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Gaussian intrinsic entanglement Ladislav Miˇsta, Jr.1 and Richard Tatham1 1Department of Optics, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic WeintroduceacryptographicallymotivatedquantifierofentanglementinbipartiteGaussiansys- tems called Gaussian intrinsic entanglement (GIE). The GIE is defined as the optimized mutual information of a Gaussian distribution of outcomes of measurements on parts of a system, condi- tioned on the outcomes of a measurement on a purifying subsystem. We show that GIE vanishes onlyonseparablestatesandexhibitsmonotonicityunderGaussianlocaltrace-preservingoperations and classical communication. In the two-mode case we compute GIE for all pure states as well as forseveralimportantclassesofsymmetricandasymmetricmixedstates. Surprisingly,inallofthese cases, GIE is equal to Gaussian Rényi-2 entanglement. As GIE is operationally associated to the 7 secret-key agreement protocol and can be computed for several important classes of states it offers 1 a compromise between computable and physically meaningful entanglement quantifiers. 0 2 n Sinceitsdiscovery,entanglementhastransitionedfrom [12] a a mere paradoxical feature of quantum mechanics [1] to J a powerful resource for communication and computing. I(A;B ↓E)=Ei→nfE˜[I(A;B|E˜)], (1) 2 The natural quest to discover all facets of entanglement revealedthe need not only to verifyits presence but also where I(A;B|E˜) is the conditional mutual information ] to quantify it. Some crucial properties of entanglement and the infimum is taken over all channels E E˜. The h → p suchasmonogamy[2]arequantitativeandthereforecan- intrinsicinformationisanupperbound(notalwaystight - notbedescribedwithoutintroducingentanglementmea- [13]) on the rate at which a secret key can be generated t n sures. Further, entanglement measures are required for from the investigated distribution and what is more, it a the characterization of entangling gates [3] and set the is conjectured that it is equal to a secret key rate in u bounds one has to surpass in experiments into some key the modified key agreement protocol called public Eve q quantuminformationprotocolssuchasentanglementdis- scenario [14, 15]. [ tillation [4]. The intrinsic information (1) can be used to quan- 1 tify entanglement in a quantum state ρ . This is ac- AB Existing entanglement measures either possess a good v complished by projective measurements in some bases 2 operational meaning or are computable but not both. A and B , and a generalized measurement with 3 The first kind of measure is best exemplified by the dis- {| i} {| i} a generating set E on subsystems A,B and E of 3 tillable entanglement [5], which quantifies the pure-state {| i} a purification Ψ of the state, Tr Ψ Ψ = ρ . If 0 entanglementonecandistillfromasharedquantumstate | i E| ih | AB the state ρ is entangled (separable) and the ba- 0 AB but is difficult to compute. At the opposite extreme is . sis (set) A and B ( E ) is chosen suitably, 1 thelogarithmicnegativity[6,7],whichiscomputablefor {| i} {| i} {| i} the obtained distribution P(A,B,E) = A B E Ψ 2 0 anystatebutlacksacoherentoperationalinterpretation. |h |h |h | i| has strictly positive (zero) intrinsic information for any 7 However, to assess the utility of a given entangled state 1 choice of the set (basis) E ( A and B ) [16]. inpracticaltasks,weneedtodevelopentanglementmea- {| i} {| i} {| i} : Thus,tofaithfullymapentanglementontosecretcorrela- v sures which are both computable and physically mean- tions the optimized intrinsic information [16], µ(ρ ) = i ingful. Unfortunately, except for the entanglement of AB X inf sup [I(A;B E)] , has to be taken, formation [5], which quantifies how much pure-state en- {|Ei,|Ψi}{ {|Ai,|Bi} ↓ } r which exhibits some properties of an entanglement mea- a tanglement one needs to create a shared quantum state sure, namely equality to the von Neumann entropy on andwhichcanbe computedfortwoqubits[8]andGaus- pure states and convexity. It might seem that µ is a sian states [9, 10], no such measure is currently known. goodcandidateforthesoughtentanglementmeasurebut One wayto probe the gapis to quantify entanglement ithastwodrawbacks. First,itisnotknownwhetheritis in the context of classical secret key agreement [11]. In non-increasing under local operations and classical com- thiscryptographicprotocol,threerandomvariablesA,B munication (LOCC) as it should [17]. Second, so far it and E distributed according to P(A,B,E) are held by has been computed only for a two-qubit Werner state two honest parties, Alice and Bob, and an adversary, [16]. Eve. Alice and Bobare connected by a public