality[12]. Thebottleneckforunleashingthepower
ciatedtoaninfinite-dimensionalHilbertspace = F F
of non-Gaussian CV quantum technology has remained the 1 2
H ⊗
[3]. HereF istheFockspaceofeachindividualmodei,de-
quantitativecharacterizationof entanglementin states which i
deviate fromGaussianity, a crucialstep to evaluateand con- scribedbytheladderoperatorsaˆi, aˆ†i satisfyingthecanonical
troltheirusefulnessforapplicationsandthebona-fidenessof commutationrelations[aˆ , aˆ†]=δ . Wecancollectthefield
i j ij
2
quadrature operators into the vector Xˆ = qˆ ,pˆ ,qˆ ,pˆ , ψ = ∞ c(k) n k,n k ,with[11]
with qˆ = aˆ + aˆ† and pˆ = aˆ aˆ†. The{e1xac1t de2scr2ip}- | ki n=k n | − − i
i i i i i − i P
tion ofa genericCV (non-Gaussian)state requiresarbitrary-
n (1 T2λ2)2k+1
ordermomentsofthecanonicaloperators.Ingeneral,thefirst c(k) =(Tλ)n−k − , (1)
momentsX¯ ( Xˆ1 , Xˆ1 , Xˆ2 , Xˆ2 ) canbe adjustedby n (cid:18) k (cid:19)s2F1(−k,−k;1;T2λ2)
≡ h i h i h i h i
localdisplacementswithoutaffectingentanglement:theywill
where F denotes the Gauss hypergeometricfunction. We
besettozerowithoutlossofgenerality. Two-modeGaussian 2 1
defineaneffectivesqueezingparameterrsuchthat
states are henceforth completely specified by the 4 4 real
×
symmetriccovariancematrix(CM)σofthesecondmoments
σ = Xˆ Xˆ +Xˆ Xˆ /2 Xˆ Xˆ [3]. TheCMelements Tλ z =tanhr.
ij i j j i i j ≡
h i −h ih i
for an arbitrarytwo-mode CV state can be efficiently recon-
structedinthelabbymeansofhomodynedetections[17,19]. Thus|ψki=Nk−1/2aˆk1aˆk2|ψ0(r)i,where
Werecallthatanyphysicaltwo-modeCM
∞
n
= [c(0) k!]2
σ = α1 γ Nk n=k n (cid:18)k(cid:19)
γT α X
(cid:18) 2(cid:19) =k!2 F k+1,k+1;1;tanh2r cosh−2rtanh2kr.
2 1
canbeconvertedbylocalunitariesintoa standardform[20]
(cid:0) (cid:1)
whereα = diag a , a andγ = diag γ , γ witha The logarithmic negativity of the PSS states for arbitrary k
i i i x p i
1,γ γ 0.{ } { } ≥ can be computedin simple closed formvia the formula[11]
x p
Ent≥ang|lem|≥ent of a pure bipartite state ψ is universally EN(ψk)=2log[ ∞n=kc(nk)]andreadsinournotation
| i
quantifiedby the Von Neumannentropyof the reducedden-
P
sity matrix of each subsystem (entropy of entanglement E) E (ψ )= log[(1 z)2k+2 F k+1,k+1;1;z2 ].
N k 2 1
[21]. The optimal convex-roof extension of E for a mixed − − (2)
state ̺ definesthe entanglementof formation[21] EF(̺) = NoticethatEN(ψk)increaseswithk(cid:0): iterationoftheph(cid:1)oton
inf{pi,ψi} ipiE(ψi),wheretheinfimumrunsoverallpure- subtraction process further enhances the entanglement com-
state decoPmpositions of ̺ = ipi|ψiihψi|. Another popu- pared to the original Gaussian instance. We aim at an esti-
larmeasureofentanglementforpureandmixedstates,whose mate of the entropy of entanglement of PSS for any k, and
P
computationisingeneralsimpler,isthelogarithmicnegativity specificallyataccurateboundsonthisuniversalentanglement
[21,22]EN =log tr̺Ti [23],wherethepartialtransposi- quantifier which can be measured experimentally with high
| |
tion̺Ti isobtainedfrom̺bytransposingthedegreesoffree- efficiency.WewillnowderivetheCMofPSSstatesinclosed
(cid:2) (cid:3)
domofonesubsystemonly.ForpurestatesE E. Anim- formandshow thatit containsenoughinformation,straight-
N
≥
portantresultinCV entanglementtheoryisthe‘extremality’ forwardlyaccessibleinthelab,forthedesiredtask.
