Measurement-device-independent entanglement detection for continuous-variable systems
MMeasurement-device-independent entanglement detectionfor continuous-variable systems
Paolo Abiuso, ∗ Stefan B¨auml, and Daniel Cavalcanti
ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute ofScience and Technology, 08860 Castelldefels (Barcelona), Spain
Antonio Ac´ın
ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute ofScience and Technology, 08860 Castelldefels (Barcelona), Spain andICREA - Institucio Catalana de Recerca i Estudis Avan¸cats,Passeig Lluis Companys 23, 08010 Barcelona, Spain (Dated: January 27, 2021)We study the detection of continuous-variable entanglement, for which most of the existing meth-ods designed so far require a full specification of the devices, and we present protocols for entangle-ment detection in a scenario where the measurement devices are completely uncharacterised. Wefirst generalise, to the continuous variable regime, the seminal results by Buscemi [PRL 108, 200401(2012)] and Branciard et al. [PRL 110, 060405 (2013)], showing that all entangled states can bedetected in this scenario. Most importantly, we then describe a practical protocol that allows for themeasurement-device-independent certification of entanglement of all two-mode entangled Gaussianstates. This protocol is feasible with current technology as it makes only use of standard opticalsetups such as coherent states and homodyne measurements.
I. INTRODUCTION
Entanglement is the main resource for a broad rangeof applications in quantum information science, amongwhich are quantum key distribution [1], quantum com-putation [2], and quantum metrology [3]. It is thereforecrucial to develop methods to detect entanglement thatare reliable and practical. The most common method todetect entanglement is given by entanglement witnesses[4]. However, to be reliable this technique requires a per-fect implementation of the measurements. Indeed, smallcalibration errors can lead to false-positive detection ofentanglement [5, 6], which can be critical when usingthe wrongly detected entangled state for quantum infor-mation purposes. A possible way of circumventing thisproblem is to move into the so called device-independent(DI) scenario [7]. In this framework measurements donot need to be characterised, since entanglement is de-tected through the violation of Bell inequalities, whichonly use the statistics provided by the experiment, with-out making any assumptions on the real implementa-tion. The DI scenario is however stringent from an ex-perimental point of view, requiring low levels of noiseand high detection efficiencies. This is why other ap-proaches requiring intermediate level of trust on the de-vices have been developed. In particular, there existmethods that do not require any characterization of themeasurement implemented for entanglement detection,known as measurement-device-independent (MDI) [8, 9].Here, we consider the problem of entanglement detec-tion in continuous-variable (CV) systems. While the ap- ∗ [email protected] proach based on entanglement witnesses is well under-stood, very little is known about methods not requiringa full characterization of measurement devices. A fullyDI approach is complex because of the difficulty of find-ing useful Bell tests for continuous-variable states. Forinstance, in the Gaussian regime, which is the most feasi-ble experimentally, DI entanglement detection is impos-sible because no Bell inequality can be violated [7], henceintermediate approaches are necessary. The main goal ofof this work is to provide methods for MDI entanglementdetection in CV systems. We first demonstrate that, inprinciple, all entangled states can be detected in this sce-nario. Then, we describe a protocol that can detect theentanglement of all two-mode Gaussian states and doesnot require the characterisation of the measurements per-formed. Our protocol only relies on the use of trusted,well-calibrated, sources of coherent states, the easiest toprepare in the lab.An entanglement detection scenario where two parties,Alice and