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, y and Matteo Fadel5, z 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-Universit´ePSL, CNRS, Sorbonne Universit´e,Coll`ege 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 entan- glement but not as general as Bell nonlocality [3]. Being eas- ier to generate and detect than nonlocality renders EPR steer- ing a valuable resource for a variety of quantum information tasks [4–6], such as quantum teleportation [7,8], one-side de- vice 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 steer- ing, 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 jh[H; M]iρj 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 maxM2span(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 2 span(X) that is optimally tailored to the conditional state B cal and convenient witness for EPR steering in split nonclas- ρbjY . 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 BjA X −2 B ment within these accessible higher-order observable ensem- (χ ) [A; H; X; Y]:= p(bjY) max χ [ρbjY ; H; M] : M2span(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(bjY)ρB , which are determined by the local probability dis- spin-squeezing parameters approximate the conditional QFI bjY tribution p(bjY) 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 bjY 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 BjA B F [A; H; Y]:= p(bjY)FQ[ρbjY ; 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 jh[H; M]iρB j FBjA[A; H; Y] ≥ (χ−2)BjA[A; H; X; Y] ≥ ; Phase estimation and the spin-squeezing parameter.– In VarBjA[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.