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Einstein–Podolsky–Rosen steering measure for two-mode continuous variable states Ioannis Kogias1, 2, ∗and Gerardo Adesso1, † 1School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom 2ICFO - The Institute of Photonic Sciences, Av. Carl Friedrich Gauss, 3 08860 Castelldefels (Barcelona), Spain compiled: March 11, 2015 Steering is a manifestation of quantum correlations that embodies the Einstein-Podolsky-Rosen (EPR) paradox. While there have been recent attempts to quantify steering, continuous variable systems remained elusive. We introduce a steering measure for two-mode continuous variable systems that is valid for arbitrary states. The mea- sure is based on the violation of an optimized variance test for the EPR paradox by quadrature measurements, and admits a computable and experimentally friendly lower bound only depending on the second moments of the state, which reduces to a recently proposed quantifier of steerability by Gaussian measurements. We further show that Gaussian states are extremal with respect to our measure, minimizing it among all continuous variable states with fixed second moments. As a byproduct of our analysis, we generalize and relate well-known EPR-steering criteria. Finally an operational interpretation is provided, as the proposed measure is shown to quantify a guaranteed key rate in semi-device independent quantum key distribution. 1. Introduction Almost 80 years have passed since the landmark pa- per of Einstein-Podolsky-Rosen (EPR) [1] on a para- doxical manifestation of quantum correlations which Schr¨odinger later termed quantum steering [2, 3], yet the topic is more timely than ever. From one-sided device independent entanglement verification [4] and quan- tum key distribution [5, 6] to signifying secure quan- tum teleportation [7] and performing entanglement- assisted subchannel discrimination [8], Einstein’s scru- tinized notion of steering finds increasingly many ap- plications in non-classical tasks after its recent formu- lation as a distinct type of asymmetric nonlocality by Wiseman and co-workers [4, 9], thus making it a sub- ject of intense research [10]. Steering, in a modern quantum information lan- guage [4, 9], can be understood as the task of two dis- tant parties, say Alice and Bob, in which Alice tries to convince Bob that the quantum state ˆρAB they share is entangled, by remotely creating quantum ensembles on Bob’s site that could not have been created without shared entanglement. Given that Bob does not trust Alice and her announced measurements, we say that Alice can steer Bob’s state (and thus convince Bob), or equivalently that the state ˆρAB is “A →B” steer- able, if and only if (iff) the probabilities of all possible joint measurements cannot be written in the factoriz- ∗john k 423@yahoo.gr † Gerardo.Adesso@nottingham.ac.uk able form [4]: P (A, B|a, b, ˆρAB) = X λ PλP (A|a, λ) P (B|b, ˆρλ) , (1) where the lower-case letters a ∈MA and b ∈MB de- note local observables for Alice and Bob, while A and B their corresponding outcomes. Violation of (1) im- plies the failure of a local hidden state model to explain the measurement statistics. As one can see from Eq. (1), steering is an asymmetric form of nonlocality that sits in-between entanglement [11] and Bell nonlocality [12– 14]. Not all entangled states are steerable, and not all steerable states are Bell nonlocal. In order for steering to be useful one should first be able to detect it in experiments [15–24]. The first at- tempt to create an experimental criterion that captures the essence of the EPR paradox [10] in a continuous variable setting was made in the 80’s by M. Reid [25], whose criterion is commonly known as Reid’s crite- rion and which was shown later to be only a special case of an EPR-steering test in the sense of (1) [26]. Today our knowledge about the detection and distri- bution of steering has significantly advanced [27–29], with a plethora of effective criteria derived [26, 30–32] and phenomena like steering monogamy identified in multi-party scenarios [29]. Besides a yes/no answer to the question of steerability given by various steer- ing criteria, however, one is interested in how much a state is steerable for practical purposes. Only quite re- cently, the quantification of steering was put forward by researchers [8, 33, 34] to assess how much a quan- tum state’s statistics deviate from (1), and thus how useful it can be for tasks that use steering as their re- arXiv:1411.0444v2 [quant-ph] 10 Mar 2015