Bob, do not assume a particular descriptionof their measurement but use trusted sources of stateswas first introduced by Buscemi [8]. Namely, let us con-sider that Alice and Bob can produce states ψ µA (cid:48) and ψ νB (cid:48) according to some distributions P A (cid:48) ( µ ), P B (cid:48) ( ν ) re-spectively. Alice and Bob can use these states as in-puts to their measurement devices, which return mea-surement outcomes a and b respectively. These outcomesoccur with probability P ( a, b | ψ µA (cid:48) , ψ νB (cid:48) ) = Tr[ M aAA (cid:48) ⊗ N bBB (cid:48) ( ψ µA (cid:48) ⊗ ρ AB ⊗ ψ νB (cid:48) )], where M aAA (cid:48) and N bBB (cid:48) areunknown measurement operators defining a Positive-Operator Valued Measure (POVM). The main goal ofAlice and Bob is to determine if ρ AB is entangled basedon the knowledge of ψ µA (cid:48) , ψ νB (cid:48) , P A (cid:48) ( µ ), P B (cid:48) ( ν ), and P ( a, b | ψ µA (cid:48) , ψ νB (cid:48) ). Besides the calibration issue discussedbefore, this scenario is motivated by cryptographic tasks a r X i v : . [ qu a n t - ph ] J a n in which Alice and Bob do not trust the provider of themeasurement devices they are using.For finite dimensional Hilbert spaces, Buscemi hasshown that any entangled state ρ AB can be certified inthis scenario, but his proof is not constructive [8]. Theauthors of [9] have shown how to construct MDI entan-glement witness from standard entanglement witnesses.A different route was considered in [10, 11], where thequestion was formulated as a convex optimisation prob-lem that can be efficiently solved numerically.In what follows we first generalise the results of [8, 9]and show that the entanglement of every CV entangledstate can in principle be detected in a MDI scenario. Wethen move to the experimentally relevant case of Gaus-sian states and operations and show a MDI protocol thatis able to certify the entanglement of all two-mode Gaus-sian entangled states. This protocol is feasible with cur-rent technology in that it only requires the productionof coherent states and the implementation of homodynemeasurements. Moreover, our approach provides an in-teresting connection between MDI entanglement detec-tion and quantum metrology. II. REDUCTION TO PROCESS TOMOGRAPHY
In this section, we show that it is possible to detect theentanglement of any entangled state in a MDI scenariowhere Alice and Bob use coherent states as trusted inputs(the proof is presented for two-mode bipartite states butit is easily generalizable to n -modes, see App. B).Suppose Alice and Bob are in possession of trustedsources producing coherent states | α (cid:105) A (cid:48) and | β (cid:105) B (cid:48) ,respectively, according to some distribution. Theshared entangled state is ρ AB . The systems AA (cid:48) and BB (cid:48) are then projected onto respective two-modesqueezed vacuum (TMSV) states, i.e. the measurement {| Φ ( r ) (cid:105)(cid:104) Φ ( r ) | , − | Φ ( r ) (cid:105)(cid:104) Φ ( r ) |} is performed on both AA (cid:48) and BB (cid:48) ( r is the squeezing parameter). Conditioned on α and β , the probability of both measurements obtainingoutput ‘1’, corresponding to the projector | Φ ( r ) (cid:105)(cid:104) Φ ( r ) | ≡ Φ ( r ) , can be expressed as P ρ (1 , | α, β )= Tr (cid:104)(cid:16) Φ ( r ) AA (cid:48) ⊗ Φ ( r ) BB (cid:48) (cid:17) ( | α (cid:105)(cid:104) α | A (cid:48) ⊗ ρ AB ⊗ | β (cid:105)(cid:104) β | B (cid:48) ) (cid:105) = Tr (cid:104) M ( r ) A (cid:48) B (cid:48) | α (cid:105)(cid:104) α | A (cid:48) ⊗ | β (cid:105)(cid:104) β | B (cid:48) (cid:105) , (1)where we have defined M ( r ) A (cid:48) B (cid:48) := Tr AB (cid:104)(cid:16) Φ ( r ) AA (cid:48) ⊗ Φ ( r ) BB (cid:48) (cid:17) ( ρ AB ⊗ A (cid:48) B (cid:48) ) (cid:105) , (2)which is a POVM element by construction. Our mainobservation in this section is that non-separability of thePOVM element defined by (2) is equivalent to the under-lying state being entangled. Namely, we have Proposition 1
For any r > , the POVM element M ( r ) A (cid:48) B (cid:48) defined by eq. (2) is entangled if and only if ρ AB is entangled. Proof.