ofGaussianstates: foranyCV state ̺withsecondmoments Denoting by Oˆ an observable involving normally ordered
givenby σ, the correspondingGaussian state ̺G definedby combinationsof ladder operatorson modes 1 and 2, we ob-
thesameCMhassmallerentanglement(quantifiedbyacon- serve that ψ Oˆ ψ = −1 ψ aˆ†kaˆ†kOˆaˆkaˆk ψ , that
tinuousandstronglysuperadditivemeasure)than̺[15]. Itis h k| | ki Nk h 0| 1 2 1 2| 0i
is, we can evaluate expectation values of relevant operators
alsoknownthatforarbitrarytwo-modeGaussianstatesE is
F in terms of normally ordered higher moments computed on
computable,additive,andstronglysuperadditive[24,25].
the unperturbed Gaussian twin beams. Recalling that, for
ψ (r) ,thenormallyorderedcharacteristicfunctionreads[2]
0
| i
χ(N)(α ,α∗,α ,α∗)=exp sinhr[(α α +α∗α∗)coshr
0 1 1 2 2 { 1 2 1 2 −
III. COVARIANCEMATRIXANDENTANGLEMENTOF (α 2+ α 2)sinhr] ,wecanwritethe(normallyordered)
1 2
| | | | }
PUREPHOTONSUBTRACTEDSTATES momentgeneratingformula:
The starting point for the definition of “ideal” PSS is a Mjm = (aˆ†)j(aˆ )i(aˆ†)m(aˆ )l (3)
il ψ0h 1 1 2 2 iψ0
puretwo-modesqueezedGaussianstate(or‘twinbeam’inthe ( 1)j+m ∂i+j+l+m
λTophte=icCaltMalannσhgru0afgoaern)t,dh|iψtsh0es(trap0toe)siiits=ivaelPrre0andiλysnit√nhe1st−saqnuλde2ae|rnzdi,nfngoir,mdwegwhreeitrehe. = ∂−α1i∂α∗1j∂α2l∂α∗2mχ(0N)(α1,α∗1,α2,α∗2)(cid:12)(cid:12)(cid:12)αα∗11,,22==00 .
0 (cid:12)
a = a = cosh2r , γ = γ = sinh2r . The beam
1 2 0 x − p 0 TheformulaEq.(3)canbereadilyappliedtocomputethesec-
1 (2) of ψ is let to interfere, via a beam splitter with
| 0i ondmomentsofthecanonicaloperatorsonournon-Gaussian
transmittivityT (preferablyT closetounity),withavacuum
states. After some algebra, the CM σ of PSS of the form
mode 1′ (2′). The output is a four-mode Gaussian state of k
ψ turnsoutto be thatof a symmetrictwo-modesqueezed
modes 1,1′,2,2′. A photon-number-resolving detection of | ki
thermal state, automatically in standard form, with a =
exactly k photons in each of the two beams 1′ and 2′, con- 1
a a(k) = −1(Mk,k +Mk+2,k+2Mk+1,k+Mk,k),
ditionally projects the state of modes 1,2 into a pure sym- 2 ≡ Nk k+2,k k,k k+1,k k,k
metric non-Gaussian state [26], given in the Fock basis by and γ = γ γ(k) = −1(Mk,k +Mk+1,k+1 +
x − p ≡ Nk k+1,k+1 k,k
3
14 HaL k =1 14 HcL k =10
12 12
10 10
E 8 E 8
6 6
4 4
2 2
0 0
0 1 2 3 4 5 0 1 2 3 4 5
r r
14 HbL k =3 14 HdL mixed, k =1
12 12
10 10
E 8 E 8
6 6
4 4
2 2
0 0
0 1 2 3 4 5 0 1 2 3 4 5
r r
FIG. 1: (color online). Upper (solid) and lower (dotted) bounds on the entanglement of PSS obtained from second moments, plotted as
functionsof thesqueezing r. Theentanglement of formationisindividuated withinthenarrow shaded regionbetweenthebounds. Panels
[(a)–(c)]depictidealPSS|ψki,whoselogarithmicnegativityisshownaswellforreference(thindashedline).Panel(d)depictsamorerealistic
PSS̺k¯=1modeledasabinomialmixtureoftheGaussiantwinbeamandthefirstfourpurePSS,̺k¯=1 = Pnk=0(1−p)n−kpk`nk´|ψkihψk|,
withn=p−1 =4;noticehowonlyslightmodificationsoccurtotheboundscomparedtothecorrespondingidealcase(a).
