Detecting Einstein-Podolsky-Rosen Steering in Non-Gaussian Spin States from Conditional Spin-Squeezing Parameters

Detecting Einstein-Podolsky-Rosen steering in non-Gaussian spin states from conditional spin-squeezing parameters

Jiajie Guo,1 Feng-Xiao Sun,1 Daoquan Zhu,1 Manuel Gessner,2, ∗ Qiongyi He,1, 3, 4, † and Matteo Fadel5, ‡ 1State Key Laboratory for Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics, & Collaborative Innovation Center of Quantum Matter, Peking University, Beijing 100871, China 2Laboratoire Kastler Brossel, ENS-UniversitéPSL, CNRS, Sorbonne Université,Collège de France, Paris, France 3Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China 4Peking University Yangtze Delta Institute of Optoelectronics, Nantong 226010, Jiangsu, China 5Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland (Dated: June 25, 2021) We present an experimentally practical method to reveal Einstein-Podolsky-Rosen steering in non-Gaussian spin states by exploiting a connection to quantum metrology. Our criterion is based on the quantum Fisher information, and uses bounds derived from generalized spin-squeezing parameters that involve measurements of higher-order moments. This leads us to introduce the concept of conditional spin-squeezing parameters, which quantify the metrological advantage provided by conditional states, as well as detect the presence of an EPR paradox.

Introduction.– Einstein-Podolsky-Rosen (EPR) steering over-squeezed was first termed by Schrodinger¨ [1] to describe the contradic- spin state tion to local complementarity in the EPR paradox [2]. As an intermediate correlation, EPR steering is stronger than entanglement but not as general as Bell nonlocality [3]. Being eas- ier to generate and detect than nonlocality renders EPR steering a valuable resource for a variety of quantum information tasks [4–6], such as quantum teleportation [7,8], one-side device independent QKD [9–12], quantum secret sharing [13– 15] and assisted quantum metrology [16]. splitting Typically, EPR steering is revealed from the violation of a measurement conditional state criterion based on a local uncertainty relation [4, 17]. For this direction reason, such criteria are often expressed in terms of variances A B of linear operators, and therefore best suited to reveal steering in Gaussian states, where the correlations are fully described by first and second-order moments. Recently, non-Gaussian states were shown to have more competitive advantages in several quantum information protocols [18–21]. However, their nontrivial correlations appear in higher-order moments steering of physical operators, leading to the failure of steering criteria limited to linear observables. To detect non-Gaussian steering, some approaches have taken higher-order moments into account [22, 23]. For example, nonlinear correlations in a FIG. 1. Illustration of the investigated protocol. A non-Gaussian three-photon down-conversion process with quadratic steer- (over-squeezed) spin state is prepared in an ensemble of particles, arXiv:2106.13106v1 [quant-ph] 24 Jun 2021 ability index were considered in Ref. [24]. A steering crite- that are then distributed to form subsystems A and B. Because of quantum correlations, a measurement on A projects B into one of rion derived from Hillery and Zubairy’s multimode entangle- several highly sensitive conditional states. The knowledge of A’s ment criterion [25] has been investigated to detect steering in measurement setting and result allow B to make the best use of its multipartite scenario [26] and further extended to a higher- state by optimising his local measurement. With the criteria we pro- order version in a two-well BEC ground state [27]. Never- pose, steering between the two subsystems can be concluded. theless, these methods are specifically tailored to particular states, and a general steering criterion for non-Gaussian states is still highly desirable to further unlock their potential appli- provements to overcome the standard quantum limit and are cations. nowadays routinely prepared in a variety of platforms, from Nonclassical spin states are many-body quantum states of solid state systems to atomic ensembles [28]. Recent stud- great interest for fundamental studies as well as for practical ies have in particular explored the metrological potential of applications. For example, squeezed spin states have attracted non-Gaussian spin states, both in theory [29, 30] and experi- increasing attention in quantum metrology for precision im- ment [31–35]. 2