Let us assume ρ AB is separable, i.e. ρ AB = (cid:88) µ p µ ρ µA ⊗ σ µB . (3)We can then define M ( r ) µA (cid:48) :=Tr A (cid:104) Φ ( r ) AA (cid:48) ( ρ µA ⊗ A (cid:48) ) (cid:105) ,N ( r ) µB (cid:48) :=Tr B (cid:104) Φ ( r ) BB (cid:48) ( σ µB ⊗ B (cid:48) ) (cid:105) , (4)which are POVM elements by construction. It is easyto see that M ( r ) A (cid:48) B (cid:48) = (cid:80) µ p µ M ( r ) µA (cid:48) ⊗ N ( r ) µB (cid:48) , which is sep-arable. It remains to be shown that the POVM elementdefined by Eq. (2), which can be rewritten as [12] M ( r ) A (cid:48) B (cid:48) = (1 − λ ) λ ˆ n A +ˆ n B ρ AB λ ˆ n A +ˆ n B (5)(where λ = tanh r and ˆ n X = a † X a X the number operatoron mode X ), is entangled for all entangled ρ AB . In fact,suppose there exists an entanglement witness W suchthat Tr[ ρW ] < ρ (cid:48) W ] ≥ ρ (cid:48) separable.From W we can obtain a Hermitian operator ˜ W suchthat Tr[ M ( r ) A (cid:48) B (cid:48) ˜ W ] <
0, whereas for any separable POVMTr[ (cid:80) µ p u ( M µA ⊗ N µB ) ˜ W ] ≥
0. Consider in fact˜ W = λ − ˆ n A − ˆ n B W λ − ˆ n A − ˆ n B . (6)It is then easy to see thatTr[ M ( r ) A (cid:48) B (cid:48) ˜ W ] = (1 − λ )Tr[ ρW ] < . (7)For separable POVMs, on the other hand, it holds (cid:88) µ p µ Tr[ ˜ W ( M µA ⊗ N µB )] = (cid:88) µ p µ Tr[ W ( ˜ M µA ⊗ ˜ N µB )] , (8)where ˜ M µA = λ ˆ n A M µA λ ˆ n A , ˜ N µB = λ ˆ n B N µB λ ˆ n B . The op-erators ˜ M µA and ˜ N µB are manifestly positive semidefinite,meaning that under a proper renormalization they canbe seen as states, thus generating a separable state ρ (cid:48) such that Tr[ ρ (cid:48) W ] ≥
0. This implies (cid:88) µ p µ Tr[ W ( ˜ M µA ⊗ ˜ M µB )] ≥ , (9)which finishes the proof.We show in App. A that the violation of the derivedwitness (7) scales as 1 /N , where N is the energy scale(number of photons) defined by the original witness,which is to be compared with the 1 /d scaling of the finite-dimensional case [9] ( d being the Hilbert space dimensionof ρ AB ).As a consequence of Proposition 1, Alice and Bobcan certify the entanglement of ρ AB in a MDI way, iftheir output statistics allow them to fully reconstructthe POVM element M ( r ) A (cid:48) B (cid:48) . As in the case of discretevariables [10], this can be achieved by means of processtomography [13, 14]. It has been shown that the the setof all coherent states form a tomographically completeset via the Glauber-Sudarshan P-representation [15, 16].Further it has been shown that discrete sets of coherentstates can form tomographically complete sets, as well[17, 18]. As a special case of process tomography with co-herent states, it has been shown that POVM elements canbe fully reconstructed by their output statistics [19–21].Once Alice and Bob have reconstructed M ( r ) A (cid:48) B (cid:48) , they candetermine whether it is non-separable using an entangle-ment criterion. We also note that, for a given entangledstate, if the witness W is known, all that is necessaryis to evaluate Tr[ M ( r ) A (cid:48) B (cid:48) ˜ W ], which might not require fulltomography. In summary, we have the following Corollary 1