communi- In this Letter we propose a quantifier of bipartite en- cation channel and their goal is to generate a secret key, tanglement called intrinsic entanglement (IE) defined as thatis acommonstringofrandombits aboutwhichEve has practically no information. For the key agreement to be possible it is necessary that Alice and Bob share E↓(ρAB)= sup inf [I(A;B E)] . (2) correlations which cannot be distributed by public com- {|Ai,|Bi}(cid:26){|Ei,|Ψi} ↓ (cid:27) munication, i.e., secret correlations. A useful quantifier The IE contains a reverse order of optimization in com- ofsecretcorrelationsisthesocalledintrinsicinformation parison with the quantifier µ and hence E µ due to ↓ ≤ 2 themax-mininequality[18]. ThemainadvantageofIEis ceptional measure of Gaussian entanglement which has that one cancompute it easierthanµ aswe show below. cryptographic interpretation and many important prop- WefocusonIEforanimportantclassofGaussianstates. erties, and which can be computed in many cases. These states are the backbone of quantum information We consider quantum systems with infinite- technologiesbasedoncontinuousvariables[19]andoccur dimensional Hilbert state space, e.g., light modes. as ground or thermal state of any bosonic quantum sys- An n-mode system is characterized by a vector of temina“linearized”approximation[20]. Gaussianstates quadratures ξ = (x ,p ,...,x ,p )T whose components 1 1 n n canbealsoeasilyprepared,manipulatedandmeasuredin obey the canonical commutation rules [ξ ,ξ ] = i(Ω ) j k n jk manyexperimentalplatformsencompassinglight,atomic with Ω = n iσ , where σ is the Pauli-y matrix. n ⊕j=1 y y ensembles, trapped ions or optomechanical systems [21]. Gaussian states are fully described by a covariance ma- Unfortunately, evaluation of IE for Gaussian states in- trix (CM) γ with entries γ = ξ ξ +ξ ξ 2 ξ ξ jk j k k j j k h i− h ih i volvescomplexoptimizationovergenerallynon-Gaussian and by a vector of first moments ξ , which is irrelevant h i measurements and states, and thus some simplifications in the present entanglement analysis and so is assumed are needed. First, we restrictAlice and Bobto Gaussian to be zero. We use Gaussian unitary operations which measurements. We do that because a scheme generating areforn modesrepresentedatthe levelofCMsbyareal distributionP representsthefirststageofaquantumkey 2n 2n symplectic matrix S fulfilling SΩ ST =Ω . We n n × distribution protocol (with an individual attack) and in restrict ourselves to Gaussian measurements described protocols based on Gaussian states honest parties typi- by the positive operator valued measure [28] cally perform Gaussian measurements [19]. Second, we assumethatitisoptimaltouseaGaussianmeasurement Π(d)=(2π)−nD(d)Π D†(d), (3) 0 and channel on Eve’s side. This assumption is plausi- ble as Gaussian attacks are optimal [22] with respect to which satisfies the completeness condition the lower bound on the secret key rate for all important Π(d)d2nd = 11, where d2nd = Πn dd(x)dd(p). Gaussian protocols. HRe2rne, the seed element Π is a norml=a1lizeld dlen- R 0 sity matrix of a generally mixed n-mode Gaus- Here we thus investigate the so called Gaussian IE sian state with zero first moments and CM Γ, (GIE)definedbyEqs.(1)and(2), whereallstates,mea- D(d) = exp( idTΩ ξ) is the displacement opera- surements and channels are Gaussian. We show that − n for a Gaussian state ρAB, the GIE is equal to the opti- tor, and d=(d(1x),d(1p),...,d(nx),d(np))T ∈R2n is a vector mized mutual information of a distribution of outcomes of measurement outcomes. of Gaussian measurements on subsystems A and B of a Simplification of GIE.—Initially we show that the as- conditional state obtained by a Gaussian measurement sumptionofGaussianityofallstates,measurements,and on subsystem E of a Gaussian purification of the state the channel E E˜, considerably simplifies the quantity → ρAB. Next,weprovethatGIEisfaithful,i.e.,itvanishes (2). Assume that ρAB ≡ ρA1...ANB1...BM of Eq.(2) is an iffρAB isseparable,anditdoesnotincreaseunderGaus- (N + M)-mode Gaussian state with CM γAB. Let us sian local trace-preserving operations and classical com- furtherassumethat Ψ ABE isaGaussianpurificationof | i munication (GLTPOCC). Finally, we compute GIE ana- thestatewithCMγπ,whichcontainsK purifyingmodes lyticallyforseveralimportantclassesofmixedtwo-mode E1,E2,...,EK. Considernow,thatthesubsystemsA,B symmetricandasymmetricGaussianstates. Astheopti- and E are distributed among Alice, Bob and Eve, who muminGIEisalwaysreachedbyfeasiblehomodyneand carryoutlocalGaussianmeasurements(3)characterized heterodyne detection, it is anexperimentallymeaningful by covariance matrices (CMs) ΓA,ΓB and