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2 source [35]. Two measures of steering have been pro- posed in particular for finite-dimensional systems, the so-called steering weight [33], and the steering robust- ness [8]. While both measures are not amenable to an- alytical evaluation and can only be computed numeri- cally by semidefinite programming, the steering robust- ness has a nice operational interpretation in the con- text of subchannel discrimination [8]. For continuous variable systems, a computable steering quantifier spe- cific to Gaussian states and measurements has also been very recently proposed [34]. In this paper we present an accessible approach to the quantitative estimation of steerability for bipartite two-mode continuous variable states. We examine re- cent experimental criteria for steering [26], the so-called EPR-Reid variance criteria whose applicability extends to all (Gaussian and non-Gaussian) states, and analyze their maximal violation by optimal local quadrature ob- servables for Alice and Bob, in order to capture the largest possible departure from (1) for a given state. Hence we define (in Section 2) a suitable measure of steering for an arbitrary two-mode state, and we prove that it admits an analytically computable lower bound that captures the degree of steerability of the given state by Gaussian measurements. The lower bound coin- cides with the Gaussian steering measure introduced in a previous work [34], whose usefulness is here gen- eralized from the Gaussian domain to arbitrary states. We prove Gaussian states to be in fact extremal [36], as they are minimally steerable among all states with the same covariance matrix, according to the measure proposed in this paper. As a corollary of our analy- sis, we show (in Section 3) that a necessary and suf- ficient condition for steerability of Gaussian states un- der Gaussian measurements obtained by Wiseman et al. based on covariance matrices [4, 9], remains valid as a sufficient steering criterion for arbitrary non-Gaussian states, and amounts to Reid’s criterion [10, 25] when optimal Gaussian local observables are chosen for the latter. We conclude (in Section 4) with a summary of our results and an outlook of currently open questions motivated by the present analysis. 2. A steering measure for two-mode states based on quadrature measurements In general [37], a measure of steering should quantify how much the correlations of a quantum state depart from the expression in Eq. (1). Since a manifestation of these correlations can be observed by the violation of suitable EPR-steering criteria, one can get a quantita- tive estimation of the degree of steerability in a given state by evaluating the maximum violation of a cho- sen steering criterion as revealed by optimal measure- ments. One expects that the higher the violation (i.e., the amount of correlations), the more useful the state will be in tasks that use quantum steering as a resource. In this paper we consider an arbitrary state ˆρAB of a two-mode continuous variable system. The rele- vant steering criteria to our work will be the so-called multiplicative variance EPR-steering criteria [26], of which Reid’s criterion [25] is a special case. Following [10, 25, 26], let us consider a situation where Bob mea- sures two canonically conjugate observables on his sub- system, ˆxB, ˆpB with corresponding outcomes XB, PB, and Alice tries to guess Bob’s outcomes based on the outcomes of measurements on her own subsystem. If, say, the outcome of Alice’s measurement is XA, corre- sponding to a local observable ˆxA, we can denote by Xest (XA) Alice’s inference of Bob’s measurement out- come XB. The average inference variance of XB given Alice’s estimator Xest (XA) is defined by ∆2 infXB = D [XB −Xest (XA)]2E , (2) where the average is taken with respect to the joint probability distribution P (XA, XB) and over all out- comes XA, XB. One can show [10] that the optimal estimator minimizing the inference variance ∆2 infXB is the mean Xest (XA) = ⟨XB⟩XA evaluated on the con- ditional distribution P (XB|XA). Substituting in (2) we obtain the minimal inference variance of XB by mea- surements on A, ∆2 minXB = X XA P (XA) ∆2 (XB|XA) , (3) where ∆2 (XB|XA) is the conditional variance of XB calculated from P (XB|XA). Clearly, from the proper- ties stated above, it holds that ∆2 infXB ≥∆2 minXB. Sim- ilarly we can define an inference variance ∆2 infPB for ˆpB and its corresponding minimum ∆2 minPB given respec- tively by analogous formulas to (2) and (3), but condi- tioned on PA instead of XA. In [10, 26] it was shown that a bipartite state ˆρAB shared by Alice and Bob is steerable by Alice, i.e. “A →B” steerable, if the condi- tion ∆2 minXB ∆2 minPB ≥1, (4) is violated. Notice that the criterion (4) is independent of Al- ice’s and Bob’s first moments, since displacements of the form XA(B) →XA(B) + dA(B) leave the inference variances (of both position and momentum) invariant as can be easily seen from the definition (3). Therefore, first moments will be assumed to be zero in the rest of the paper without any loss of generality. We remark that the EPR-steering criterion (4) is ap- plicable to arbitrary states and is valid without any as- sumption on the Hilbert space of Alice’s subsystem, as Bob just needs to identify two distinctly labelled mea- surements performed by Alice [26]. However, in or- der to keep our analysis accessible, we will further as- sume that Alice’s allowed measurements are restricted to be quadrature ones, i.e., projections on the eigen- basis of (generally rotated) canonically conjugate op- erators ˆxθ A and ˆpθ A, such that [ˆxθ A, ˆpθ A] = i in natural units. Although quadrature measurements are not gen- eral and not necessarily optimal to detect steerability in
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3 all states, they are convenient from a theoretical point of view and can be reliably implemented in laboratory by means of homodyne detections. One immediately sees that the product of variances in (4) is not invariant under local unitary operations (apart from displacements) by Alice and Bob, thus a state might be detected as more or less steerable if some local change of basis is implemented. In order to capture steerability in an invariant way, one can consider the maximum violation of (4) that a quan- tum state ˆρAB can exhibit, by minimizing the prod- uct ∆2 minXB ∆2 minPB over all local unitaries Ulocal = UA ⊗UB for A and B applied to the state. We then propose to quantify the “A →B” steerability of an arbitrary two-mode continuous variable state ˆρAB detectable by quadrature measurements, via the mea- sure SA→B (ˆρAB) = max 0, −1 2 ln F , (5) where F = min {Ulocal} ∆2 minXB ∆2 minPB. (6) The measure naturally quantifies the amount of vi- olation of an optimized multiplicative variance EPR- steering criterion of the form (4) for an arbitrary state ˆρAB. As one would expect from any proper quanti- fier of quantum correlations, the measure enjoys local unitary invariance by definition, and it vanishes for all states which are not “A →B” steerable. Calculating SA→B in an analytical manner for an ar- bitrary state is still a difficult task. In general, given a quantum state, the minimization in F involves both Gaussian and non-Gaussian local unitaries for Alice and Bob, which correspond to violations of (4) by Gaus- sian and non-Gaussian quadrature measurements, re- spectively. It is possible, though, to obtain a com- putable lower bound to SA→B if one constrains the op- timization to Gaussian unitaries only. The lower bound, presented in the next subsection, will then provide a quantitative indication of the “A →B” steerability of ˆρAB that can be demonstrated by Gaussian measure- ments on Alice’s subsystem. 2.A. Lower bound A short introduction of the reader to Gaussian states is first intended [38]. An arbitrary bipartite Gaussian state ˆρG AB is determined, up to local displacements, by its sec- ond moments, i.e., it is specified by the covariance ma- trix (CM) σAB, which can be written in the block form σAB = A C CT B . (7) Here, A and B are the marginal CMs corresponding to the reduced states of Alice and Bob respectively, while C encodes intermodal correlations. For two- mode states, A, B, and C are 2 × 2 matrices. The matrix elements of the CM, defined by (σAB)ij