For every entangled state ρ AB , if | α (cid:105) and | β (cid:105) are chosen from tomographically complete sets, Al-ice and Bob can certify the entanglement of ρ AB in ameasurement-device-independent way. The result presented in this section suffers from severalpractical problems in their realization: firstly, they relyon performing the POVM that projects on the two modesqueezed states defined in Eq. (1). A typical scheme forsuch measurement involves photodetection (see App. A),which has in general low efficiency and high cost. Sec-ondly, the full tomography (or the generation of ˜ W )could be in general experimentally inefficient. There-fore, the previous proof is mostly a proof-of-principle re-sult. Next, we show that feasible schemes for MDI en-tanglement detection are possible. In fact, we proposean experimentally-friendly MDI entanglement detectionprotocol which is based solely on homodyne measure-ments and can detect all two-mode Gaussian entangledstates. III. MDI ENTANGLEMENT WITNESS FORALL TWO-MODE GAUSSIAN STATES
In this section we present a practical method for MDIentanglement certification of Gaussian states that can beimplemented using readily available optical components.Our method is inspired by the entanglement witness in-troduced in a seminal paper by Duan et al. [22] (see alsoSimon [23]). In that work, it was proven that the in-equality (cid:104) EW κ (cid:105) ≡ (cid:10) ∆ ˆ u κ (cid:11) + (cid:10) ∆ ˆ v κ (cid:11) ≥ κ + κ − , (10)where (cid:104) ∆ ˆ O (cid:105) is the variance of the operator ˆ O , andˆ u κ = (cid:18) κ ˆ x A − ˆ x B κ (cid:19) , ˆ v κ = (cid:18) κ ˆ p A + ˆ p B κ (cid:19) , (11) (i) holds for any two-mode separable state, real number κ , where κ > A and B [24], while (ii) for any entangled Gaussian state thereexist a value of κ and pairs of quadratures such thatEq. (10) is violated.Our main result is an experimentally-friendly methodfor MDI entanglement detection inspired by the wit-ness (10) and given by the following proposition: Proposition 2
Let | α (cid:105) and | β (cid:105) be coherent states pre-pared by Alice and Bob according to the Gaussian proba-bility distribution p ( α ) = 1 πσ e −| α | /σ . (12) Consider the setup in Fig. (1) in which uncharacterizedlocal measurements are applied jointly on these states andhalf of an unknown state ρ AB , producing as a result tworeal numbers ( a , a ) for Alice and ( b , b ) for Bob. Forall local measurements and all separable states one has (cid:104) MDIEW κ (cid:105) ≡ (cid:10) U κ (cid:11) + (cid:10) V κ (cid:11) ≥ κ + κ − σ σ , (13) where U κ and V κ are U κ ≡ κa − b κ − κα x − β x κ √ ,V κ ≡ κa + b κ − κα p + β p κ √ . (14) For any two-mode entangled Gaussian state, there existlocal measurements acting jointly on the state and theinput coherent states violating inequality (13) . Looking at its definition, the operational meaning ofthe witness is clear: Alice and Bob results should besuch that their difference and sum, weighted by κ , areas close as possible to the same difference and sum ofthe quadratures of the coherent states, divided by √ Proof of (13) . To prove the inequality we need tominimize the value of the witness over all separable states ρ AB = (cid:80) i p i ρ ( i ) A ⊗ ρ ( i ) B . We can restrict the analysis toproduct states because the witness is linear on the state.It follows that the output probability factorises p ( a , a , b , b | α, β ) = Tr (cid:2) M Aa ,a | α (cid:105)(cid:104) α | (cid:3) Tr (cid:2) M Bb ,b | β (cid:105)(cid:104) β | (cid:3) , (15)and we are left with two independent POVMs on theinput states to be optimised. Using (cid:104) U κ (cid:105) + (cid:104) V κ (cid:105) ≥(cid:104) ∆ U κ (cid:105) + (cid:104) ∆ V κ (cid:105) and the fact that the distribution iscompletely factorised