ΓE, respec- quantitywhichstaysinlinewithotheroptimizedquanti- tively. As a result, the participants share a zero-mean tiessuchasGaussianquantumdiscord[23–25],wherethe GaussiandistributionP(dA,dB,dE)ofmeasurementout- optimumisalsooftenattainedbyhomodyningorhetero- comes dA,dB and dE with a classical covariance matrix dyning. Remarkably,wefindfurther,thatthecalculated (CCM) [29] Σ expressed with respect to the AB E par- | GIE is always equal to an important measure of Gaus- titioning as sianentanglementcalledGaussianRényi-2(GR2)entanglement [26], which is defined as a convex roof of the Σ= γAB +ΓA⊕ΓB γABE α β , (4) pure-state Rényi-2 entropy of entanglement. The GR2 (cid:18) γATBE γE +ΓE (cid:19)≡(cid:18)βT δ (cid:19) entanglement is a proper and natural measure of Gaus- sian entanglement being monotonic under all Gaussian where γAB,γABE and γE are blocks of the CM γπ ac- LOCC (GLOCC) and monogamous. Additionally, the cording to the same partitioning. In what follows, we GR2 entanglement is additive on two-mode symmetric analyze GIE defined by Eq. (2), where the role of the states and finds interpretation in terms of a phase-space distribution P(A,B,E) is playedby the Gaussiandistri- sampling entropyfor Wigner function [26,27]. Ourfind- bution P(dA,dB,dE) and the optimization is performed ingsleadustoaconjecture,thatGIEandGR2entangle- overGaussianchannelsE E˜ andCMs γπ andΓA,B,E. → ment are equal on all Gaussian states. If the conjecture First, we identify the conditional mutual information is true,allpropertiesofthe latterquantityextendto the I(A;B E) of Eq. (1) for the distribution P(d ,d ,d ). A B E | former and vice versa, thereby providing us with an ex- According to definition [30] I(A;B E) is the standard | 3 mutual information I (A;B) of the conditional distribu- channelcanbeincorporatedintoEve’smeasurement,we c tion P(d ,d d ) averaged over the distribution of the can omit the minimization with respect to the channels A B E | variable d . The distribution is Gaussian with a CCM without loss of generality. Furthermore, as for any pu- E given by the Schur complement [31, 32] of CCM (4) rification and measurement on subsystem E there is a measurementon subsystemE ofa fixed purificationgiv- 1 σ = γ +Γ Γ γ γT , (5) ing the same conditionalmutual information (6), we can AB AB A⊕ B− ABEγE +ΓE ABE further restrict ourselves in the definition of GIE to a fixed purification and minimization with respect to all where the inverse is to be understood generally as the CMs Γ . Consequently, GIE simplifies to pseudoinverse. Making use of the formula for mutual E information of a bivariate Gaussian distribution [33], we EG(ρ )= sup inff(γ ,Γ ,Γ ,Γ ), (8) arrive at Ic(A;B)=f(γπ,ΓA,ΓB,ΓE), where ↓ AB ΓA,ΓBΓE π A B E f(γ ,Γ ,Γ ,Γ )= 1ln detσAdetσB (6) where f is given in Eq. (6), γπ is CM of a fixed purifica- π A B E 2 detσ tion and the infimum (supremum) is taken over all CMs (cid:18) AB (cid:19) Γ (Γ ) of measurements on subsystem E (A and B). E A,B with σA,B being local submatrices of CCM (5). From Faithfulness.—WefirstprovethatGIEvanishesiffρAB Eq.(6)itthenfollowsthatIc(A;B)isindependentofdE is separable. The proofcloselyfollowsasimilarprooffor and hence I(A;B E)=Ic(A;B)=f(γπ,ΓA,ΓB,ΓE). intrinsic information given in Ref. [16]. The “only if” | We next provethat the channel E E˜ in Eq. (1) can part has been proved in Ref. [35]. It follows from the → beintegratedintoEve’smeasurement. Again,weassume fact, that any separable Gaussian state has a Gaussian a Gaussian channel [34], d˜E = XdE +y, mapping the purificationwhich canbe projectedonto a productstate 2K 1 vector dE of Eve’s measurement outcomes onto of subsystems A and B by a suitable measurement on a ne×w L 1 vector d˜E. Here X is a real matrix and subsystem E. Hence, after the measurement the CCM y = (y1,y×2,...,yL)T is a random vector obeying a zero (5)reducestoσAB =(γA+ΓA) (γB+ΓB)foranylocal ⊕ mean Gaussian distribution with CCM Y with elements Gaussian measurements on subsystems A and B, where Yij =2hyiyji. The channel transforms the CCM (5) to γA,B denotelocalCMsoftheproductstate. Thisimplies thatthemutualinformation(6)vanishesforanyCMsΓ 1 A σÃB = α−βXTXδXT +Y XβT, (7) and ΓB, and therefore E↓G(ρAB)=0 as required. The “if” part can be proved by contradiction. Let whereα,β andδareblocksofCCM(4). Withthehelpof E↓G(ρAB) = 0 for some entangled state ρAB. Then, for the singularvalue decomposition[31]ofmatrixX,CCM any CMs ΓA and ΓB there is a CM ΓE such that the (7) can be recast after some algebra into the form (5) mutual information in Eq. (6) vanishes. This is equiva- with CM Γ replaced with a different CM (see [35] for lenttothestatisticalindependenceofthevariablesobey- E