Mk,k+1+Mk+1,k). Explicitly: quantifies the maximum quadrature correlation between the
k+1,k k,k+1
twomodes[28]. Itistemptingtopostulatethat,forPSS ψ
k
| i
a(k) = (coshr)2+4k[ F (k+1,k+1;1;tanh2r)]−1 with an arbitrary degreek of deGaussification, a functionof
2 1
[2(k+1)2sinh2r F ( k, k;2;tanh2r) γ(k) may yield an overestimate of the actual entanglement
2 1
× − − (which, we remark, is nontrivially encoded in higher-order
+ F ( k, k;1;tanh2r)]; (4)
2 1 correlationstoo). Wecanturnthisblurrybitofintuitioninto
− −
2tanhr(k+1) F (k+1,k+2;1;tanh2r) thefollowing
γ(k) = 2 1 .(5)
F (k+1,k+1;1;tanh2r) Theorem1. Forallk 0,
2 1 ≥
Eup log[1+2γ(k)] E (ψ ) E(ψ ). (7)
We will now show how to extract from the CM elements a k ≡ ≥ N k ≥ k
lowerandanupperboundonE(ψ ).
k Proof. Therightmostinequalityholdsbydefinition. Herewe
Lower bound.—The extremalityof Gaussian states [15] en-
sketch the (quite technical) proofof the leftmost one, which
tailsthat[27]
is one of the main results of this paper. The simple cases
k = 0, 1canbeprovenbyinspection,hencewespecifyhere
Elow E (̺G) E(ψ ), (6)
k ≡ F k ≤ k to arbitrary k 2. Using Eq. (2) and Eq. (5), and expo-
≥
where ̺G is the mixed Gaussian state with CM σ . Ex- nentiatingbothsidesoftheinequality,theproblemreducesto
k k proving that F(k)(z) F k+1,k+1;1;z2 + 4z(k +
p(1li+cxit)l2y,logEklo(w1+x)=2 g[(a1(−kx))2−loγg(k()1]−wx)h2er.e [24] g(x) = 1)2F1 k+2,k+1;1≡;z22 1−(cid:0)(1 −z)−2k−2 ≥ (cid:1)0 where we
4x 4x − 4x 4x recall that tanhr z (0, 1). We can write F(k)(z) as
Upperbounhd.—ThielogarithmicnhegativitiyofpurePSSisal- a powe(cid:0)r series, F≡(k)(z)∈=(cid:1) ∞ f(k)zm, where f(k) =
ready an upper bound for E, however it is defined in terms m=1 m m
ofallthemomentsofthestate,anditsexperimentaldetermi- (1−(−1)m) 2k+m+ 21 P+1 14(−1+(−1k)m)+k+m2 2 −
nation (in real conditions requiring a complete state tomog- 2k+m+1 . Weobservethatf(k) 0andf(k) 0 j 1,
raphy)becomes rather demandingeven when the states take (cid:2) 2k+1 (cid:0) (cid:1)2j−1(cid:3)≥(cid:0) 2j ≤ ∀(cid:1)≥
andmoreoverf(k) f(k) j > k+1. Thisentailsthat
veryspecialforms[9]. Thestrengthofourinvestigationisto (cid:0) (cid:1) 2j−1 ≥ − 2j ∀
by truncating the power series at m = 2k+2 we discard a
derive a slightly looser upper bound but which is a function
ofthesecondmomentsonlyofthePSS,i.e. ofσk. We sim- positive remainder: F(k)(z) ≥ F˜(k)(z) ≡ 2mk=+12fm(k)zm.