between the two sides it followsthat the value of the witness is lower bounded by theminimum of12 (cid:18) κ + 1 κ (cid:19) (cid:16)(cid:68) ∆ (cid:104) √ a − α x (cid:105)(cid:69) + (cid:68) ∆ (cid:104) √ a − α p (cid:105)(cid:69)(cid:17) . (16) FIG. 1. Experimental setup for the MDI entanglement detec-tion of a 2-mode state. Fiduciary coherent states are preparedby the parties and measured together with the correspondingsubsystems of the unknown state ρ AB . To compute the boundof the entanglement witness (13), measurements should beseen as uncharacterized black boxes producing the outputs.To obtain a violation, the following specific measurements areimplemented: the coherent states are mixed with the respec-tive modes of ρ AB in a 50:50 beam splitter and homodynemeasurements of ˆ x and ˆ p are performed on the two outputs. That is: in the absence of correlations, the best Alice andBob can do is to separately perform the optimal measure-ments to estimate the input coherent states.Minimizing the expression in the second parenthesisin (16) looks essentially like a metrology problem. Alower bound, in turn, can be obtained using a multi-parameter Bayesian version of the quantum Cram´er-Raobound [25]. The simultaneous estimation of the posi-tion and momentum quadratures has been studied thor-oughly and is optimized for coherent states by measur-ing ˆ x and ˆ p on two different modes after a 50:50 beam-splitter [26, 27]. In particular, assuming a Gaussian priordistribution, like in our case, the minimal sum of vari-ances is equal to σ / (1+ σ ) [26], which proves the boundfor the entanglement witness.To prove the violation claimed in Proposition 2, it suf-fices to show that for any entangled two-mode Gaussianstate there exist local measurements and values of ( κ, σ )that lead to it. Consider now the optical setup depictedin Fig. 1. Alice and Bob, upon receiving their respectivesubsystems of ρ AB , first mix them with local coherentstates in a balanced beam splitter, and then measure theposition and momentum quadratures in the output ports.The output observables are thusˆ A = ˆ x α + ˆ x A √ , ˆ A = ˆ p α − ˆ p A √ , ˆ B = ˆ x β + ˆ x B √ , ˆ B = ˆ p β − ˆ p B √ . (17)The quadratures ˆ x A , ˆ p A , ˆ x B and ˆ p B are those used inthe standard witness (10). Observables (17) are used todefine the measurement outputs needed for the compu-tation of our MDI witness (13). More precisely, consider first the case in which the average values of the state’squadratures are null, (cid:104) ˆ x A (cid:105) = (cid:104) ˆ p A (cid:105) = (cid:104) ˆ x B (cid:105) = (cid:104) ˆ p B (cid:105) = 0.Then, by taking ( a , a , b , b ) equal to the statistical out-put of ( ˆ A , ˆ A , ˆ B , ˆ B ) respectively, it follows by substi-tuting in (14) [28] (cid:10) U κ (cid:11) = (cid:42)(cid:32) κ ˆ x α + ˆ x A √ − κ ˆ x β + ˆ x B √ − κα x − β x κ √ (cid:33) (cid:43) = 12 (cid:18) κ (cid:104) ∆ ˆ x α (cid:105) + (cid:104) ∆ ˆ x β (cid:105) κ + (cid:10) ∆ ˆ u κ (cid:11)(cid:19) = 12 (cid:18) κ + κ − (cid:10) ∆ ˆ u κ (cid:11)(cid:19) . (18)Similarly, (cid:10) V κ (cid:11) = (cid:16) κ + κ − + (cid:10) ∆ ˆ v κ (cid:11)(cid:17) , and conse-quently we find that in the proposed scheme (cid:104) MDIEW κ (cid:105) = 12 (cid:18) κ + κ − (cid:104) EW κ (cid:105) (cid:19) . (19)The generalization to states that have non-zero averagesof the quadratures is obtained by simply offsetting theoutputs accordingly as follows: a is the output of ˆ A −(cid:104) ˆ x A (cid:105) / √ a of ˆ A + (cid:104) ˆ p A (cid:105) / √ κ such that (cid:104) EW κ (cid:105) < κ + κ − , which implies from Eq. (19) that inthe proposed scheme (cid:104) MDIEW κ (cid:105) < κ + κ − . It is thensufficient to choose σ large enough to violate (13). IV. TWO-MODE SQUEEZED STATE CASE,WITH NOISE TOLERANCE