the explicit formof the new CM). Therefore,a Gaussian ing the respective bivariate Gaussian distribution with measurement on Eve’s system followed by a Gaussian CCM σAB [30], which implies that σAB = σA σB. ⊕ channel on outcomes of the measurement is equivalent Hence, for a Gaussian measurement ΠE(dE) with CM to another Gaussian measurement, which concludes the ΓE the corresponding (unnormalized) conditional state proof. TrE[Ψ ΨΠE(dE)] factorizes. By integrating the latter | ih | Further simplification follows from the invariance of state over all measurementoutcomes dE and taking into CCM (5) under a change of which purification state is accountthe completenessconditionforthe measurement used, accompanied by a corresponding change to Eve’s ΠE(dE) one gets an expression of the state ρAB in the measurement. Namely, for any two CMs γ and γ¯ of formofaconvexmixture ofproductstates andtherefore π π purifications with K-mode and K¯-mode purifying sub- the state is separable, which is a contradiction. Thus, system E and E¯, where K K¯, there is a symplec- equalityE↓G(ρAB)=0 implies separabilityofρAB which tic matrix on subsystem E¯ w≤hich brings CM γ¯ to CM accomplishes the proof. π γπ [11⊕(K¯−K)] [35–37]. As a result, for CM γ¯π and Monotonicity.—ForGIEtobeagoodGaussianentan- CM⊕Γ¯E¯ of a measurement on subsystem E¯, which pos- glement measure it should not increase under GLOCC [17, 38]. This means, that if such an operation maps sess CCM σ¯ , Eq. (5), there is for CM γ a CM Γ of AB π E E aninputGaussianstateρ ontoanoutputGaussian a measurement on subsystem E giving σAB =σ¯AB, and AinBin state ρ , then vice versa [35]. Accordingly, there is a free choice over AoutBout which Gaussian purification of ρAB to work with. EG(ρ ) EG(ρ ). (9) The proposed quantity GIE is defined as the condi- ↓ AinBin ≥ ↓ AoutBout tional mutual information I(A;B E˜) given in Eq. (6), We here outline the proof of inequality (9) for the sub- | where σAB is replaced with σÃB, Eq. (7), which is first set of GLOCC given by GLTPOCC (see [35] for the de- minimized with respect to all Gaussian channels E E˜ tailed proof). First, we construct a suitable purification → and subsequently CMs Γ and γ of measurements and of the output state ρ . For this purpose, we use E π AoutBout purifications, respectively, and then maximized with re- realization of the operation by a continuous-variable E specttoallpure-stateCMsΓ andΓ . AsanyGaussian teleportation protocol [39], where a quantum channel is A B 4 aGaussianstateχrepresentingtheoperation[32,40,41] χ onto a product state and Ψ onto the state | i | iAinBinEin (seeFig.1(a)). Here,theinputstateρ isteleported ρ , and hence the right-hand side of the first AinBin AinBin|Ein via input subsystems A and B of the state χ inequality is equal to the mutual information of out- 1 1 A1B1A2B2 to the output subsystems A and B by Bell measure- comes of measurements with CMs Γ and Γ on 2 2 Aout Bout ments on composite subsystems (A A ) and (B B ). the state ( )(ρ ), where are opera- in 1 in 1 EA ⊗EB AinBin|Ein EA,B The measurements comprise separate measurements of tionswithzerodisplacementsduetotheindependenceof the difference of the x-quadratures and the sum of the mutualinformationfromdisplacements. Since operation p-quadratures on each corresponding pair of modes. Af- , j = A,B, is trace-preserving, it can be realized by a j E ter the measurementsandsuitable displacementsof sub- Gaussian unitary operation U on input system j and j in systems A and B , the output state ρ is ob- vacuum ancilla j , followed by discarding of a part of 2 2 AoutBout anc tained on subsystems A and B . Now, by replac- outputsystem, j ,andaddingclassicalGaussiannoise out out disc ing states ρ and χ with their purifica- [43] (see Fig. 1(b)). The noise can be integrated into a tions Ψ AinBinand χA1B1A2B2 , respectively, and new measurement with CM Γ¯ and we can also work | iAinBinEin | iA1B1A2B2Eχ jout teleporting the respective parts of the purifications, we with a measurement on a larger system (j j ) with out disc get the sought purification Φ of the out- CMΓ¯ Γ ,becauseitnevergivesasmallermutual | iAoutBoutEχEin jout⊕ jdisc put state ρ (see Fig. 1(a)). As the operation information than the original measurement [30]. More- AoutBout over, unitary U can be integrated into a new measure- j (a) (b) ment with CM Γ on system (j j ), which yields jinjanc in anc thesamemutualinformationwhichcanbefurtherrewrit- ten in terms of some measurements with CMs Γ˜ and Ain Γ˜ on state ρ [35]. Therefore, the second in- Bin AinBin|Ein equality holds. Finally, CMs Γ˜ and Γ˜ cannot give Ain Bin a largermutual informationthan optimal CMs Γ and Ain Γ , andthus the last inequality is fulfilled, which com- Bin pletes the monotonicity proof. Computability.