Tr ˆRi ˆRj + ˆRj ˆRi ˆρG AB , are expressed via the vector ˆR = (ˆxA, ˆpA, ˆxB, ˆpB)T that conveniently groups the phase-space operators ˆxA(B), ˆpA(B) for each mode. The canonical commutation relations these operators satisfy can be compactly expressed as [ ˆRj, ˆRk] = i(ΩAB)jk, where ΩAB = ΩA ⊕ΩB is the symplectic matrix, with ΩA = ΩB = 0 1 −1 0 [38]. The CM of any (Gaussian or non-Gaussian) physical state needs to satisfy the bona fide condition σAB + i (ΩA ⊕ΩB) ≥0 . (8) Gaussian operations are defined as those which pre- serve the Gaussianity of the states they act upon. To obtain a lower bound for the steering measure SA→B (ˆρAB) in terms of second moments, we first re- mind the reader that with no loss of generality one can assume vanishing first moments of ˆρAB (see discussion below Eq. (4)). We will show that, for arbitrary states ˆρAB with corresponding CM σAB, the product of in- ference variances ∆2 infXB ∆2 infPB, defined as in (2), ac- quires its minimum value when σAB is expressed in the so-called standard form ¯σAB = ¯A ¯C ¯CT ¯B , (9) in which the submatrices ¯A
diag (a, a), ¯B
diag (b, b), and ¯C = diag (c1, c2) take a diagonal form. The standard form can always be obtained for any state by suitable local unitary operations [39, 40] and is unique up to a sign flip in c1 and c2, as its elements can be recast as functions of four local invariants of the CM [41]. Let us begin by considering a steerable ˆρAB that violates (4), so that SA→B (ˆρAB) > 0. We use the fact that ∆2 infXB ≥∆2 minXB, when a linear estimator Xest (XA) = gxXA + dx is used in (2); after minimizing the inference variance over the real numbers gx, dx and considering vanishing first moments without any loss of generality, we find ∆2 infXB = ⟨X2 B⟩−⟨XBXA⟩2/⟨X2 A⟩ [10]. Similar considerations hold for the inference vari- ance of momentum, where an estimator of the form Pest (PA) = gpPA + dp will give ∆2 infPB = ⟨P 2 B⟩− ⟨PBPA⟩2/⟨P 2 A⟩after optimizing over the real numbers gp, dp. Since a linear estimator is optimal for inferring the variance in the case of Gaussian states [10, 25], but not anymore in the general case, the inequality ∆2 infXB∆2 infPB ≥∆2 minXB∆2 minPB will be true for all states (with equality on Gaussian states). Hence, F in
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4 (5) can be upper bounded as follows, F = min {UG}∪{UnG} ∆2 minXB∆2 minPB ≤ min {UG}∪{UnG} ∆2 infXB∆2 infPB ≤min {UG} ∆2 infXB∆2 infPB, (10) where we have decomposed the set of local unitaries {Ulocal} into Gaussian {UG} and non-Gaussian {UnG} ones. The product of inference variances in (10) is in- tended as evaluated from the optimal linear estimator as detailed above [10], namely ∆2 infXB∆2 infPB = �⟨X2 B⟩−⟨XBXA⟩2/⟨X2 A⟩ × �⟨P 2 B⟩−⟨PBPA⟩2/⟨P 2 A⟩ , (11) Since an upper bound on F will give us the desired lower bound on SA→B, what remains is to compute this upper bound, i.e., the rightmost quantity in (10), which only depends on the CM elements of the state. Note that the product of inference variances (11), using linear estimators, defines what is well-known in the literature as Reid’s criterion [25], ∆2 infXB∆2 infPB ≥1 , (12) whose violation is sufficient to detect “A →B” steer- ability of a general two-mode state based on second or- der moments. Local Gaussian unitaries (that do not give rise to dis- placements) acting on states ˆρAB, translate on the level of CMs as local symplectic transformations Slocal = SA ⊕SB, acting by congruence: σAB 7→SlocalσABST local [38, 42]. In order to compute min{Slocal} ∆2 infXB∆2 infPB we can, with no loss of generality, consider a CM ¯σAB in standard form, apply an arbitrary local symplectic operation Slocal to it, then evaluate ∆2 infXB ∆2 infPB on the transformed CM Slocal¯σABST local, and finally mini- mize this quantity over all possible matrices SA(B). To perform the minimization we parametrize the matrix elements of SA(B) in the following convenient way, SA(B) =