At last, to illustrate the feasibility of our scheme, weapply it to the case of ρ AB being a TMSV state. Byincluding noise tolerance, we pave the way for an exper-imental realization of our MDI entanglement witness.A TMSV state can be described as the mixing of twosingle-mode squeezed states (one squeezed in ˆ p and onein ˆ x ) [29]. In the Heisenberg picture, this results inˆ x A = e r ˆ x (0)1 + e − r ˆ x (0)2 √ , ˆ p A = e − r ˆ p (0)1 + e r ˆ p (0)2 √ , ˆ x B = e r ˆ x (0)1 − e − r ˆ x (0)2 √ , ˆ p B = e − r ˆ p (0)1 − e r ˆ p (0)2 √ , (20)where the superscript { ˆ x (0) , ˆ p (0) } represents operatorsacting on the vacuum. Consider now the two operatorsˆ u κ =1 = ˆ x A − ˆ x B and ˆ v κ =1 = ˆ p A + ˆ p B . From Eq. (10)we see, by choosing κ = 1, that these operators sat-isfy for any separable state (cid:104) ∆ ˆ u κ =1 (cid:105) + (cid:104) ∆ ˆ v κ =1 (cid:105) ≥
1. If we compute the above combination for the two-mode squeezed state, we obtain ˆ u κ =1 = √ e − r ˆ x (0)2 andˆ v κ =1 = √ e − r ˆ p (0)1 . Consequently, it holds (cid:104) EW k =1 (cid:105) T MSV = e − r < . (21)This is not surprising: as soon as there is squeezing r > (cid:104) MDIEW κ =1 (cid:105) = (cid:0) e − r (cid:1) .To check noise tolerance, we consider losses in themodes A and B modelled as a beam splitterˆ a A ( η A ) = (cid:112) − η A ˆ a A (0) + √ η A ˆ a (0) NA , ˆ a B ( η B ) = (cid:112) − η B ˆ a B (0) + √ η B ˆ a (0) NB , (22) where ˆ a (0) NX is a vacuum mode acting as a noise on mode X , while ˆ a X (0) is the corresponding noiseless mode. Wefocus on a natural scenario in which the source producingthe two-mode squeezed state is between Alice and Boband losses affect the two modes, not necessarily in a sym-metric way. At the same time, the same loss noise (22)applied to the input coherent states would lead only to arenormalization of α and β and can be compensated byincreasing the variance σ . Using Eqs. (20) and (22) wecan accordingly compute (cid:104) EW κ (cid:105) T MSV,η A ,η B = (cid:28) ∆ (cid:18) κ ˆ x A ( η A ) − ˆ x B ( η B ) κ (cid:19) + ∆ (cid:18) κ ˆ p A ( η A ) + ˆ p B ( η B ) κ (cid:19)(cid:29) = 12 (cid:16) κ η A + η B κ (cid:17) + e r (cid:18) κ (cid:112) − η A − √ − η B κ (cid:19) + e − r (cid:18) κ (cid:112) − η A + √ − η B κ (cid:19) . (23)Notice that it is always possible to choose κ such that κ √ − η A − κ − √ − η B = 0 and r big enough tonullify the last term of (23), and obtain a score (19) (cid:104) MDIEW κ (cid:105) = (cid:0) κ ( η A + 1) + ( η B + 1) κ − (cid:1) , which islower than the separable bound (cid:0) κ + κ − (cid:1) (cid:16) σ σ (cid:17) forlarge enough σ . In this sense the entanglement witnesswe analysed here is loss-resistant. In Figure 2 we plotthe trade-off between noise, entanglement, and varianceof the prior distributions σ for the MDI detection of en-tanglement in the case of symmetric losses ( η A = η B , κ = 1). V. DISCUSSION
