Sudden death and revival of Gaussian Einstein-Podolsky-Rosen steering in noisy channels

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.


INTRODUCTION
Nonlocality, which challenges our comprehension and intuition about the nature, is a key and distinctive feature of quantum world. Three different types of nonlocal correlations: Bell nonlocality [1], Einstein-Podolsky-Rosen (EPR) steering [2][3][4][5][6] and entanglement [7] have opened an epoch of unrelenting exploration of quantum correlations since they were introduced. The notion of EPR steering was introduced as the phenomenon that one can remotely steer the (conditional) quantum state owned by the other party through local measurements on half of an entangled state [2,3]. According to the hierarchy of nonlocality, EPR steering stands between entanglement and Bell nonlocality [8]. The existence of steering does not imply the violation of any Bell inequality, while the violation of at least one Bell inequality immediately implies steering in both directions which is referred as two-way steering [9]. Remarkably, there is a situation that a quantum state may be steerable from Alice to Bob, but not vice versa, which is called the one-way steering [9][10][11][12][13][14][15][16][17]. This intriguing feature comes from the intrinsically asymmetric with respect to the two subsystems.
Besides two-mode EPR steering, the multipartite EPR steering has also been widely investigated since it has potential application in quantum network. Genuine multipartite steering exists if it can be shown that a steering nonlocality is necessarily shared among all observers [36]. The criterion for multipartite EPR steering and for genuine multipartite EPR steering has been developed [36,37]. Genuine high-order EPR steering has also been experimentally demonstrated [38]. Multipartite Gaussian EPR steering in Greenberger-Horne-Zeilinger (GHZ) state [10,39] and a four-mode cluster state [11] have been demonstrated. The experiments of multipartite Gaussian EPR steering only demonstrated EPR steering in the case of bipartite splitting at present, while genuine multipartite EPR steering for Gaussian states has not been demonstrated. Recently, the distribution of multipartite steering in a quantum network by separable states has been demonstrated [40,41].
The performance of modern quantum communication sort of relies on the interaction between quantum system and environment, which is generally described by the theory of decoherence. Decoherence, caused by the loss and noise encountered in quantum channel, can destroy fragile quantum properties of quantum states, and thus the information carried by the quantum states will be affected. Decoherence effects on entangled state has been well studied [42][43][44][45][46][47]. It has been shown that two entangled qubits become completely separable in a finite-time under the influence of vacuum which is the so called entanglement sudden death [42,43], and disentanglement occurs in Gaussian multipartite entangled states when one mode is transmitted in a noisy channel [48]. It is essential to recover entanglement when entanglement sudden death happens. A number of methods to recover the destroyed entanglement have been demonstrated, such as the non-Markovian environment [48][49][50], weak measurement [51] and feedback [52]. Since EPR steering is so intriguing and has enormous potential in quantum communication applications, it is imperative and significant to investigate the influence on Gaussian EPR steering made by decoherence. In 2015, the decoherence of steering was theoretically investigated [53]. Further more, it is unknown whether the methods used to recover entanglement can be used to recover EPR steering when sudden death of EPR steering happens.
Here, we experimentally investigate properties of Gaussian EPR steering for a two-mode squeezed state (TMSS) when it is transmitted in lossy and noisy quantum communication channels, respectively. Firstly, we investigate the influence of purity of quantum states on Gaussian EPR steering. We show that in a lossy channel, the impurity of the state leads to decrease of steerabilities and two-way steering region, but it never leads to sudden death of Gaussian EPR steering. Secondly, we find the Gaussian EPR steering totally disappear at certain noise level in a noisy quantum channel, which demonstrates sudden death of EPR steering. Thirdly, we successfully revive the Gaussian EPR steering after its sudden death by establishing a correlated noisy channel. We also confirm that the steering quantifier proposed by I. Kogias et al. and the steering criterion proposed by M. Reid are equivalent for a TMSS with Gaussian measurement. The presented results provide useful references for applying Gaussian EPR steering in noisy environment.