2 Methods derived from quantum metrology [36–40], already Var[θest] = χ [ρ, H, M]/m, where allow for the efficient detection of multiparticle entanglement Var[ρ, M] without addressing individual spins. In particular, the quan- χ2 ρ, H, M [ ]:= 2 (1) tum Fisher information (QFI) constitutes a powerful tool for |h[H, M]iρ| capturing even strongly non-Gaussian features of quantum is the spin-squeezing parameter [36]. states by probing them for their sensitivity under small pertur- For an unbiased estimation, a fundamental limit to the bations [31]. Very recently, the QFI was also used to formulate sensitivity is given by the Cramer-Rao´ bound Var[θ ] ≥ a criterion for EPR correlations [16], thus providing us with a est (mF[ρ, H, M])−1, where F[ρ, H, M] is the Fisher information powerful method for detecting EPR steering in non-Gaussian (FI) [41, 42]. By optimizing over all observables M, the states. maximum value of the FI defines the quantum Fisher infor- However, accessing the QFI is often challenging. Deter- mation (QFI), i.e. FQ[ρ, H] = maxM F[ρ, H, M], which de- mining the QFI of arbitrary mixed states requires full knowl- termines the optimal sensitivity potential of the probe state −2 edge of the quantum state. On the other hand, efficient approx- ρ [43]. In conclusion, we have χ [ρ, H, M] ≤ F[ρ, H, M] ≤ imations based on the full counting statistics demand that a FQ[ρ, H][39, 44]. For practical experiments, the achiev- −2 carefully chosen observable is measured with high resolution, able sensitivity can be optimized by maximizing χ [ρ, H, M] which is also difficult in multipartite systems. Spin-squeezing over a set of measurement operators M that can be real- parameters [36] have proven to be efficient alternatives with istically implemented [29]. Denoting with X a basis for high practical relevance, especially for Gaussian spin systems; such measurements, we can achieve the maximal sensitivity −2 suitable generalizations are also able to capture non-Gaussian maxM∈span(X) χ [ρ, H, M]. features from higher-order moments [29]. But so far they have Assisted phase estimation with conditional squeezing been limited to the detection of entanglement in a many-spin parameter.– In the assisted phase-estimation protocol [16], ensemble with collective measurements. Bob’s estimation of θ is improved by communication from Alice about her measurement setting and result, Y and b. This Here, we introduce the concept of conditional spin- information allows Bob to choose a measurement observable squeezing parameters, and based on that we propose a practi- M ∈ span(X) that is optimally tailored to the conditional state B cal and convenient witness for EPR steering in split nonclas- ρb|Y . This way, Bob can achieve on average an estimation sical spin states. For the purpose of detecting non-Gaussian sensitivity given by the conditional spin-squeezing parame- steering, spin observables involving higher-order moments are ter [45] taken into consideration. An optimization of the measure- −2 B|A X −2 B ment within these accessible higher-order observable ensem- (χ ) [A, H, X, Y]:= p(b|Y) max χ [ρb|Y , H, M] . M∈span(X) bles leads to conditional nonlinear spin-squeezing parameters, b (2) whose potential to detect steering in a wider class of non- Here, we introduced the definition of assemblages A(b, Y) = Gaussian states is explored. We demonstrate that conditional p(b|Y)ρB , which are determined by the local probability dis- spin-squeezing parameters approximate the conditional QFI b|Y tribution p(b|Y) for results b conditioned on Alice’s measure- criterion [16] and as we increase the order of the measured ment observable Y and Bob’s conditional state ρB . Note that moments this approximation ultimately converges to the QFI b|Y the ultimate limit in phase estimation for a specific measure- criterion. In addition, we also prove that the conditional spin ment Y for Alice is expressed by the conditional Fisher infor- squeezing parameters detect a larger class of steerable cor- mation [16] relations than Reid’s criterion [4, 17]. As a detailed study, X we analyze their performance using analytical results for split B|A B F [A, H, Y]:= p(b|Y)FQ[ρb|Y , H] , (3) one-axis-twisted states, where a hierarchy of criteria is clearly b shown. Our work provides an experimentally practical tool to witness non-Gaussian steering, which helps to further investi- In fact, we can obtain a chain of inequalities (see Supplemen- gate the quantum information of non-Gaussian spin states and tary Information Sec. IA) paves a way to exploiting their promising potential. 2 |h[H, M]iρB | FB|A[A, H, Y] ≥ (χ−2)B|A[A, H, X, Y] ≥ , Phase estimation and the spin-squeezing parameter.– In VarB|A[A, M, Y] a typical phase estimation protocol, a generator H imprints an (4) unknown parameter θ on quantum state ρ. An observable M is which results in a hierarchy of EPR steering criteria that will then measured on the probe state ρ(θ) and an estimator θest for be discussed below. θ is constructed as a function of the measurement results. This Connection to EPR steering.– Besides the estimation of protocol is repeated m times and, for unbiased cases, the vari- the phase θ, we could be interested in estimating its generator ance of the estimator Var[θest] represents the deviation of the H. As θ and H are conjugate variables, the uncertainty princi- estimate θest to the parameter θ. A simple estimator known as ple prevents their simultaneous knowledge with arbitrary pre- the method of moments is constructed from the average value cision [46]. By making use of EPR steering from Alice to of M and yields, in the limit m 1 the phase uncertainty Bob in the assisted metrology protocol, an inference of these 3 properties can be realized below the local uncertainty limit, while in (8) only a single M is used for the entire assemblage. −1 which is given by Var[θest]Var[Hest] ≥ (4m) [16]. Here, As we will see below, this leads in particular to an increased Var[θest] and Var[Hest] are inference variances. Based on the potential to reveal non-Gaussian EPR steering in a wider class additional information of Alice’s measurement setting X and of states, especially when the set X contains higher-order mo- result a, Bob uses the estimator hest(a) to predict the result ments of the collective spins. h of his local measurement H with the inference variance Reduction to linear-estimate Reid’s criterion.– If Bob’s P 2 Var[Hest]:= a,h p(a, h|X, H)(hest(a) − h) . A lower bound estimator hest(a) depends linearly on Alice’s measurement re- for Var[Hest] is given by the conditional variance [4] sult a and takes the form hest(a) = ga + d, optimal estimates are obtained by minimizing the inference variance Var[Hest]. B|A X B Var [A, H, X]:= p(a|X)Var[ρa|X, H] . (5) Based on that, a well-known linear-estimate Reid’s criterion a commonly used in experiments is