In this work we have promoted the task ofmeasurement-device-independent entanglement certifica-tion to the continuous-variable regime. We first gen-eralised the result by Buscemi and proved that allcontinuous-variable entangled state can in principle bedetected in this scenario. Then, we showed a simple testable to detect the entanglement of all two-mode Gaussianentangled states. Most importantly, the test only relieson the preparation of coherent states and uses standardexperimental setups, thus being readily available withcurrent technology.Our work also opens up a series of interesting direc-tions. From a general perspective, our works opens thepath to the use of CV quantum systems for MDI tasksbeyond entanglement detection, such as randomness gen-eration or secure communication. Another possible re-search direction would be to investigate the possible gen-erality of the connection between entanglement detectionand metrology exploited here. In particular, can all MDIentanglement tests be translated into a parameter esti-mation problem? We are therefore confident that our σ = σ = σ = σ = σ = FIG. 2. Obtainable value (19) of (cid:104)
MDIEW κ =1 (cid:105) , for a two-mode squeezed vacuum state with squeezing parameter r andunder the presence of losses with parameter η A = η B ≡ η (cf. (23)). The area under each σ -curve defines the range ofparameters for which MDI entanglement is certified by theinequality (13), which assumes a Gaussian prior for the inputstates with width σ . results will motivate further studies on the field of quan-tum information with continuous variables. ACKNOWLEDGMENTS
This work was supported by the Government of Spain(FIS2020-TRANQI and Severo Ochoa CEX2019-000910-S), Fundaci´o Cellex, Fundaci´o Mir-Puig, Generalitat deCatalunya (CERCA, AGAUR SGR 1381 and Quantum-CAT). A.A. is supported by the ERC AdG CERQUTEand the AXA Chair in Quantum Information Science. P.A. is supported by “la Caixa” Foundation (ID 100010434,Grant No. LCF/BQ/DI19/11730023). S.B. acknowl- edges funding from the European Union’s Horizon 2020research and innovation program, grant agreement No.820466 (project CiViQ). D.C. is supported by a Ramony Cajal Fellowship (Spain). [1] A. K. Ekert, Quantum cryptography based on bell’s the-orem, Phys. Rev. Lett. , 661 (1991).[2] R. Jozsa and N. Linden, On the role of entanglement inquantum-computational speed-up, Proc. R. Soc. Lond. A , 2011–2032 (2003).[3] G. T´oth and I. Apellaniz, Quantum metrology froma quantum information science perspective, Journal ofPhysics A: Mathematical and Theoretical , 424006(2014).[4] O. G¨uhne and G. 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Friis, Bayesianparameter estimation using gaussian states and measure-ments, arXiv preprint arXiv:2009.03709 (2020).[28] We use that coherent states have minimum variances (cid:104) ∆ ˆ x α,β (cid:105) = (cid:104) ∆ ˆ p α,β (cid:105) = .[29] S. L. Braunstein and P. Van Loock, Quantum infor-mation with continuous variables, Reviews of ModernPhysics , 513 (2005). Appendix A: On the feasibility of Proposition 1
The proof of principle realisation of a MDI entanglement witness proposed in Sec. II is based on the possibility ofperforming the POVM defined by the projection on the two-mode squeezed vacuum state Φ ( r ) . Here we show that inprinciple this is realisable using photodetectors, single mode squeezing and a beam splitter. Indeed, it is known thata two-mode squeezed vacuum can be obtained by applying the following unitaries on a two-mode vacuum [29]: | Φ ( r ) (cid:105) = ˆ B ˆ S ( r )1 ˆ S ( − r )2 | (cid:105) . (A1)Here ˆ S ( r ) X