—GIEcanbecalculatedanalyticallyfor several classes of two-mode Gaussian states. Without loss of generality [35] we can take CMs of the states in the standard form [9, 44] a11 κ γ = (10) FIG. 1: (a) Construction of the purification of the output AB κ b11 (cid:18) (cid:19) state ρAoutBout via teleportation. BMj: Bell measurement on subsystem (jinj1), Dj: displacement of subsystem j2. (b) with κ = diag(kx,−kp), where kx ≥ kp ≥ 0. We evalu- DecompositionofoperationEj anditsintegrationintoamea- ate GIE both for symmetric states with a=b as well as surement. forsomeasymmetricstates. First,we calculateaneasier computable upper bound on GIE obtained by reversing is GLTPOCC, the state χ is a Gaussian mixture of the order of optimization in its definition (8). Next, we E displaced Gaussian product states χ (r) χ (r), find for some fixed measurements on modes A and B a A1A2 ⊗ B1B2 where r characterizes the displacement, which represent measurementonsubsystemE givingminimalf whichat products of local Gaussian trace-preserving operations the same time saturates the bound (see Appendix [45] [34, 42], (r) (r). Consequently, we can take the for details). It turns out, that for all symmetric (asym- A B E ⊗E purification χ inthe formforwhichthereis metric)statesconsideredhere,GIEisachievedbydouble | iA1B1A2B2Eχ a measurement with CM Γ˜ on subsystem E , which homodyne detection on modes A and B, and homodyne Eχ χ projects it onto the product states. detection (heterodyne detection, i.e., projectiononto co- Let γk and Γ , j = A,B,E, now denote CMs of herent states) on subsystem E. We have found GIE for the discπussed pujkrification and optimal measurements the following three sets of states: for the state ρ , k = in,out, i.e., EG(ρ ) = (i) Symmetric GLEMS [55].— The states ( ρ(1)) AkBk ↓ AkBk ≡ AB fqu(γeπnk,cΓeAokf,tΓhBekf,oΓllEowk)i.ngThcheaiinneqoufainlietyqu(a9l)itiisest:hen a conse- haave1/o(nae+uknpi)tasnydmaplseucbtiscysetiegmenEvaliusesin[5g6l]e-wmhoednec.eFkoxra=ll − the states GIE reads as [45] EG(ρ ) f(γout,Γ ,Γ ,Γ˜ Γ ) ↓ AoutBout ≤ π Aout Bout Eχ ⊕ Ein a f(γin,Γ˜ ,Γ˜ ,Γ ) EG(ρ ). EG ρ(1) =ln . (11) ≤ π Ain Bin Ein ≤ ↓ AinBin ↓ AB  a2 k2 (cid:16) (cid:17) − p Thefirstinequalityissatisfiedbecausef cannotdecrease q  by replacing the optimal measurement having CM Γ If k = k symmetric GLEMS satisfy a2 k2 = 1 and by a (generally suboptimal) product measurement wEiotuht therxefore pthey reduce to pure states ( ρp−).pEquation CM Γ˜ Γ . Next, the latter measurement projects (11) then gives EG(ρp )=ln(a) [35].≡ AB Eχ ⊕ Ein ↓ AB 5 (ii) Symmetric squeezed thermal states [57].—The quantifier of Gaussian entanglement GIE which com- states ( ρ(2)) fulfil the condition k =k k and they promises between computability and operational signifi- are enta≡nglAedBiff a k < 1 [9, 44]. Fxor allpt≡he entangled cance. Closed formulae for GIE for two classes of sym- states which satisfy−a 2.41 GIE is equal to [45] metric states have been obtained, Eqs. (11) and (12), ≤ which canbe compactly written as EG(ρ )=ln [ν˜ + ↓ AB { − EG ρ(2) =ln (a−k)2+1 , (12) (ν˜−)−1]/2} if ν˜− <1 and E↓G(ρAB)=0 if ν˜− ≥1, where ↓ AB 2(a k) ν˜ = (a k )(a k ). Interestingly, this is nothing (cid:16) (cid:17) (cid:20) − (cid:21) − − x − p but GR2 entanglement for symmetric states [9, 26]. As whereas for separable states EG(ρ(2)) = 0 by faithful- the GIEpfor some asymmetric states, Eq.(13), also coin- ↓ AB ness. cideswiththeGR2entanglement[45]weconjecture,that (iii) Asymmetric squeezed thermal GLEMS.—The the two quantities are equivalent. The confirmation or states ( ρ(3)) fulfill the condition k = k k and refutationoftheconjectureaswellasanalysisoftheother possess≡oneAunBitsymplectic eigenvalue.xForalplt≡he states propertiesofGIEisleftforfutureresearch. Wehopethat for which √ab 2.41 GIE is given by [45] the presentresults willstimulate further studies ofphys- ≤ ically meaningful computable entanglement measures. a+b EG ρ(3) =ln . 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A 67, 052311 (2003). 1 Supplementary Information Gaussian intrinsic entanglement Ladislav Miˇsta, Jr. and Richard Tatham Appendix A: GIE for two-mode Gaussian states Here σ = diag(1, 1) is the diagonal Pauli-z matrix, z OI×J is the I J −zero matrix, 11 is the 2 2 identity × × For a Gaussian state ρ of two modes A and B GIE matrix and S is a symplectic matrix, i.e., a 4 4 real AB × is defined explicitly as matrix satisfying the symplectic condition EG(ρ )= sup inff(γ ,Γ ,Γ ,Γ ), (A1) SΩST =Ω, (A9) ↓ AB π A B E ΓA,ΓBΓE where where 2 1 detσ detσ 0 1 f(γ ,Γ ,Γ ,Γ )= ln A B (A2) Ω= J, J = , (A10) π A B E 2 detσ 1 0 (cid:18) AB (cid:19) i=1 (cid:18)− (cid:19) M with that brings the CM γ to the Williamson normal form AB [56] σ = γ +Γ Γ , (A3) AB AB|E A B ⊕ where σA,B are local submatrices of σAB and ΓA and SγABST =diag(ν1,ν1,ν2,ν2). (A11) Γ are single-mode CMs of Gaussian measurements on B Hereν ν 1arethesocalledsymplecticeigenvalues modes A and B, respectively. Here, 1 2 ≥ ≥ ofCMγ andR=1,2is the numberof the symplectic AB γ = γ γ 1 γT (A4) eigenvaluesstrictlygreaterthanone. TheuseofEq.