plyobservethatfork = 0,namelyforGaussiantwinbeams, Letusnowdefineaparametricclassofhypergeometricsums,
E (ψ ) = arcsinh[γ(0)]. Thestandard-formparameterγ(0) S(k) = 2k+2 2k+l−m+1 f(k). By meansPof Zeilberger’s
N 0 l m=1 l m
P (cid:0) (cid:1)
4
algorithm[29]onecanverifythatSl(k) >0∀k≥2, l≥0. We 2.0 Dkmax Ukmax
will now show that F˜(k)(z) S(k)z2k+2 0 to conclude 1.5
≥ 0 ≥
theproof.WetaketheratioR(k)(z)=[F˜(k)(z)]/[S(k)z2k+2] 1.0
0
and expand it in power series around z = 1−: R(k)(z) = 0.5
1+ ∞ ( 1)l[S(k)/S(k)](z 1)l. Buttrivially(z 1)l = 0.0 k
l=1 − l 0 − − 1 2 3 4 5 6 7 8 91011121314151617181920
( 1)l(1 z)l, and being z 1 the alternating sign is can-
c−ellePdto−yieldR(k)(z) 1. ≤ (cid:3)
FIG.2: (coloronline). Scalingasafunctionofkoftheasymptotic
≥
Wehaveshownthat,quiteremarkably,thesimplemeasure- absoluteerror∆mkaxontheestimateofentanglementviasecondmo-
ment of the CM of a pure PSS enables to pin down the en- ments (dark bars) and of the asymptotic entropic non-Gaussianity
tropyofentanglementquantitativelywithinanalyticalapriori Υmkax(lightbars)forpurek-photon-subtractedstates|ψki.
bounds. In fact, onecan appreciatehow close the lowerand
upper bounds (both functions of the second moments only)
aretoeachotherforvariousvaluesofk inFig.1[(a)–(c)]. A type detectors which are only able to discriminate the vac-
crucialfactisthattheabsoluteerror∆k =(Ekup−Eklow)/2on uum from a bunch of an undefined number of photons[11].
the entanglementquantificationasymptotically saturates (for This means that a more appropriate description of a class
r )toaconstantvalue of PSS must be in terms of statistical mixtures of the form
→∞ ∆max = 1[log(4+8k) 1], ̺thk¯e=‘averkagpek’|ψnkuimhψbke|r,owfhpehreotwonescsaunbdtreaficnteedk¯pe≡r beakmpk[k26a]s.
k 2 − P P
For these mixed states not even the logarithmicnegativityis
being ∆ ∆max for any finite r. Accordingly, since the available in closed form (and its numerical evaluation on a
k ≤ k
actual value of the entanglement (measured by the average computer requires several days for any given squeezing de-
between lower and upper bound) diverges linearly with the gree[11]),letalonetheentanglementofformation. Remark-
squeezing r, the relative error δ = (Eup Elow)/(Eup + ably, our bounds can be immediately extended to pinpoint
k k − k k
Elow) on the estimate of E(ψ ) from the CM vanishes for theentanglementofformationofmixedsymmetricPSSfrom
k k
r 0, rendering our method rigorously accurate. We no- the sole knowledge of second moments. We first observe
≫
ticethat,ingeneral,theerror∆ increaseswithk,althoughit that(havingzerofirstmoments)theCM transformslinearly:
k
staysoftheorderoffewunits–onascalerangingtoinfinity σk¯ = kpkσk, hence it can be computed analytically for
t–raecvteionnfsorpberigbkea(me.)g,.th∆uks≤sca4rcfeolryuapffteocktin=g1th0e00qupahloittyonofsuthbe- tahneycporrorPbeaspboilnitdyindgistGriabuustsioiann{sptkat}efwroimthECqMs.(σ4k¯,5,)a.cTchoerdEinFgotof
estimate [see Fig. 1]. We believethata physicalexplanation theextremalitytheorem[15],standsasalowerboundforthe
forthescalingof∆k isrootedinthefactthatwithincreasing EF of the non-Gaussian mixed PSS ̺k¯ [27]. On the other
k the PSS ψ are increasingly more non-Gaussian, hence hand, denotingby γ(k¯) the (1,3) elementof the mixed-state
k
| i
there is more information not retrievable from second mo- CMσk¯,γ(k¯) = kpkγ(k),theupperboundEq.(7)isimme-
muaetinntsgothnely.enTthroispiacrgnuomn-eGntaucassniabneitmya[d1e6]quΥantoitfatiψveb,ywehviaclh- diatelyextendedPtothemixedcase:log[1+2γ(k¯)]≥EF(̺k¯).