Analogously, we can obtain Var[θest] with a different choice 2 |h[H, M]iρB | for Alice’s measurement settings (denoted by Y) and Bob’s ∆ := − 4Var[X − gH] ≤ 0 . (9) 4 Var[Y + g0 M] measurement M. The uncertainty bound is a witness of EPR steering, whose violation implies that Alice’s measure- Note that in a Gaussian system, where quantum correlations ments can steer Bob’s states to overcome Bob’s local phase- are well characterized with first and second-order moments, generator uncertainty relation. the best estimator equals the optimized linear estimator [4]. Consider now a local hidden state (LHS) model [3], de- In this case, we have Var[X − gH] = VarB|A[A, H, X], which scribed by a classical random variable λ with probability dis- leads to ∆3 = ∆4 for Gaussian states. tribution p(λ). The assemblage in this case can be written as We define the maximum value of the left-hand side of the A P | B (a, X) = a,λ p(a X, λ)p(λ)σλ . In Ref. [16], a steering cri- above criteria as δi = maxH∈span(H),M∈span(X) ∆i, respectively terion based on the QFI is proposed. For any LHS model, it (see Supplementary Information Sec. IB,IIB,IIIC,IIID). As a holds result, a hierarchy of criteria reads

B|A B|A ∆1 := F [A, H, Y] − 4Var [A, H, X] ≤ 0 , (6) δ1 ≥ δ2 ≥ δ3 ≥ δ4 . (10) independently of the choices of Alice’s measurement. In the following, we will compare these criteria for a scenario As the conditional squeezing parameter provides a lower of experimental relevance. bound to the conditional FI (4), this allows us to formulate Split spin-squeezed state.– Squeezed states play a key the following steering criterion: For any assemblage A that role in measurement sensitivity enhancement, overcoming the admits a LHS model, we have standard quantum limit in quantum metrology [28, 36, 47–

−2 B|A B|A 49]. Experimentally, these are routinely adopted in optical and ∆2 := (χ ) [A, H, X, Y] − 4Var [A, H, X] ≤ 0 . (7) atom interferometers, e.g. Ramsey spectroscopy [28], atom clocks [50] and gravitational-wave detection [51]. Many ex- This criterion is one of the main results of this work. The vio- periments have demonstrated the preparation of spin squeez- lation of (7) reveals useful EPR steering in the assisted metro- ing in atomic systems [32, 52–55]. Here, we focus on spin logical protocol. squeezed states prepared via one-axis twisting (OAT) dynam- Reduction to Reid’s criterion.– From the previous chain ics [56], one of the paramount approaches to generate squeez- of inequalities (4), it is immediate to recover Reid’s crite- ing via atomic collisions. rion [4, 17] (here expressed in a linearized form) Initially, the atomic ensemble is in a coherent spin state, 2 |h i B | which consists of N spins polarized along the x direction. [H, M] ρ B|A ∆3 := − 4Var [A, H, X] ≤ 0 . (8) Then, the time evoution of the OAT Hamiltonian H = ~χS 2 VarB|A[A, M, Y] z can be parametrized by µ = 2χt, and the state reads As defined in Eq. (5), the conditional variance represents the s N ! average of individual variances for Bob’s conditional states, 1 X N −i µ (N/2−k)2 |ψ(µ)i = √ e 2 |ki . (11) determined as the minimized inference variance [4]. For lin- N k 2 k=0 ear observables, Reid’s criterion is very powerful for Gaus- sian states; in a continuous variable setting, it can be shown to Here, k labels the basis of Dicke states |ki. For short inter- be necessary and sufficient for steering detection by Gaussian action times, OAT results in nearly-Gaussian spin squeezed measurements [3], while it may fail to detect steering in non- states, however, as time increases the state becomes over- Gaussian cases. Because of the right-hand side in Eq. (4), we squeezed and significantly non-Gaussian [39]. obtain ∆2 ≥ ∆3, that means the criterion ∆2 we proposed con- In order to use such states for assisted phase-estimation pro- tains all LHS models based on Reid’s uncertainty relations. A tocols, we spatially separate each of the two spin up/down crucial advantage of ∆2 over Reid’s criterion ∆3 is the possi- modes into two parts, A and B, which can be described by bility to adapt the measurement observable M to each condi- a beam splitter transformation, so that a four-mode split spin- tional state ρb|Y individually [see the maximization in Eq. (2)], squeezed state is finally obtained [57]. This state has also been 4

FIG. 2. Steering detection for spilt spin states |Φ(µ)i with total atomic number N=20. (a) A hierarchy of criteria with optimized measurement (2) (2) (2) operators, where the dashed lines represent the optimized second-order criteria δ2 and δ3 involving nonlinear spin operators ensemble S . (b) Comparison among ﬁrst terms of optimized criteria δ1,2,3, which is also the chain of inequalities in (4).