1 (1−uA(B)vA(B))wA(B) vA(B) (1−uA(B)vA(B))wA(B) uA(B)wA(B) wA(B) ! (13) where the symplectic condition SA(B)ΩA(B)ST A(B) = ΩA(B) has been taken into account and the real vari- ables uA(B), vA(B), wA(B) are now independent of each other. Performing the (unconstrained) minimization over the variables uA(B), vA(B) we were able to obtain analytically the global minimum of the product (11) with respect to Gaussian observables, min {UG} ∆2 infXB ∆2 infPB = det M B σ , (14) which also constitutes the upper bound for F in (10). Here the local symplectic invariant det M B σ
b −c2 1 a b −c2 2 a is the determinant of the Schur com- plement of A in σAB, defined for any two-mode CM (7) as [9, 34] M B σ = B −CT A−1C . (15) The minimum (14) can be obtained from every state us- ing the following parameters that determine the local symplectic operations (13), (uA, vA, uB, vB) = c1vB c2 , −ab+c2 1 ab−c2 2 c2vB c1 , −ab+c2 1 ab−c2 2 vB, vB , ∀vB, wA(B). It is evident from (14) that the minimum product of inference variances (11) is achieved, in par- ticular, when evaluated for a standard form CM ¯σAB. Substituting F ≤det M B σ in (5), a lower bound for the proposed steering measure of an arbitrary two- mode state ˆρAB is obtained, SA→B (ˆρAB) ≥GA→B (σAB) , (16) where we recognize the Gaussian steering measure in- troduced in [34], GA→B (σAB) = max 0, −1 2 ln det M B σ . (17) The lower bound GA→B solely depends on local sym- plectic invariant quantities that uniquely specify the CM of the state. As is known [41], these invariant quan- tities can be expressed back with respect to the original elements of the CM which one can measure in labora- tory, e.g. via homodyne tomography [43]. Henceforth, the lower bound that we obtained is both analytically computable and, also, experimentally accessible in a routinely fashion for any (Gaussian or non-Gaussian) state, since only moments up to second order are in- volved. In the following we discuss some useful properties that the steering measure SA→B and its lower bound GA→B satisfy, and show how these results can be used to link and generalize existing steering criteria. 2.B. Properties In a recent work [34] the present authors introduced a measure of EPR-steering for multi-mode bipartite Gaussian states that dealt with the problem of “how much a Gaussian state can be steered by Gaussian measure- ments”. This measure GA→B was defined as the amount of violation of the following criterion by Wiseman et al. [4, 9], σAB + i (0A ⊕ΩB) ≥0. (18) Violation of (18) gives a necessary and sufficient con- dition for “A →B” steerability of Gaussian states by Gaussian measurements. We recall from the original papers [4, 9], where the details can be found, that for two modes the condition (18) is violated iff det M B σ < 1,
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5 hence equivalently iff GA→B (σAB) > 0, where the Gaussian steering measure is defined in (17). In a two- mode continuous variable system, a non-zero value of Gaussian steering GA→B > 0 detected on a CM σAB, which implies a non-zero value of the more general measure SA→B > 0 due to (16), constitutes therefore not only a necessary and sufficient condition for the steerability by Gaussian measurements of the Gaussian state ˆρG AB defined by σAB, but also a sufficient condi- tion for the steerability of all (non-Gaussian) states ˆρAB with the same CM σAB. While SA→B is hard to study in complete gener- ality, its lower bound however has been shown to satisfy a plethora of valuable properties. In [34] we showed that Gaussian steering acquires for two modes a form of coherent information [44], GA→B (σAB) = max{0, S (A) −S (σAB)}, with the R´enyi-2 entropies S (σ) = 1 2 ln (det σ) [45] replacing the standard von Neumann ones. Thanks to this connection GA→B (σAB) was shown to satisfy various properties that we repeat here without proof: (a) GA→B (σAB) is convex and ad- ditive; (b) GA→B (σAB) is monotonically decreasing un- der Gaussian quantum operations on the (untrusted) party Alice; (c) GA→B (σAB) = E (σp AB) for σp AB pure, and, (d) GA→B (σAB) ≤E (σAB) for σAB mixed, where E denotes the Gaussian Renyi-2 entropy measure of en- tanglement [45]. In the light of the recently developed resource theory of steering [35] properties (a) and (b) should be satisfied by any proper measure of steering, while properties (c) and (d) should be satisfied by any quantifier that respects the hierarchy of quantum cor- relations. The present paper, thus, validates all the al- ready established properties of GA→B as an indicator of steerability by Gaussian measurements, and extends them to arbitrary states. Interestingly, (16) suggests that by accessing only the second moments of an arbitrary state, one will not overestimate its steerability according to our measure. We can make this observation rigorous by showing that the steering quantifier SA→B satisfies an impor- tant extremality property as formalized in [36]. Namely, the Gaussian state ˆρG AB defined by its CM σAB mini- mizes SA→B among all states ˆρAB with the same CM σAB. This follows by recalling that the Reid product (11), which appears in (10), is independent from the (Gaussian versus non-Gaussian) nature of the state, and that linear inference estimators are globally optimal for Gaussian states as mentioned above [10]. This entails that the middle term in (10) can be recast as min {UG}∪{UnG}(∆2 infXB∆2 infPB)ˆρAB
min {UG}∪{UnG}(∆2 infXB∆2 infPB)ˆρG AB
min {UG}∪{UnG}(∆2 minXB∆2 minPB)ˆρG AB = F(ˆρG AB) , (19) where, for the sake of clarity, we have explicitly indi- cated the states on which the variances are calculated: ˆρAB denotes an arbitrary two-mode state, and ˆρG AB cor- responds to the reference Gaussian state with the same CM. Therefore, combining Eqs. (5), (10), (16), and (19), we can write the following chain of inequalities for the “A →B” steerability of an arbitrary two-mode state ˆρAB, SA→B (ˆρAB) ≥SA→B �ˆρG AB ≥GA→B (σAB) . (20) The leftmost inequality in (20) embodies the desired ex- tremality property [36] for our steering measure. This is very relevant in a typical experimental situation, where the exact nature of the state ˆρAB is mostly unknown to the experimentalist. Then, thanks to (20) we rest as- sured that, by assuming a Gaussian nature of the state under scrutiny, the experimentalist will never overesti- mate the EPR-steering correlations between Alice and Bob as quantified by the measure defined in (5). Finally, coming to operational interpretations for our proposed steering quantifier SA→B, we show that it is connected to the figure of merit of semi-device indepen- dent quantum key distribution [6], that is, the secret key rate. In the conventional entanglement-based quantum cryptography protocol [46], Alice and Bob share an ar- bitrary two-mode state ˆρAB, and want to establish a se- cret key given that Alice does not trust her devices. By performing local measurements (typically homodyne detections) on their modes, and a direct reconciliation scheme (where Bob sends corrections to Alice) they can achieve the secret key rate [6] K ≥max ( 0, ln
2 e p ∆2 infXB∆2 infPB !) . (21) Notice that the secret key rate depends on the expres- sion in (11), which is not unitarily invariant. Therefore, it can be optimized over local unitary operations. In the case where ∆2 infXB∆2 infPB takes its minimum value for the given shared ˆρAB, the lower bound on the cor- respondingly optimal key rate Kopt can be readily ex- pressed in terms of the “A →B” steerability measure, yielding Kopt ≥max 0, SA→B (ˆρAB) + ln 2 −1
. (22) Thus, SA→B quantifies a guaranteed key rate for any given state. If a reverse reconciliation protocol is used (in which Alice sends corrections to Bob) the quanti- fier SB→A of the inverse steering direction enters (22) instead. Thus, one sees that the asymmetric nature of steering correlations can play a decisive role in com- munication protocols that rely on them as resources. In the cryptographic scenario discussed, if the shared state ˆρAB is only one-way steerable, say A →B, then a reverse reconciliation protocol that relies on SB→A is not possible. A looser lower bound to the key rate (22) can also be expressed in terms of GA→B by using (16),