Experimental scheme
The principle The state we use in the experiment is a deterministically prepared TMSS, whose quantum correlations between quadratures are expressed by ∆ 2 (x A +x B ) = ∆ 2 (p A −p B ) = 2e −2r , wherex =â +â † andp = â −â † /i are the amplitude and phase quadratures of an optical mode respectively, r is the squeezing parameter ranging from 0 to infinite which correspond to no squeezing and the ideal squeezing respectively. Under this definition, the noise of the vacuum state is normalized to 1. All Gaussian properties of a TMSS can be described by its covariance matrix with T is a vector composed by the amplitude and phase quadratures of bipartite optical beams [18]. Thus the TMSS covariance matrix can be par-tially expressed as (the cross correlations between different quadratures of one mode are taken as 0) (2) For the theoretical covariance matrix elements of the TMSS we used in the experiment, A = αI, B = βI, C = γZ, I and Z are the Pauli matrices and in which V s = e −2r and V as = e 2r + δ are the squeezing and anti-squeezing fluctuation of the initial squeezing state, respectively. The existence of δ is caused by classical and uncorrelated noise in quantum resource and leads to the impurity of quantum resource. To partially reconstruct all relevant entries of its associated covariance matrix we perform 6 different measurements on the output optical modes. These measurements include the amplitude and phase quadratures of the output optical modes △ 2x B , and the cross correlations ∆ 2 (x A +x B ) and ∆ 2 (p A −p B ). The covariance elements are calculated via the identities [54] Cov As an example, the partially reconstructed covariance matrix of the experimentally prepared pure TMSS is With these theoretical and experimentally partially reconstructed covariance matrix, we can quantify the Gaussian EPR steering of quantum states. The steerability of Bob by Alice (A → B) for a two-mode Gaussian state with Gaussian measurement can be quantified by [55] G A→B (σ AB ) = max 0, The quantity G A→B vanishes iff the state described by σ AB is nonsteerable by Gaussian measurements. The steerability of FIG. 1: The principle of experiment. a, Schematic of EPR steering in lossy channel. One mode of the two-mode squeezed state (TMSS)Â is kept by Alice, the other mode of TMSSB is distributed to Bob over a lossy channel which is experimentally mimicked by a beam-splitter, the vacuum mode couples into system from the other input port of the beam-splitter. The transmission efficiency of the lossy channel is η L . b, Schematic of disappearance and revival of EPR steering. ModeB is distributed to Bob over a noisy channel with an excess noise higher than the vacuum noise. The transmission efficiency of the noisy channel is η N . An ancillary beam with noise correlated to noisy channel couples with the transmitted mode on a beam-splitter with transmission coefficient T . c, The experimental setup. A pure -3 dB TMSS at the sideband frequency of 3 MHz is generated by a nondegenerate optical parametric amplifier (NOPA). Alice and Bob perform homodyne measurement on their states respectively. An ancillary beam is used to revive the EPR steering. PBS, polarization beam splitter; LO, local oscillator; EOM, electro-optic modulator; HD, homodyne.
Alice by Bob (B → A), quantified by G B→A , can be obtained by replacing the role of A by B.
It is worth mentioning that the first criterion for the demonstration of the EPR paradox was proposed by M. Reid in 1989 [56], and it was linked to the concept of steering lately [57]. For a two-mode Gaussian state, the quantifier as given by Eq. (4) is equivalent to the Reid's criterion which is based on Heisenberg uncertainty relation, i.e., a state is steerable from Alice to Bob if the following relation is violated: where V X B |X A and V P B |P A are the conditional variances of Bob's measurements conditioned on Alice's results, and For steering from Bob to Alice, the roles of A and B are swapped. This criterion does not require the assumption of Gaussian states or Gaussian systems to be valid. The PPT criterion [64] is applied to characterise the entanglement, which is a sufficient and necessary condition for a Gaussian TMSS. The PPT value can be determined by where Γ = det A + det B − 2 det C. When the PPT value is less than 1, the two modes are entangled. At first, we investigate the effect of purity of quantum state on EPR steering. The purity of a quantum state ̺ is µ (̺) = Tr̺ 2 [18], which varies in the range of 1 D ≤ µ ≤ 1, where D = dim H for a given Hilbert space H. The minimum is reached by the totally random mixture, the upper bound is saturated by pure states. In the limit of CV systems (D → ∞ ), the minimum purity tends to zero. The purity of any N-mode Gaussian state is only function of the symplectic eigenvalues υ i of covariance matrix σ [58], which is given by Regardless of the number of modes, the purity of a Gaussian state is fully determined by the global symplectic invariant det σ alone [58].