prepared experimentally [58]. In the bipartite Dicke basis, it Reid’s criterion, i.e., ∆4, with non-linear operators is intro- can be written as duced. Here, we also extend our conditional spin-squeezing s parameter based criterion ∆2 and general Reid’s criterion ∆3 N NA N−NA ! ! ! 1 X X X N N N − N to nonlinear version, where they can be further optimized by |Φ(µ)i = A A N taking into account a set X of higher-order measurements for 2 NA kA kB NA=0 kA=0 kB=0 Bob (i.e. M). When M is a product of up to n linear spin −i µ (N/2−k −k )2 (n) × e 2 A B |k i |k i . (12) A NA B N−NA observables, optimized nonlinear δ2,3 are still upper bounded by the Fisher criterion δ1, but this bound becomes increas- ∈ { } | i where Nα is the number of particles in α A, B , kα Nα rep- ingly tight as n grows larger. To be concrete, let us start (2) resents the Dicke states with kα spins down and Nα − kα spins with the second-order criteria, X = S , where an ensem- up along the z direction. ble of linear and symmetric quadratic spin operators S(2) = 2 2 2 1 1 1 The properties of split spin states can be characterized by (S x, S y, S z, S x, S y , S z , 2 {S x, S y}, 2 {S x, S z}, 2 {S y, S z}) is intro- α P (i) local collective spin observables, that is S = i∈α σ /2, duced. We refer to Ref. [34] for an experimental measurement where σ(i) is the vector of Pauli matrices acting on particle of such observables. Moreover, it was shown in Ref. [30] how i. such observables may become accessible by a second OAT Measurement optimisation.– The sharpest formulations evolution before the measurement of a linear spin observable. (2) (2) of the above criteria are obtained by optimizing the measure- For our second-order criteria ∆2 and ∆3 , Alice’s measurement observables X, Y for Alice, and H, M for Bob, respec- ments X, Y and Bob’s generator H for the phase imprinting (1) tively. In linear cases, S = (S x, S y, S z) is used to describe evolution are still linear, but the measurement operator for all the local collective spin operators for Alice and Bob. The Bob takes into account second-order operators M = m · S(2), (1) m ∈ R9 linear measurements that appear in the ﬁrst-order criterion ∆2 with . Analogously, we can also obtain higher-order have been extensively used in experiments [28]. We consider criteria, leading to another chain of inequalities the squeezing direction of the split spin squeezed states to de- δ(1) ≤ δ(2) ≤ δ(3) ≤ · · · ≤ δ , (13) ﬁne the z axis and the anti-squeezing direction y. Therefore, i i i 1 Alice’s measurement settings X and Y can be also restricted for both criteria i = 2, 3. to the yz plane. To the optimize measurement directions of As illustrated in Fig.2(a), for N = 20 split spin states, Alice’s and Bob’s observables, we construct the moment ma- we obtain analytically optimized criteria δi evolved with trix [29] and covariance matrix for Bob’s conditional states OAT squeezing parameter µ, where both the hierarchy re- (see Supplementary Information Sec. IIB for details). lations (10) and (13) are shown clearly. At small squeez- For the metrological characterization of non-Gaussian spin ing levels µ, the evolution generates near-Gaussian split spin- states, higher-order moments of physical observables are of squeezed states and all criteria detect steering and tend to great importance [29]. Note that in [24], the linear-estimate converge. However, if only linear operators are considered, 5

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SUPPLEMENTARY INFORMATION

I. FISHER CRITERION

A. Deﬁnition: Fisher criterion

As proposed in [16], a formulation of an EPR criterion can be derived from the quantum conditional variance

B|A X B Var [A, H]:= min p(a|X)Var[ρ | , H] . (14) Q X a X a and the quantum conditional Fisher information

B|A X B F [A, H]:= max p(b|Y)FQ[ρ | , H] . (15) Q Y b Y b For any assemblage A that admits a LHS model, the following bound holds:

B|A B|A FQ [A, H] ≤ 4VarQ [A, H] . (16) Also here, note that the quantum conditional variance (14) and the quantum conditional Fisher information (15) are deﬁned as an optimisation over Alice’s measurement. For simplicity, however, we always compute a conditional Fisher Information by choosing a speciﬁc measurement Y for Alice

B|A X B F [A, H, Y]:= p(b|Y)FQ[ρb|Y , H] , (17) b and also identify the conditional variance as calculated based on a speciﬁc measurement X for Alice with the symbol

B|A X B Var [A, H, X]:= p(a|X)Var[ρa|X, H] , (18) a

B|A B|A B|A B|A In summary, we then have FQ [A, H] ≥ F [A, H, Y] and VarQ [A, H] ≤ Var [A, H, X]. So after choosing speciﬁc measurement setting for Alice, the Fisher criterion (16) can be written as:

B|A B|A ∆1 := F [A, H, Y] − 4Var [A, H, X] ≤ 0 , (19) its violation indicates EPR steering from Alice to Bob.

