Sudden Death and Revival of Gaussian Einstein–Podolsky–Rosen Steering

www.nature.com/npjqi ARTICLE OPEN Sudden death and revival of Gaussian Einstein–Podolsky–Rosen steering in noisy channels ✉ Xiaowei Deng1,2,4, Yang Liu1,2,4, Meihong Wang2,3, Xiaolong Su 2,3 and Kunchi Peng2,3 Einstein–Podolsky–Rosen (EPR) steering is a useful resource for secure quantum information tasks. It is crucial to investigate the effect of inevitable loss and noise in quantum channels on EPR steering. We analyze and experimentally demonstrate the influence of purity of quantum states and excess noise on Gaussian EPR steering by distributing a two-mode squeezed state through lossy and noisy channels, respectively. We show that the impurity of state never leads to sudden death of Gaussian EPR steering, but the noise in quantum channel can. Then we revive the disappeared Gaussian EPR steering by establishing a correlated noisy channel. Different from entanglement, the sudden death and revival of Gaussian EPR steering are directional. Our result confirms that EPR steering criteria proposed by Reid and I. Kogias et al. are equivalent in our case. The presented results pave way for asymmetric quantum information processing exploiting Gaussian EPR steering in noisy environment. npj Quantum Information (2021) 7:65 ; https://doi.org/10.1038/s41534-021-00399-x INTRODUCTION teleportation30–32, securing quantum networking tasks33, and 1234567890():,; 34,35 Nonlocality, which challenges our comprehension and intuition subchannel discrimination . about the nature, is a key and distinctive feature of quantum Besides two-mode EPR steering, the multipartite EPR steering world. Three different types of nonlocal correlations: Bell has also been widely investigated since it has potential application nonlocality1, Einstein–Podolsky–Rosen (EPR) steering2–6, and in quantum network. Genuine multipartite steering exists if it can entanglement7 have opened an epoch of unrelenting exploration be shown that a steering nonlocality is necessarily shared among 36 of quantum correlations since they were introduced. The notion of all observers . The criterion for multipartite EPR steering and for 36,37 EPR steering was introduced as the phenomenon that one can genuine multipartite EPR steering has been developed . remotely steer the (conditional) quantum state owned by the Genuine high-order EPR steering has also been experimentally 38 other party through local measurements on half of an entangled demonstrated . Multipartite Gaussian EPR steering in 10,39 state2,3. According to the hierarchy of nonlocality, EPR steering Greenberger–Horne–Zeilinger state and a four-mode cluster 11 stands between entanglement and Bell nonlocality8. The existence state have been demonstrated. The experiments of multipartite of steering does not imply the violation of any Bell inequality, Gaussian EPR steering only demonstrated EPR steering in the case while the violation of at least one Bell inequality immediately of bipartite splitting at present, while genuine multipartite EPR implies steering in both directions, which is referred as two-way steering for Gaussian states has not been demonstrated. Recently, steering9. Remarkably, there is a situation that a quantum state the distribution of multipartite steering in a quantum network by may be steerable from Alice to Bob, but not vice versa, which is separable states has been demonstrated40,41. called the one-way steering9–17. This intriguing feature comes The performance of modern quantum communication sort of from the intrinsically asymmetric with respect to the two relies on the interaction between quantum system and environ- subsystems. ment, which is generally described by the theory of decoherence. The one-way EPR steering has been observed for both Decoherence, caused by the loss and noise encountered in continuous variable (CV)9–13 and discrete variable (DV) sys- quantum channel, can destroy fragile quantum properties of tems14–17. Gaussian states and Gaussian operations play a central quantum states, and thus the information carried by the quantum role in the analysis and implementation of CV quantum states will be affected. Decoherence effects on entangled state has – technology18,19. When considering generally bipartite Gaussian been well studied42 47. It has been shown that two entangled states and using Gaussian measurements, the steerability between qubits become completely separable in a finite-time under the the submodes is referred as Gaussian steering. What is more influence of vacuum which is the so called entanglement sudden interesting, it has been shown that the direction of Gaussian EPR death42,43, and disentanglement occurs in Gaussian multipartite steering can be manipulated in noisy environment12. The entangled states when one mode is transmitted in a noisy temporal quantum steering and spatio-temporal steering have channel48. It is essential to recover entanglement when entangle- also been investigated20–22. Very recently, the EPR steering using ment sudden death happens. A number of methods to recover hybrid CV and DV entangled state23 and remote generation of the destroyed entanglement have been demonstrated, such as – Wigner-negativity through EPR steering24 are demonstrated. As the non-Markovian environment48 50, weak measurement51, and applications, EPR steering has been identified as an essential feedback52. Since EPR steering is so intriguing and has enormous resource for one-sided device-independent quantum key distribu- potential in quantum communication applications, it is imperative tion25–27, quantum secret sharing28,29, secure quantum and significant to investigate the influence on Gaussian EPR 1Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, China. 