In practice, the loss and noise in the generation system will lead to generation of impure quantum states [47,59]. For example, the phonon noise in the three-color entangled state prepared by a non-degenerate optical parametric oscillator (OPO) is a type of classical and uncorrelated noise which leads to an impure three-color entangled state. Scattered light by thermal phonons inside a second-order nonlinear crystal is the source of additional phase noise observed in OPOs [47,59]. For our TMSS, the relationship between purity µ (̺) and δ is The scheme of the transmission of Gaussian EPR steering in a lossy channel for pure and impure TMSSs is shown in Fig.  1a. ModeB of the TMSS is distributed in a lossy channel, the mode after transmission is given byB L = √ η LB + 1 − η Lv , where η L andv represent the transmission efficiency of quantum channel and the vacuum mode induced by loss into the quantum channel, respectively. The partially reconstructed covariance matrix of the TMSS distributed over lossy channel in the experiment we have Secondly, the transmission of Gaussian EPR steering in a noisy channel is investigated, as shown in Fig. 1b, where the excess noise higher than the vacuum noise exists. ModeB of the TMSS is distributed in the noisy channel, the mode after transmission is given byB N = √ η NB + (1 − η N ) gN + 1 − η Nv , whereN and g are the Gaussian noise in the channel with transmission efficiency of η N and the magnitude of noise, respectively. The partially reconstructed covariance matrix of the TMSS distributed over noisy channel is Finally, we demonstrate the revival of Gaussian EPR steering by establishing a correlated noisy channel, where an auxiliary channel carrying on the noise correlated with that of the noisy channel is introduced. In today's communication system, it is relevant to consider channels with correlated noise (i.e. non-Markovian environment), because the time and space correlated noise exist naturally [60][61][62]. Furthermore, it has been experimentally found that correlated noisy channel can be established by two bounded fibers [63], thus the method we use in experiment to built the non-Markovian environment is applicable and practical significant. As shown in Fig. 1b. The auxiliary noisy channel based on an ancillary coherent stateĉ an , which is modulated to carry the noise information correlated with that in the noisy quantum channel, sayingĉ ′ an =ĉ an + √ g anN (g an is the magnitude of the correlated noise and is experimentally adjustable). Thus there exists correlation between the quantum system and environment. The noisy transmitted modeB N couples with the auxiliary noisy channel on a revival beam-splitter with transmission efficiency T . One of the output mode we need from the revival beam- − Tĉ an , when the adjustable g an and T satisfy it can be described aŝ Thus the excess noise in modeB R can be removed totally, and the steerability betweenB R andÂ can be revived to some degree. After the revival operation, the covariance matrix of the revived state is Experimental set-up The experimental set-up is shown in Fig. 1c. We use a non-degenerate optical parametric amplifier (NOPA) to prepare the TMSS. NOPA is pumped by a continuous-wave intracavity frequency-doubled Nd:YAP-LBO laser with two output wavelengths at 540 nm and 1080 nm. The NOPA consists of an α-cut type-II KTP crystal which is front face coated (work as the input coupler with transmittance of 21.2% and 0.04% at 540 nm and 1080 nm, respectively) and a concave mirror (work as the output coupler with transmittance of 0.5% and 12.5% at 540 nm and 1080 nm, respectively). Via the frequency down-conversion process of the pump field at 540 nm inside the NOPA which operated at deamplification condition [the relative phase between the pump laser and the injected signal equals to (2n + 1) π], the TMSS at 1080 nm we want can be generated. Experimentally, the TMSS we prepared is a nearly pure -3.0 dB state at sideband frequency of 3 MHz.
The loss in the quantum channel is experimentally mimicked by a beam-splitter, which is composed by a half-wave plate and a polarization beam-splitters (PBS), the vacuum mode couples into system from the other input port of the beam-splitter. In the noisy environment, to couple the transmitted modeB and the noise-modulated coherent beam (auxiliary noisy channelĉ ′ an ), the beam-splitter (revival beamsplitter) is composed by a PBS, a half-wave plate and another PBS as shown in Fig. 1c, where the transmission efficiency η N and T are adjusted by setting the half-wave plate. The correlated noises between the quantum system and environment are generated by same signal generator experimentally. We reconstruct the covariance matrix of the quantum state with the homodyne detection systems, and quantify its steering properties.

RESULTS
The effect of purity on EPR steering Figure 2 shows the effect of purity of the TMSS on EPR steering in a lossy channel, where the steerabilities quantified by two criteria given by Eqs. (4) and (5) are shown in Fig.  2a and 2b, respectively. For a pure −3 dB TMSS, one-way steering appears in a lossy channel, and the critical point of one-way steering property occurs at η L = 0.5. For comparison, four different purities of the TMSSs at 0.8, 0.6, 0.4 and 0.2 are considered, which show that the maximum steerability and the phenomenon of one-way EPR steering property happens earlier with the decreasing of purity. This confirms that the impurity of the TMSS leads to decrease of steerabilities and two-way steering region. However, the steerability from Alice to Bob never disappear until the purity tends to zero, which means that the impurity of the TMSS does not lead to sudden death of Gaussian EPR steering. The results in Fig. 2a and 2b confirm that two criteria given by Eqs. (4) and (5) are equivalent for a TMSS with Gaussian measurement in a lossy channel.