B. Calculation: Fisher criterion

Decomposing the Hamiltonian H in terms of a vector of observables H as H = nT H with real coeﬃcient vector n, the conditional variance can be written as

And analogously, we also have VarB|A[A, H, X] = nT ΓB|A[A, H, X]n, which allows us to optimize the witness over the Hamilto- nian H analytically:

B|A B|A max ∆1 = 4 max Var [A, H, Y] − Var [A, H, X] H∈span(H) n B|A B|A = λmax(Γ [A, H, Y] − Γ [A, H, X]). (24)

Since the squeezing direction lies in the yz-plane, we also restrict Alice’s measurement direction to this plane, and represent it by the angle φ on the yz-plane: cos(φ)y + sin(φ)z. We label the measurement direction φX for X , φY for Y. After optimizing also over Alice’s measurements, we thus obtain the optimized witness

B|A B|A δ1 := 4 max λmax(Γ [A, H, Y] − Γ [A, H, X]) . (25) φX ,φY

B|A For the comparison of the hierarchy (4), we limit the optimization to the ﬁrst term, i.e., 4 maxφY λmax(Γ [A, H, Y]).

C. A chain of inequalities

It holds for arbitrary measurements M that [39]

2 |h[H, M]iρ| F [ρ, H] ≥ . (26) Q Var[ρ, M] In analogy to results in [16], based on the Cauchy-Schwarz inequality and the deﬁnitions of the conditional Fisher information and the conditional variance, we obtain the chain of inequalities

2 X |h[H, M]iρB | B|A A ≥ | b|Y F [ , H, Y] p(b Y) B b Var[ρb|Y , M] P 2 | b p(b|Y) h[H, M]iρB | ≥ b|Y P B b p(b|Y)Var[ρb|Y , M] 2 |h[H, M]iρB | = P B b p(b|Y)Var[ρb|Y , M] 2 |h[H, M]iρB | = . (27) VarB|A[A, M, Y] We will use this result to lower bound the Fisher criterion when using speciﬁc measurements for Alice. Note that this relation (27) leads to the hierarchy of criteria.

II. CONDITIONAL SPIN-SQUEEZING PARAMETER BASED CRITERION

A. Deﬁnition: Conditional spin-squeezing parameter based criterion

Rather than using the QFI, which corresponds to the maximal sensitivity achievable with an optimal estimation strategy and observable, we can quantify Bob’s sensitivity assuming the estimation of the phase parameter with the method of moments based on the data obtained from a speciﬁc collective spin observable. In this case, we arrive at the quantum conditional spin-squeezing parameter (where we consider the same measurement M for diﬀerent results of Alice, even if in full generality one can have Ma|X)

−2 The bound FQ[ρ, H] ≥ χ [ρ, H, M] for all M implies that

B|A −2 B|A FQ [A, H] ≥ (χQ ) [A, H, X] . (30) This can be used to introduce a steering criterion weaker than the Fisher criterion but in general stronger than the Reid’s criterion, namely

−2 B|A B|A (χQ ) [A, H, X] ≤ 4VarQ [A, H] . (31) For a speciﬁc choice of the measurement Y, we deﬁne conditional spin-squeezing parameters

−2 B|A X −2 B (χ ) [A, H, X, Y]:= p(b|Y) max χ [ρb|Y , H, M] , (32) M∈span(X) b such that, according to the ﬁrst line in Eq. (27) we have

B|A −2 B|A FQ [A, H, Y] ≥ (χ ) [A, H, X, Y] . (33) This can be used to introduce the conditional spin-squeezing parameter based steering criterion

−2 B|A B|A ∆2 := (χ ) [A, H, X, Y] − 4Var [A, H, X] ≤ 0 . (34)

(n) (n) Furthermore, the criterion ∆2 can be extended to a higher-order version ∆2 , if nonlinear measurement settings X are taken into consideration

(n) −2 B|A (n) B|A ∆2 := (χ ) [A, H, X , Y] − 4Var [A, H, X] ≤ 0 . (35)

(n) With higher-order measurement settings, ∆2 will get closer to the Fisher criterion ∆1, and will be eventually equal to ∆1 when X becomes a complete basis of operators [29].

B. Calculation: Conditional spin-squeezing parameter based criterion

Expanding H and M in terms of vectors of observables as H = nT H and M = mT X, the ﬁrst term of (34) can be written as

M[ρ, H, X]:= CT [ρ, H, X]Γ−1[ρ, X]C[ρ, H, X] , (38)

For the conditional spin squeezing parameter, we obtain

−2 B|A X T B (χ ) [A, H, X, Y] = p(b|Y) n M[ρb|Y , H, X]n b = nT MB|A[A, H, X, Y]n , (39) where we introduced the conditional moment matrix

B|A X B M [A, H, X, Y] = p(b|Y)M[ρb|Y , H, X] . (40) b 10

Next, we choose an optimal Hamiltonian H, i.e., the vector n to maximize the steering witness. Together with (21), we obtain max (χ−2)B|A[A, H, X, Y] − 4VarB|A[A, H, X] = max nT MB|A[A, H, X, Y]n − 4nT ΓB|A[A, H, X]n H∈span(H),M∈span(X) n B|A B|A = λmax(M [A, H, X, Y] − 4Γ [A, H, X]) . (41) At last we choose optimal measurements for Alice. In analogy to our previous derivation we obtain