represents a single-mode squeezing operation with squeezing parameter r on mode X , which satisfies ˆ S ( r ) † X =ˆ S ( − r ) X . The following unitary ˆ B is induced by a 50:50 beamsplitter. Therefore, for any two mode state | ψ (cid:105) , theprojection on Φ ( r ) can be simulated by a photodetection preceded by the corresponding inverse unitary (cid:104) Φ ( r ) | ψ (cid:105) = (cid:104) | ˆ S ( − r )1 ˆ S ( r )2 ˆ B † | ψ (cid:105) . (A2)We notice as well that the energy scale defined by the original entanglement witness W of ρ AB sets a bound on theviolation of the POVM-entanglement-witness (7) proposed in the main text. Recall that ˜ W = λ − ˆ n A − ˆ n B W λ − ˆ n A − ˆ n B ,where ˆ n X = a † X a X is the photon-number operator of mode X . Therefore, for ˜ W = to be bounded, W needs to havefinite energy as well. Suppose W (cid:46) e − ( n A + n B ) /N for large energies. The value of N is an upperbound on the energyscale of W . Then, from (6) we see that for ˜ W to be bounded as well, it is necessary that λ − e − /N < < λ < λ are e − /N < λ < , (A3)and so 1 − e − /N > − λ > . (A4)Confronting this last bound with (7) we notice that for large energies O ( N ) of the original witness, the proposedviolation is dampened by a factor ∼ N . Appendix B: Generalization of Proposition 1 to N modes
Proposition 1 can be easily generalised to states having any number of modes. To be explicit how this can be done,we show the generalisation to a bipartite state having two modes on Alice side, A , A , and one mode on Bob side, B . In such a case we substitute Eq. (1) by P ρ (1 , , | α , α , β ) = Tr (cid:104)(cid:16) Φ ( r ) A A (cid:48) ⊗ Φ ( r ) A A (cid:48) ⊗ Φ ( r ) BB (cid:48) (cid:17) (cid:0) | α (cid:105)(cid:104) α | A (cid:48) ⊗ | α (cid:105)(cid:104) α | A (cid:48) ⊗ ρ A A B ⊗ | β (cid:105)(cid:104) β | B (cid:48) (cid:1)(cid:105) = Tr (cid:104) M ( r ) A (cid:48) A (cid:48) B (cid:48) | α (cid:105)(cid:104) α | A (cid:48) ⊗ | α (cid:105)(cid:104) α | A (cid:48) ⊗ | β (cid:105)(cid:104) β | B (cid:48) (cid:105) , (B1)where it is defined M ( r ) A (cid:48) A (cid:48) B (cid:48) := Tr A A B (cid:104)(cid:16) Φ ( r ) A A (cid:48) ⊗ Φ ( r ) A A (cid:48) ⊗ Φ ( r ) BB (cid:48) (cid:17) (cid:0) ρ A A B ⊗ A (cid:48) A (cid:48) B (cid:1)(cid:105) . (B2)That is, for the generalization, each party generates a number of coherent states equal to the number of modes of hisside of the partition; projections on TMSV states are performed accordingly. The derivation then follows in the sameway as presented in the main text, that is, noticing that M ( r ) A (cid:48) A (cid:48) B (cid:48) = (1 − λ ) λ ˆ n A +ˆ n A +ˆ n B ρ A A B λ ˆ n A +ˆ n A +ˆ n B , (B3)and defining the entanglement witness ˜ W for M ( r ) A (cid:48) A (cid:48) B (cid:48) in terms of the original entanglement witness W of ρ A A B ,˜ W = λ − ˆ n A ! − ˆ n A − ˆ n B W λ − ˆ n A − ˆ n A − ˆ n B . (B4)It follows that Tr[ M ( r ) A (cid:48) A (cid:48) B (cid:48) ˜ W ] = (1 − λ ) Tr[ ρ A A B W ] < , (B5)while for separable POVMs, it holds (cid:88) µ p µ Tr[ ˜ W ( M µA A ⊗ N µB )] = (cid:88) µ p µ Tr[ W ( ˜ M µA A ⊗ ˜ N µB )] , (B6)where ˜ M µA A = λ ˆ n A +ˆ n A M µA A λ ˆ n A +ˆ n A , ˜ N µB = λ ˆ n B N µB λ ˆ n B . Following the same reasoning as in the main text,the operators ˜ M µA A and ˜ N µB are positive, and can be seen as representing unnormalized states, thus generating aseparable state ρ (cid:48) such that Tr[ ρ (cid:48) W ] ≥≥