k k Theprooffollowsfromtheconcavityofthelogfunction,the
| i
simply amountsin this case to the Von Neumannentropyof
convexity of the entanglement of formation, and obviously
the associated Gaussian state with CM σk. We obtain, for Theorem1.Namely,log[1+2γ(k¯)] p log[1+2γ(k)]
r 0, ≥ k k ≥
≫ kpkE(ψk)≥EF(̺k¯). ThebehavioProftheboundsinsuch
1 k k+√2k+1+1 morerealisticconditionsisshowninFig.1(d)foraninstance
Υmkax = 2 log 2 +√2k+1log k . Pwith k¯ = 1. We observe, in general, that for reasonable
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) modeling of the mixture (e.g., p following a binomial dis-
k
Foranyr,thenon-Gaussianityandtheabsoluteerrorarevery tribution) the looseningof the boundscomparedto the ideal
close to each other, with Υ ∆ , and exhibit the same cases with k = k¯ is negligible: our scheme is efficient
k k
≥ ⌊ ⌋
scalingwithk (seeFig.2). Thisfascinatingconnectionadds and robust against the specific source of imperfection con-
insight to our analysis and leads us to sum up the results sideredhere(theusageofnon-phon-number-resolvingdetec-
achieved so far as follows. Entanglement in ideal photon- tors).Weplantodeepenourinvestigationinthefuturefollow-
subtracted states can be measured from the covariance ma- ingtheexperimentalprogressesinthegenerationof(generally
trix up to a narrow error that scales with the states’ non- non-symmetric)PSSstates[26],thusproperlymodelingother
Gaussianity. sourcesof imperfections(e.g. mismatches or dark counts in
thephotonconditioning)thatariseinpracticaldemonstrations
[30],inordertotesttherobustnessandreliabilityofourtech-
IV. GENERALIZATIONTOMIXEDSTATESAND niquesforestimatingentanglementinfullyrealisticsituations.
FURTHERREMARKS Inthiscontext,letusbrieflycommentonthedirectimple-
mentation of our results in experiments. Once a two-mode
In most practical implementations, efficient photon- PSSisprepared,oneneedstomeasurethefullCMbyhomo-
number-resolving detectors are not available and the condi- dyne detections as in [17, 19], transformit in standard form
tional generation of PSS is achieved by means of “on/off” (i.e. extract the symplectic invariants a ,a ,γ ,γ ) [3, 20],
1 2 x p
5
andthen(uponverificationthatthestandard-formCMhasthe periments, specifically for the important class of (realistic)
structurepredictedhere: a benchmarkforthestate engineer- photon-subtractedstates [10, 11]. This is the start of a pro-
ing) readily evaluate our lower and upper bounds to ensure gramwhichwillcontinuewiththesystematicinvestigationof
anaccurateestimate ofthe entanglementofformationofthe quantitativeentanglementwitnessesforotherclassesofnon-
producednon-Gaussianstate. Gaussianstatesintermsoflow-ordermoments.Wehopewith
ourresultto stimulate advancesin theengineeringandchar-
acterizationofnon-Gaussianresources,andtheirexploitation
V. CONCLUSION fordemonstrationsimpossibletoachievewithGaussianstates
only, in order to explore the actual limits that quantum me-
In this paper, in the spirit of [15, 28], we have gone be- chanicsposesontheaccessandmanipulationofinformation.
yond the conventional belief that out of Gaussian states the
covariancematrix playsa marginalrole in CV entanglement
quantification. On the contrary, we demonstratedthat clever Acknowledgments
exploitation of such an easily accessible component of the
state, bears extremely useful and precise information on the IthankC.Rodo´,A.Sanpera,F.Dell’Anno,F.Illuminati,V.
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