B|A B|A δ2 := max λmax(M [A, H, X, Y] − 4Γ [A, H, X]) . (42) φX ,φY

For Alice, the optimal measurement direction φX, φY can be found by searching in the yz-plane (φX,Y ∈ [0, π]). And for Bob, the optimal phase generator’s direction nopt is the eigenvector corresponding to the maximum eigenvalue in (42). Finally, the −1 B B optimal measurement direction mopt for the other measurement operator M is mopt = αΓ [ρb|Y , X]C[ρb|Y , H, X]nopt, with a real normalization constant α [29]. B Note that here we optimize M for each conditional state ρb|Y individually, i.e., Bob may choose to measure a diﬀerent observable, depending on the information that is provided by Alice. Below when we introduce Reid’s criterion, we will provide a diﬀerent optimization, where only one optimal M may be chosen for one Alice’s measurement Y. Besides the hierarchy (27), this is the main advantage of ∆2 compared to Reid’s criterion ∆3. We obtain a higher-order versions (35) by replacing Bob’s linear measurement setting X with nonlinear one X(n)

(n) B|A (n) B|A δ2 := max λmax(M [A, H, X , Y] − 4Γ [A, H, X]) . (43) φX ,φY

(1) (1) B|A (1) When only the linear collective spin operators are involved X = S , the ﬁrst term of δ2 is maxφY λmax(M [A, H, X , Y]. (n) (n) B|A (n) And for n-order collective spin operators X = S (n = 2, 3, 4, ...), it becomes maxφY λmax(M [A, H, X , Y].

III. REID’S CRITERION

A. Deﬁnition: Reid’s criterion

In Reid’s criterion [4], based on the information of Alice’s measurement X and result a, Bob chooses an estimator hest(a) to predict the value and measures H to yield the result h. The average deviation is called inference variance

X 2 Var[Hest]:= p(a, h|X, H)(hest(a) − h) , (44) a,h where p(a, h|X, H) is the joint probability. The other inference variance Var[Mest] can be obtained analogously by measurement settings Y and M. The sharpest formulation of Eq. (44) is thus obtained by minimizing the estimation error. The optimal estimator hest(a) = B P B 2 2 Tr{ρa|X H} attains the lower bound Var[Hest] ≥ a p(a|X)Var[ρa|X, H], where Var[ρ, H] = hH iρ − hHiρ is the variance with B|A hOiρ = Tr{ρO}. Optimising over Alice’s measurement setting X leads to the quantum conditional variance VarQ [A, H](14), B|A and any ﬁxed choice of X yields its upper bound Var [A, H, X](18). In summary, we then have (for any Hest)

B|A B|A VarQ [A, H] ≤ Var [A, H, X] ≤ Var[Hest] . (45) Based on the quantum conditional variance, Reid’s criterion for an EPR paradox consists of a violation of the local uncertainty limit 2 |h[H, M]iρB | VarB|A[A, H]VarB|A[A, M] ≥ . (46) Q Q 4 If we chose speciﬁc measurement settings X, Y for Alice, we have

The inequality ∆3 ≤ 0 holds for all non-steerable assemblages, i.e., those that admit an LHS model, thus steering is detected by a violation of this inequality. In the simplest situations, in particular for Gaussian states, it is optimal to consider a linear estimator, i.e., hest(a) = ga + d, This yields an extensively used linear-estimate Reid’s criterion [17]

2 |h[H, M]iρB | ∆ := − 4Var[X − gH] ≤ 0 . (49) 4 Var[Y + g0 M]

B. Discussion on estimators and variances in Reid’s criterion.

In this section, we will distinguish between MSE (mean squared error) and the variance. We consider the remote estimation of measurement results h performed on Bob’s system from the results a that are acquired from local measurements on Alice’s system. For each estimator hest(a) we deﬁne the mean squared error

X 2 MSE[Hest]:= p(a, h|X, H) (hest(a) − h) , (50) a,h and the variance

X 2 Var[Hest]:= p(a, h|X, H) (hest(a) − hhesti) , (51) a,h P where hhesti = a p(a|X)hest(a) is the average of the estimator. For unbiased estimators, we have hhesti = h and MSE[Hest] = Var[Hest]. In general we have the relation

2 2 MSE[Hest] = Var[Hest] + hHesti − 2hHestHi + hH i . (52)

Let us now focus on a linear estimator constructed from the eigenvalues a of Alice’s measurement observable X for the estimation of the result h of Bob’s measurement of H. We thus take hest(a) = ga, where g ∈ R is a constant. We obtain

2 MSE[Hest] = h(gX − H) iρAB , (53) and

X 2 A Var[Hest] = p(a, h|X, H) (ga − ghXi) = Var[ρ , gX] , (54) a,h where hhesti = ghXi. It is clear that

AB Var[ρ , gX − H] ≤ MSE[Hest] , (55) with equality when ghXi = hHi. B B A/B A/B Speciﬁcally, we focus on the observables H = S y and M = S z and we denote by y and z the measurement results, i.e., A/B A/B B B A A the eigenvalues of S y and S z . We estimate Bob’s result z for S z from Alice’s result z for S z using the linear estimator B A A zest(z ) = −gzz . This will lead to