(A7) AB|E AB − ABEγ +Γ ABE on the right-hand side (RHS) of Eq. (A4) further yields E E for the CM γ the expression AB|E isaCMofaconditionalstateρ [32]ofmodesAand AB|E B obtainedby a Gaussianmeasurementwith CM Γ on E γ =S−1γ(0) (S−1)T (A12) purifying subsystem E of the minimal purification [46] AB|E AB|E of the state ρ , where γ is a CM of the state ρ AB AB AB with and γ and γ denote the blocks of the CM of the ABE E purification ( γ ) expressed with respect to the AB E 1 ≡ π | γ(0) = γ(0) γ(0) (γ(0) )T, (A13) splitting, i.e., AB|E AB − ABEγ(0)+Γ ABE E E γ γ γπ =(cid:18)γATABBE AγBEE (cid:19). (A5) whereγA(0B) deno(t0e)stheWilliamsonnormalform(A11)of CM γ , i.e., γ =diag(ν ,ν ,ν ,ν ). If the Gaussian state ρ is a pure state ( ρp), the AB AB 1 1 2 2 AB In order to calculate the GIE we now need to express ≡ off-diagonal block γ is a zero matrix and the GIE ABE CM (A13) as well as the symplectic matrix S appearing then can be calculated easily. In the main text as well in Eq. (A12) in terms of the elements of the CM γ . AB as in Ref. [35] it was shown, that in this case the GIE For this purpose it is convenient to express the CM in a coincideswiththeGaussianRényi-2(GR2)entanglement block form with respect to the AB splitting, ( EG(ρp)) [26], | ≡ 2 A C E↓G(ρp)=E2G(ρp)= 21ln(detγA), (A6) γAB =(cid:18)CT B (cid:19). (A14) where γ is a CM of the reduced state of mode A of the Owing to the invariance of GIE (A1) under the Gaus- A state ρp. sian local unitary operations [35] we can without loss of In what follows, we focus on calculation of GIE for generality assume CM (A14) to be in the standard form mixedtwo-modeGaussianstates. The statespossessthe [44], blocks γ and γ of the form [35] ABE E a 0 c 0 x γABE =S−1γA(0B)E, γE =γE(0), (A7) γ = 0 a 0 cp (A15) AB  c 0 b 0  x where 0 c 0 b  p    γA(0B)E = RiO=1 νi2−1σz , γE(0) = R νi11.(A8) withcx ≥|cp|≥0. Sincestateswithcxcp ≥0aresepara- (cid:18)L 2(2p−R)×2R (cid:19) i=1 ble[44]andthuspossesszeroGIE[35],incalculationswe M 2 can restrict ourself only to CMs satisfying c c < 0. In- quantum state if and only if a2 k2 1 and that the x p − x ≥ troducing new more convenient parameters k c and CM corresponds to an entangled state if and only if 1> x x ≡ k c = c ,wearriveatthefollowingstandard-form (a k )(a k ) [9]. p p p x p ≡| | − − − CM which we shall consider in what follows [9]: From Eq. (A17) it further follows that the symplectic eigenvalues of CM (B2) read explicitly as a 0 k 0 x 0 a 0 k γAB =kx 0 b −0p , (A16) ν1 = (a+kx)(a−kp), 0 k 0 b q  − p  ν2 = (a−kx)(a+kp). (B3) where kx kp >0. q ≥ As for the symplectic matrix S, we calculate it using The symplectic eigenvalues of CM (A16) can be cal- the method of Ref. [47]. Here, we seek the matrix in the culated conveniently from the eigenvalues of the matrix form of a product S = 2 V∗ WT, where iΩγAB whichare of the form ν1, ν2 [6]. In terms of ⊕i=1 {± ± } parameters a,b,k and k they read explicitly as x p (cid:0) 1 i (cid:1) i V = − (B4) √2 1 1 ∆ √D (cid:18) (cid:19) ν = ± , (A17) 1,2 s 2 and W contains in its columns the eigenvectors of the matrix iΩγ which are chosen such that S is real, it AB where satisfies the symplectic condition (A9) and it does not ∆ = a2+b2 2k k , mix position and momentum quadratures. Thus we find x p − the symplectic matrix S that brings the CM (B2) to the D = ∆2 4detγAB Williamsonnormalform (A11) to be the following prod- − = a2 b2 2+4(ak bk )(bk ak ). uct x p x p − − − (A18) (cid:0) (cid:1) S =(SA SB)UBS. (B5) ⊕ Similarly, we can express the symplectic matrix S Here, whichbringsCM(A16)toWilliamsonnormalform(A11) in terms of parameters a,b,kx and kp. This can be done 1 11 11 U = (B6) usingeitheramethodofRef.[47]oramethodofRef.