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6 Δinf 2 XB Δinf 2 PB det Μσ B 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 r EPR-steering parameter Fig. 1. (Color online) We illustrate the performance of Reid’s [25] and Wiseman et al.’s [4] EPR-steering criteria for the steer- ing detection of a pure two-mode squeezed state with squeez- ing r, with CM transformed from the standard form by the ap- plication of a local symplectic transformation parameterized as in (13), with uA(B) = vA(B)/(1+v2 A(B)), wA(B) = 1+v2 A(B) (in the plot, we choose vA = 0.16 and vB = 0.19). The criteria are represented by their figures of merit, namely the product of conditional variances (dashed blue line) for Reid’s criterion (12) and the determinant det MB (solid orange line) for Wise- man et al.’s criterion (18). The two-mode squeezed state is steerable for all r > 0, but the aforementioned criteria detect this steerability only when their respective parameters give a value smaller than unity (straight black line). As one can see, we have det MB < 1 for all r > 0 and independently of any local rotations, while Reid’s criterion detects steerability only for a small range of squeezing degrees and is highly affected by local rotations. If the state is sufficiently rotated out of the standard form, the unoptimized Reid’s criterion will not be able to detect any steering at all. in case one wants to study the advantage that Gaussian steering alone gives for the key distribution, or one just wants to get an estimate. 3. Reid, Wiseman, and a stronger steering crite- rion Finally, we discuss the implications of our work on ex- isting EPR-steering criteria [4, 25]. The second order EPR-steering criteria by Reid (12) and Wiseman et al. (18), are perhaps the most well-known ones for continu- ous variable systems. Although a comparison between them has been issued before in a special case (two- mode Gaussian states in standard form) [9], they ap- pear to exhibit quite distinct features in general [26]. On one hand, Wiseman et al.’s criterion (18), defined only in the Gaussian domain, is invariant under local sym- plectics and provides a necessary and sufficient condi- tion for steerability of Gaussian states under Gaussian measurements. On the other hand, Reid’s criterion (12) is applicable to all states but is not invariant under lo- cal symplectics and as a result it cannot always detect steerability even on a Gaussian state. As an illustra- tive example, we show in Fig. 1 the performance of the two criteria for steering detection in a pure two-mode squeezed state, locally rotated out of its standard form. One can clearly see that Wiseman et al.’s criterion is su- perior to the non-optimized Reid’s one, which fails to detect steering in the regimes of very low or very high squeezing [47]. However, it was previously argued [26] that Wise- man et al.’s stronger condition could not qualify as a general steering criterion, and could not be used in an experimental scenario where sources of non- Gaussianity may be present, since the derivation of the criterion and its validity were limited strictly to the Gaussian domain, while general EPR-steering criteria should be defined for all states and measurements. The exact connection established by (14) between Wiseman et al.’s figure of merit, det M B σ , and Reid’s product of inference variances (11), makes us realize now that the two criteria are just two sides of the same coin; i.e., Wiseman et al.’s criterion represents the best perfor- mance of Reid’s criterion when optimal Gaussian ob- servables are used for the latter. As a byproduct of this connection, we have thus upgraded the validity of Wiseman et al.’s criterion to arbitrary two-mode contin- uous variable states. Namely, our results imply that a violation of (18) on any state ˆρAB with CM σAB is sufficient to certify its “A →B” steerability, as de- tectable in laboratory by optimal quadrature measure- ments. This condition can be thus regarded, to the best of our current knowledge, as the strongest experimen- tally friendly EPR-steering criterion for arbitrary two- mode states involving moments up to second order. 