2State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan, China. 3Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, China. ✉ 4These authors contributed equally: Xiaowei Deng, Yang Liu. email: [email protected] Published in partnership with The University of New South Wales X. Deng et al. 2 steering made by decoherence. In 2015, the decoherence of The steerability of Bob by Alice (A → B) for a two-mode Gaussian steering was theoretically investigated53. Furthermore, it is state with Gaussian measurement can be quantified by55: unknown whether the methods used to recover entanglement 1 det A can be used to recover EPR steering when sudden death of EPR GA!Bðσ Þ¼ ; : AB max 0 ln σ (4) steering happens. 2 det AB Here, we experimentally investigate the properties of Gaussian A!B The quantity G vanishes if the state described by σAB is EPR steering for a two-mode squeezed state (TMSS) when it is nonsteerable by Gaussian measurements. The steerability of Alice transmitted in lossy and noisy quantum communication channels, by Bob (B → A), quantified by GB!A, can be obtained by replacing respectively. First, we investigate the influence of purity of the role of A by B. quantum states on Gaussian EPR steering. We show that in a It is worth mentioning that the first criterion for the lossy channel, the impurity of the state leads to decrease of demonstration of the EPR paradox was proposed by M. Reid56, steerabilities and two-way steering region, but it never leads to and it was linked to the concept of steering lately57. For a two- sudden death of Gaussian EPR steering. Second, we find the mode Gaussian state, the quantifier as given by Eq. (4)is Gaussian EPR steering totally disappear at certain noise level in a equivalent to the Reid’s criterion which is based on Heisenberg noisy quantum channel, which demonstrates sudden death of EPR uncertainty relation, i.e., a state is steerable from Alice to Bob if the steering. Third, we successfully revive the Gaussian EPR steering following relation is violated: after its sudden death by establishing a correlated noisy channel. V j V j 1; (5) We also confirm that the steering quantifier proposed by I. Kogias XB XA PB PA et al. and the steering criterion proposed by M. Reid are equivalent ’ where VXBjXA and VPBjPA are the conditional variances of Bob s for a TMSS with Gaussian measurement. The presented results measurements conditioned on Alice’s results, and provide useful references for applying Gaussian EPR steering in ¼ σ = VXBjXA VPBjPA det AB det A. For steering from Bob to Alice, the noisy environment. roles of A and B are swapped. This criterion does not require the assumption of Gaussian states or Gaussian systems to be valid. At first, we investigate the effect of purity of quantum state on RESULTS EPR steering. The purity of a quantum state ϱ is μðÞ¼ϱ Trϱ218, The principle 1 μ ¼ H which varies in the range of D 1, where D dim for a The state we use in the experiment is a deterministically prepared given Hilbert space H. The minimum is reached by the totally 1234567890():,; TMSS, whose quantum correlations between quadratures are random mixture, the upper bound is saturated by pure states. In Δ2ðÞ^ þ ^ ¼ Δ2ðÞ^ À ^ ¼ À2r ^ ¼ ^ þ → ∞ expressed byxA xB pA pB 2e , where x a the limit of CV systems (D ), the minimum purity tends to 0. ^y ^ ^ ^y The purity of any N-mode Gaussian state is only function of the a and p ¼ a À a =i are the amplitude and phase quadratures 58 symplectic eigenvalues υi of covariance matrix σ , which is given of an optical mode, respectively, r is the squeezing parameter by: ranging from 0 to infinite which correspond to no squeezing and 1 1 the ideal squeezing, respectively. Under this definition, the noise μðÞ¼ϱ Q ¼ pffiffiffiffiffiffiffiffiffiffiffi : υ (6) of the vacuum state is normalized to 1. All Gaussian properties of a i i det σ TMSS can be described by its covariance matrix: Regardless of the number of modes, the purity of a Gaussian state 0 1 σ α 0 γ 0 is fully determined by the global symplectic invariant det 58 B α Àγ C alone . B 0 0 C AC In practice, the loss and noise in the generation system will lead σAB ¼ B C ¼ (1) @ γ 0 β 0 A CT B to generation of impure quantum states47,59.

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