[48]. BS √2 11 11 Foragenerictwo-modeCM(A16)theformofthematrix (cid:18)− (cid:19) S iscomplexandthereforewedonotwriteithereexplic- is a matrix describing a balanced beam splitter and itly. In what follows, we work with particular subclasses oftheclassofgenerictwo-modeGaussianstatesforwhich S = zA−1 0 , S = zB 0 (B7) S attains a simple form which is presented explicitly in A (cid:18) 0 zA (cid:19) B (cid:18) 0 zB−1 (cid:19) the respective subsection. with zA = 4 aa+−kkxp >1 and zB = 4 aa−+kkxp > 1 are matrices correspqonding to local squeeziqng transformations in Appendix B: GIE for symmetric states quadratures x and p , respectively. A B Makinguseofthesymplecticeigenvalues(B3)andthe In this section we calculate GIE defined in Eq. (8) of symplectic matrix (B5) we can now express CM (A12) themaintextforsomesubclassesoftheclassoftwo-mode solely in terms of the parameters a,k and k and the x p symmetric Gaussianstates. The states are characterized CM Γ . However, this form of CM (A12) is not suit- E by the condition a=b whence their standard-form CMs able for analytical calculation of the GIE, because the (A15) and (A16) reduce to CM depends on CM Γ that we minimize over in the E definition of GIE, Eq. (A1), in a complicated way via a 0 c 0 x the term 1 . Although the term can be calculated γ = 0 a 0 cp (B1) γE(0)+ΓE AB  c 0 a 0  explicitly for an arbitrary two-mode Gaussian state, the x 0 c 0 a obtained form of CM (A13) still depends on elements of  p    CM ΓE in a way which is too complicated for analytical and calculations. Nevertheless, there are some subclasses of the set of symmetric states for which the CM simplifies a 0 k 0 x such that the GIE can be calculated analytically. These 0 a 0 k γAB =k 0 a −0p , (B2) states can be identified if one realizes that the dimen- x (0) sionof CM γ andtherefore also CM Γ is determined  0 −kp 0 a  by the numbeEr R, which appears in Eq. E(A8) and which respectively. For the sake of further use let us also recap denotesthenumberofsymplecticeigenvaluesofarespec- here that the matrix (B2) describes a CM of a physical tive two-mode Gaussian state which are strictly greater 3 than one. Therefore, the set of mixed two-mode Gaus- 3.0 sian states splits into two subsets containing states with R = 1 and R = 2, respectively. States with R = 1 are 2.5 moresimplebecauseforthemthematrixγ(0)+Γ isjust E E single-mode and therefore the inverse 1 is simple. γE(0)+ΓE x2.0 In the following subsection we show, that for symmetric V stateswithR=1theGIEcanbecalculatedanalytically. The next subsectionis then dedicatedto analyticaleval- 1.5 uation of GIE for a subclass of more complicated symmetric states with R=2. Specifically, in this subsection 1.0 we calculate GIE for a subclass of the symmetric states 0.5 1.0 1.5 2.0 2.5 3.0 satisfyingν =ν [57],whicharethesocalledsymmetric 1 2 V squeezed thermal states. p FIG. 2: (Color online) An example of the set M (gray area) 1. GIE for symmetric GLEMS for ν = 3. Solid red curve and dashed blue curve depict the boundary curvesVx =1/Vp and Vx =Vp, respectively. (1) Letusconsideratwo-modeGaussianstateρ witha AB CMγ(1) andR=1. Sinceν =1holdsforthisstateitis AB 2 For symmetric GLEMS we can further insert for the aGaussianstatewithapartialminimaluncertainty. This symplecticmatrixS ontheRHSofEq.(B10)thedecom- state is also known to be the Gaussian least entangled position (B5) which yields state for given global and local purities (GLEMS) [55] and it possessesthe other symplectic eigenvalueequal to γ(1) =UT S−1γ (ν)(S−1)T γsq U , (B13) AB|E BS A A|E A ⊕ B BS ν ≡ν1 = detγA(1B) (B8) where γBsq = SB−1((cid:2)SB−1)T = diag(zB−2,zB2), a(cid:3)nd where we q have used the orthogonality of the matrix (B6). as can be seen fromEq. (A11). Because R=1, the min- TheformofCM(B13)isnowsuitableforcarryingout imal purification of GLEMS contains only a single puri- of the last step of calculation of GIE, Eq. (A1), which fying mode E and therefore also the CM Γ appearing E is the optimization of the function (A2) with respect to in CM (A13) is single-mode. CMs Γ ,Γ andΓ . Here,we performthe optimization Let us assume now the CM in the form Γ = A B E E by means of the method, which was used in Ref. [35] P(ϕ)diag(V ,V )PT(ϕ), where x p for calculation of the GIE for a particular instance of symmetric GLEMS given by the two-mode reduction of cosϕ sinϕ P(ϕ)= − (B9) the three-mode continuous-variable (CV) Greenberger- sinϕ cosϕ (cid:18) (cid:19) Horne-Zeilinger (GHZ) state [49]. In this method, we with ϕ [0,π), V = τe2t and V = τe−2t, where τ 1 calculate the GIE for a Gaussianstate ρ(1) in two steps. x p AB and t ∈0. Using relation PT(ϕ) = σ P(ϕ)σ one g≥ets First, we calculate an easier computable upper bound z z ≥ after some algebra that the CM (A12) can be expressed as U ρ(1) inf sup f(γ ,Γ ,Γ ,Γ ) (B14) AB ≡ ΓEΓA,ΓB π A B E γA(1B)|E =S−1 γA|E(ν)⊕11B (S−1)T. (B10) on GIE,(cid:16)EG((cid:17)ρ(1)) U(ρ(1)). In the second step, Here, (cid:2) (cid:3) we find for↓parAtBicula≤r fixedABCMs Γ and Γ an op- A B timal CM Γ˜ which minimizes f(γ ,Γ ,Γ ,Γ ), i.e., γ (ν)=PT(ϕ)diag( , )P(ϕ), (B11) E π A B E A|E Vx Vp f(γπ,ΓA,ΓB,Γ˜E) = infΓEf(γπ,ΓA,ΓB,ΓE), and which