A B 2 MSE[Hest] = h(gzS z + S z ) iρAB , (56)

A B AB A B and if hgzS z + S z iρAB = 0, i.e., if the estimator is unbiased, this coincides with Var[ρ , gzS z + S z ]. In our cases, since the squeezing direction is on the yz-plane, the spin expectation for reduced states is

hS yiρA(B) = hS ziρA(B) = 0 . (57) so we have

A B A B hgzS z + S z iρAB = gzhS z iρA + hS z iρB = 0 . (58)

A B Similarly, we also have hgyS y + S y iρAB = 0. So it is proved that the estimator is unbiased in our case. 12

C. Calculation: general Reid’s criteiron.

The general Reid’s criterion ∆3 is deﬁned in Eq. (48), and its ﬁrst term can be written as

2 |h[H, M]iρB | max = nT M[A, H, X, Y]n , (60) M∈span(X) VarB|A[A, M, Y] where −1 M[A, H, X, Y] = C[ρB, H, X]T ΓB|A[A, X, Y] C[ρB, H, X] . (61)

Note that this optimization yields a single optimal measurement observable M, which is diﬀerent from Eq. (42). With Eq. (21), we then optimize the generator H

B|A δ3 := max λmax(M[A, H, X, Y] − 4Γ [A, H, X]) . (63) φX ,φY

Here, Bob’s optimal direction nopt is still the eigenvector corresponding to the maximum eigenvalue in Eq. (63), and the optimal B|A −1 B direction for Bob’s measurement M is mopt = βΓ [A, X, Y] C[ρ , H, X]nopt, with a real normalization constant β. By introducing higher-order measurement settings X(n) for Bob, we can obtain a nonlinear version of Reid’s criterion

(n) (n) B|A δ3 := max λmax(M[A, H, X , Y] − 4Γ [A, H, X]) . (64) φX ,φY

A A (n) (n) The ﬁrst term is maxφY λmax(M[ , H, X, Y]) for linear version δ3, and is maxφY λmax(M[ , H, X , Y]) for higher-order δ3 .

D. Calculation: linear-estimate Reid’s criterion

Linear-estimate Reid’s criterion is in Eq. (49), where the real constants g and g0 are chosen to minimize the inferred variances, respectively. In our split spin state cases, we consider X and H as local collective spin operators in the anti-squeezing direction A,B A,B 0 S y , and Y and M are local collective spin operators in squeezing direction S z , and the constants g, g are replaced with gy, gz, thus we obtain A B hS AS Bi − hS Ai hS Bi ∂Var[gyS y − S y ]ρAB y y ρAB y ρA y ρB = 0 → gy(opt) = , (65) A2 A 2 ∂gy hS i A − hS i y ρ y ρA A B A B A B ∂Var[gzS + S ]ρAB hS z iρA hS z iρB − hS z S z iρAB z z → g . = 0 z(opt) = 2 2 (66) ∂gz A A A hS z iρ − hS z iρA where ρAB is the split spin state, ρA(B) is the reduced state for Alice(Bob). Thus, the optimized linear-estimate Reid’s criterion for the split spin states is

B 2 |h i B | S x ρ A B − − AB δ4 := A B 4Var[gy(opt)S y S y ]ρ . (67) Var[gz(opt)S z + S z ]ρAB 13

IV. SPLIT SPIN STATES

Spin squeezed states (SSS) of an ensemble of N particles can be prepared from a coherent spin state polarized along the x 2 direction, through the action of the one-axis twisting (OAT) Hamiltonian H = ~χS z . The resulting state can be parametrized by µ = 2χt, where t is the interaction time, and it takes the form s N ! 1 X N −i µ (N/2−k)2 |ψ(µ)i = √ e 2 |ki . (68) N k 2 k=0 As before, the states |ki represent the Dicke states with k spins down and N − k spins up along z. In other words, k represent the number of excitations in two bosonic modes. Consider now the eﬀect of applying a beam splitter transformation to the two modes a and b, which results in a split spin states [57], which has been created experimentally [58]. In the bipartite Dicke basis, this state can be written as s N NA N−NA ! ! ! 1 X X X N NA N − NA −i µ (N/2−k −k )2 |Φ(µ)i = e 2 A B |k i |k i . (69) N A NA B N−NA 2 NA kA kB NA=0 kA=0 kB=0

A. EPR criterion

A Let us consider a measurement of S n by Alice, which yields the results NA (the number of particles in Alice side) and lA | i A (the result of the spin measurement along the chosen direction). Let lA n,NA denote the corresponding eigenstates of S n in the spin-NA/2 subspace, such that we can write