4. Conclusion In conclusion, we introduced a quantifier of EPR- steering for arbitrary bipartite two-mode continuous- variable states, that can be estimated both experimen- tally and theoretically in an analytical manner. Gaus- sian states were found to be extremal with respect to our measure, minimizing it among all continuous vari- able states with fixed second moments [36]. By fur- ther restricting to Gaussian measurements, we obtained a computable lower bound for any (Gaussian or non- Gaussian) two-mode state, that was shown to satisfy a plethora of good properties [34]. The measure pro- posed in this paper is seen to naturally quantify a guar- anteed key rate for semi-device independent quantum key distribution [6]. Finally, this work generalizes and sheds new light on existing steering criteria based on quadrature measurements [4, 25]. Nevertheless many questions still remain, comple- menting the ones posed previously in [34]. To begin with, it would be worthwhile to extend the results pre-
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7 sented here to multi-mode states and see whether a con- nection similar to (14) still holds. We also leave for further research the possibility that our quantifier (or its lower bound) may enter in other figures of merit for protocols that consume steering as a resource, like the tasks of secure quantum teleportation and teleam- plification of Gaussian states [7, 48] or entanglement- assisted Gaussian subchannel discrimination with one- way measurements [8]. Moreover, the proved con- nection of the measure with entropic quantities in the purely Gaussian scenario could be an instance of a more general property that we believe is worth investigat- ing, possibly making the link with the degree of vio- lation of more powerful (nonlinear) entropic steering tests [30, 31]. Finally, it is presently unknown whether the right- most inequality in (20) is tight; namely, whether or not non-Gaussian unitaries in the minimization of (5) can give rise to higher steerability of Gaussian states, com- pared to optimal Gaussian unitaries. This is related to the open question, first posed in [4], of whether or not there exist steerable Gaussian states which nonetheless cannot be steered by Gaussian measurements; so far, such states have not been found even by resorting to nonlinear steering criteria [30, 49]. On one hand, one would expect that Gaussian measurements are optimal for steering Gaussian states, since Gaussian operations and decompositions are indeed optimal for (provably a large class of) two-mode Gaussian states when entan- glement and discord-type correlations are considered [50–53]. On the other hand, non-Gaussian measure- ments are always required to violate any Bell inequal- ity on Gaussian states [54, 55] by virtue of their positive Wigner function, hence Gaussian measurements are in contrast completely useless for that task. Since steering is the ‘missing link’ which sits just below nonlocality and just above entanglement in the hierarchy of quan- tum correlations [4, 9], pinning down precisely the role of Gaussian measurements for steerability of Gaussian states would be particularly desirable. Here, we dare to conjecture that SA→B(ˆρG AB) = GA→B(σAB), that is, that the general measure of EPR-steering introduced in this paper would reduce exactly to the measure of Gaussian steering proposed in [34], for all two-mode Gaussian states; this would signify the optimality of Gaussian measurements for steerability of Gaussian states. How- ever, a proof or disproof of this tempting hypothesis is beyond our current capabilities, and is left here as a fu- ture challenge to the community. Acknowledgments This work was supported by the University of Notting- ham (International Collaboration Fund) and the Foun- dational Questions Institute (FQXi-RFP3-1317). We are grateful to A. Ac´ın, D. Cavalcanti, E. G. Cavalcanti, A. Doherty, Q. Y. He, A. R. Lee, M. Piani, S. Ragy, M. D. Reid, P. Skrzypczyk, and especially H. Wiseman for valuable discussions. References [1] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum- mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935). [2] E. Schr¨odinger, “Discussion of probability relations be- tween separated systems (I),” Proc. Camb. Phil. Soc. 31, 553 (1935). [3] E. Schr¨odinger, “Discussion of probability relations be- tween separated systems (II),” Proc. Camb. Phil. Soc. 32, 446 (1936). [4] H. M. Wiseman, S. J. Jones, and A. C. 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