where at the same time saturates the upper bound, i.e., U(ρ(1))=f(γ ,Γ ,Γ ,Γ˜ ). Asaconsequence,theGIE νV +1 νV +1 AB π A B E x = x , p = p (B12) for state ρ(1) then reads as V ν+V V ν+V AB x p are the eigenvalues of the CM. From the definition of EG ρ(1) =f(γ ,Γ ,Γ ,Γ˜ ). (B15) ↓ AB π A B E parameters V and V given below Eq. (B9) it then fol- x p (cid:16) (cid:17) lows that Vx 1, Vx Vp 0 and VxVp 1. Conse- In order to calculate the upper bound (B14), we first ≥ ≥ ≥ ≥ quently,atgivenν theeigenvalues(B12)lieinthesubset calculate the quantity M of the ( , )-plane such that if [1/ν,1] then p x p V V V ∈ (Vsxee∈F[i1g/.V2p).,ν], whereas if Vp ∈ (1,ν] then Vx ∈ [Vp,ν] IcG ρ(A1B)|E ≡ sup f(γπ,ΓA,ΓB,ΓE), (B16) ΓA,ΓB (cid:16) (cid:17) 4 which is the Gaussian classical mutual information q = r + ln( x /x )/2 with x = (e±2r + 2e∓2r)/3, − + ± (GCMI) of the conditional quantum state ρ(1) with wherer 0is asqueezingparameter. Minimizing there- AB|E ≥ p CM(B13)[50]. SubstitutingontheRHSofEq.(B13)for forethequantityg,Eq.(B21),exactlyasinRef.[35]and the matrix U from Eq. (B6) one finds, that the block restricting ourselves only to entangled states satisfying BS formwithrespecttotheAB splittingofCM(B13)reads the necessary and sufficient condition for entanglement, as | 1 > (a kx)(a kp), one finds, that the quantity pos- − − sessesthreecandidatesfortheminimumwhichlieonthe γ(1) = A C (B17) edges of the volume O and correspond to: AB|E C A 1. Homodyne detection of quadrature p on mode (cid:18) (cid:19) E E, i.e., Γ = Γt→+∞, where Γt diag(e2t,e−2t), or and therefore the CM is symmetric under the exchange E p p ≡ discarding of mode E, which yields ofmodes Aand B. This implies, thatthe standardform ofCM(B13)attainsthesymmetricform(B1). Denoting a+k a now the parameters of the standard form as a˜,c˜ and c˜ U ρ(1) =ln a p =ln . wecanusetheresultofRef.[35]tocalculatetheqxuantityp 1(cid:16) AB(cid:17) ra+kx! a2−kx2! (B16). Specifically, in Ref. [35] it was shown, that for p (B22) states with the symmetric standard-form CM where the parameters a˜ and c˜x satisfy inequality Here,thesecondequalityisaconsequenceoftheequality 1 1 2+ s˜ 0, (B18) a+k = , (B23) a˜ − ≥ p a k x − where s˜ ≡ a˜2−c˜2x, the GCMI (B16) is attained by which follows from the defining equality for symmetric AhomanoddyBn,efdoreptwechtiicohnsitoifsqoufadthraetfuorrems:xAandxB onmodes GLEMS, ν2 = (a−kx)(a+kp)=1. 2. Heterodyne detection on mode E, i.e., ΓE = 11, p which gives 1 a˜2 G ρ(1) = ln = ln 1 g2, (B19) Ic (cid:16) AB|E(cid:17) 2 a˜2−c˜2x − p − U2 ρ(A1B) =ln 12 zAzB+ z 1z , (B24) where g c˜x/a˜. By calculating explicitly the blocks A (cid:16) (cid:17) (cid:20) (cid:18) A B(cid:19)(cid:21) ≡ and C of CM (B17) we can express the quantities a˜ and where the parameters z and z are defined below A B g in terms of the parameters a,k and k of the original x p Eq. (B7). state andthe variables , andϕ overwhichwe carry outminimization. MakiVnpgVusxeoftheformulaa˜=√detA i.e.3,.ΓHom=oΓdty→n+e∞d,etwechteiroenΓotf qudadiarga(teu−re2t,xeE2to),nfomrowdehiEch, one finds that E x x ≡ the upper bound (B14) is equal to 1+ +2[ cosh(2q)+ sinh(2q)cos(2ϕ)] x p + − a˜= V V V V , p 2 U ρ(1) =ln a a−kx =ln a . (B20) 3 AB sa kp!  a2 k2 (cid:16) (cid:17) − − p where ± = ( x p)/2 and q = ln(zAzB), whereas for q (B25) V V ±V the quantity g one gets [35] Comparisonoftheupperbounds(B22),(B24)and(B25) 2 shows that U U and U U and therefore, the up- 1 1 3 2 3 g = a˜K2 +s a˜K2 −1 − a˜2 (B21) perbound (B14≥)is equalto U≥3, i.e., U(ρ(A1B))=U3(ρ(A1B)). (cid:18) (cid:19) Because the bound also represents at homodyne detec- with =( x p 1)/4. tions of quadratures xA and xB on modes A and B, the ForKcalcuVlaVtion−of the upper bound (B14) it remains leastmutualinformationoverallGaussianmeasurements to minimize the RHS of Eq. (B19) over all 3-tuples on mode E, the upper bound (B25) also gives GIE for ( , ,ϕ) belonging to the Cartesian product O = all symmetric GLEMS satisfying inequality (B18), p x MV V[0,π), where the set M is defined below equation × (B12). This amounts to the minimization of the RHS a oafctElyq.as(Bin21R)eofn. [t3h5e].seNtaOm,ewlyh,iccohmcpaanribseonpeorffotrhmeeqduaenx-- E↓G(cid:16)ρ(A1B)(cid:17)=ln a2−kp2. (B26) tities (B20) and (B21) with the same quantities for the q  CV GHZ state analyzed in Ref. [35] reveals, that they Note, that the RHS of the latter formula is nothing but are exactly the same. The only difference is in the pa- the mutual information of the joint distribution of out- rameter q which is for the present case defined below comes of measurements of quadratures p and p on a A B Eq. (B20), whereas for the CV GHZ state it is equal to symmetric Gaussian state with CM (B2).