XN XNA N S A = l − A |l i hl | . (70) n A 2 A n,NA A n,NA NA=0 lA=0

The event of obtaining measurement results (lA, NA) for Alice occurs with probability s NA NA N−NA ! ! ! ! A 1 X X X N N − NA NA NA −i µ (N/2−k −k )2 i µ (N/2−k0 −k )2 0 p(l , N |S ) = e 2 A B e 2 A B hl | |k i hk | |l i (71) A A n 2N 0 A n,NA A NA A NA A n,NA 2 0 NA kB kA kA kA=0 kA=0 kB=0 when 0 ≤ lA ≤ NA and zero otherwise, and produces the conditional states s A −1/2 NA N−NA ! ! ! p(l , N |S ) X X N N N − N µ 2 B A A n A A −i (N/2−kA−kB) |Φ(µ) i A = e 2 hl | |k i |k i . (72) lA,NA|S n N A n,NA A NA B N−NA 2 NA kA kB kA=0 kB=0 for Bob’s system. B Let us consider a phase shift generated on Bob’s subsystem by the observable S m. Writing − XN NXNA N − N S B = l − A |l i hl | , (73) m B 2 B m,N−NA B m,N−NA NA=0 lB=0 we obtain the following ﬁrst and second moments for the conditional states

This allows us to obtain the variance of the conditional states

B B B 2 B 2 Var[|Φ(µ) i A , S ] = h(S ) i − hS i , (77) lA,NA|S n m m lA,NA m lA,NA and an upper bound for the quantum conditional variance

Since the conditional states are pure, we have

B B B B F [|Φ(µ) i A , S ] = 4Var[|Φ(µ) i A , S ] (79) Q lA,NA|S n m lA,NA|S n m and we immediately obtain also a lower bound for the conditional quantum Fisher information as

B. Reduced quantum Fisher information and variance

The reduced state for Bob’s system reads s N NA N−NA N−NA ! ! ! ! B 1 X X X X N NA N − NA N − NA −i µ (N/2−k −k )2 i µ (N/2−k −k0 )2 0 ρ = e 2 A B e 2 A B |k i hk | . (81) 2N 0 B N−NA B N−NA 2 0 NA kA kB kB NA=0 kA=0 kB=0 kB=0

This yields ﬁrst and second moments as s N N−NA NA N−NA N−NA ! ! ! ! 1 X X X X X N N N − N N − N µ 2 µ 0 2 B A A A −i (N/2−kA−kB) i (N/2−kA−k ) hS i B e 2 e 2 B m ρ = 2N 0 2 0 NA kA kB kB NA=0 lB=0 kA=0 kB=0 kB=0 N − NA 0 × lB − hlB|m,N−N |kBiN−N hk | |lBim,N−N (82) 2 A A B N−NA A and s N N−NA NA N−NA N−NA ! ! ! ! 1 X X X X X N N N − N N − N µ 2 µ 0 2 B 2 A A A −i (N/2−kA−kB) i (N/2−kA−k ) h S i B e 2 e 2 B ( m) ρ = 2N 0 2 0 NA kA kB kB NA=0 lB=0 kA=0 kB=0 kB=0 2 N − NA 0 × lB − hlB|m,N−N |kBiN−N hk | |lBim,N−N , (83) 2 A A B N−NA A leading to the variance

B B B 2 B 2 Var[ρ , S m] = h(S m) iρB − hS miρB . (84)

To determine the quantum Fisher information, we must diagonalize the state ρB: s N NA N−NA N−NA ! ! ! ! B 1 X X X X N NA N − NA N − NA −i µ (N/2−k −k )2 i µ (N/2−k −k0 )2 0 ρ = e 2 A B e 2 A B |k i hk | 2N 0 B N−NA B N−NA 2 0 NA kA kB kB NA=0 kA=0 kB=0 kB=0

N NA ! ! 1 X X 1 N N A | i h | = N N Ψ(NA, kA) Ψ(NA, kA) 2 2 A NA kA NA=0 kA=0 XN XNA = p(NA, kA) |Ψ(NA, kA)i hΨ(NA, kA)| , (85)

NA=0 kA=0 15 where the eigenstates read s N−NA ! 1 X N − NA −i µ (N/2−k −k )2 | i 2 A B | i Ψ(NA, kA) = √ e kB N−NA , (86) N−NA kB 2 kB=0 and the corresponding eigenvalues are ! ! 1 N NA p(NA, kA) = N+N . (87) 2 A NA kA

2 B B P (pi−p j) B 2 We can now use the expression FQ[ρ , S ] = 2 |hψi|S |ψ ji| to write z i, j pi+p j z

V. OTHER RESULTS

As depicted schematically in Figs.3 and4 below, we also investigate the optimized criteria and their respective ﬁrst terms (expressing the assisted phase sensitivity) for split spin states with diﬀerent total atom numbers N, where N is assumed to be even. 16

FIG. 3. Optimized criterion for split spin states |Φ(µ)i with a total number of atoms N = 4, 6, 8, ..., 18. The hierarchy (10) and (13) are clearly (2) illustrated here. In addition, when the atom number is small (N ≤ 8), our second-order conditional spin-squeezing paramter criterion δ2 is able to reveal steering in the same range as the Fisher criterion δ1 does, and even coincides with δ1 when N = 4.

FIG. 4. First terms of optimized criteria δ1,2,3, i.e., the chain of inequalities (4), for split spin states |Φ(µ)i with a total number of atoms (2) N = 4, 6, 8, ...18. It can been seen that when N = 4, the ﬁrst term of δ2 is equal to δ1, leading to the agreement of the two criteria in this case.