In the experiment, to determine the error bar of the experimental results, we measure three covariance matrices for each quantum state and obtain the corresponding EPR steering parameters. By calculating the mean value and the standard deviation of these three EPR steering parameters, error bars are obtained. The errors of the experimental data mainly come from the phase fluctuation of the phase locking system, and we take much efforts to suppress phase fluctuation of each phase locking system to be θ ≈ 1.5 • by adjusting locking parameters.
The effect of noise on EPR steering As shown in Fig. 3a and 3b, sudden death of Gaussian EPR steering of the TMSS in a noisy channel is observed, where the variance of the excess noise is taken as 5 times of vacuum noise. Specifically, in region I ( 0.96 < η N ≤ 1), the twoway steering exists, both Alice and Bob can steer with each other. With the decreasing of η N , when 0.86 < η N ≤ 0.96 (labeled as region II), one-way steering appears, i.e. Bob can steer Alice's state while Alice cannot steer Bob's state. More seriously, when η N ≤ 0.86, the steerability between Alice and Bob totally disappeared (labeled as region III), i.e. the sudden death of the Gaussian EPR steering happens.
We also demonstrate sudden death of Gaussian EPR steering with different excess noise levels (noise levels are taken as 5, 10, 15, 20 times of vacuum noise) at fixed transmission efficiency (η N = 0.6) in Fig. 3c and 3d. We can see the two-way steering (labeled as region I) and/or one-way steering (labeled as region II) could still exist when the excess noise is relatively low. When the variance of the excess noise is higher than 0.47 (labeled as region III), the sudden death of the Gaussian EPR steering happens. The results in Fig. 3 confirm that two criteria given by Eqs. (4) and (5) are equivalent for the sudden death and revival of Gaussian EPR steering of a TMSS with Gaussian measurement.
The effect of purity of initial TMSS on EPR steering in a noisy environment is also investigated as shown in Fig. 4. Figure 4a and 4b are the three-dimensional theoretical results of Gaussian EPR steering properties of initial TMSS with different purities of 1 and 0.8, respectively. It is obvious that the direction of EPR steering is changed with different noise levels, which is the same with the result in Ref. [12]. We can see that the blue surface (G B N →A ) and light red surface (G A→B N ) have crossover, which happens with excess noise of 0.24 and 0.49 for the pure and impure TMSSs, respectively. Figure  4c shows the effect of purity of the TMSS on EPR steering in a noisy channel with excess noise of 2 times of vacuum noise. It's obvious that the sudden death of EPR steering for the impure state happens earlier and the maximum steerability is smaller than that of the pure state. The revival of EPR steering To revive the disappeared EPR steering, we fix T = 0.9 for the revival beam-splitter and adjust the magnitude of correlated noise g an . The parameters g/g an are set to 0.14, 0.18, 0.28, and 0.56 for different quantum channel efficiencies of 0.2, 0.4, 0.6, and 0.8, respectively, according to Eq. (8). After the revival operation, the disappeared Gaussian EPR steering is revived as shown in Fig. 3a and 3b. Because the excess noise in modeB R is removed totally in the revival operation, the Gaussian EPR steering is revived even at the noise level of 20 times of vacuum noise , as shown in Fig. 3c and 3d. Generally, T is better to be close to 1, because it results in adding an extra linear loss on the revived mode, this  is why the steerabilities between modeB R andÂ are not good as that between modeB L andÂ [the pure lossy transmitted regime, presented by blue dash and solid curves in Fig. 3a].

DISCUSSION
Note that the noise in quantum channel and the noise in initial TMSS have different influences on EPR steerability. With the decreasing of purity of initial TMSS, the maximum steerability decreases and the phenomenon of one-way EPR steering property happens earlier than that of the pure state (between 0.5 ≤ η ≤ 1), but the death of the EPR steering never happens. While with the increasing of noise in the quantum channel, the two-way EPR steering firstly declines to one-way EPR steering, and the sudden death of the Gaussian EPR steering finally occurs. Thus we can distinguish where the excess noise exists (whether in the communication channel or in the initial TMSS) by the Gaussian EPR steerability easily.
The sudden death of Gaussian EPR steering is different from that of the Gaussian entanglement, which is characterised by the positive partial transposition (PPT) criterion [64] (see Methods for details). Figure 5 shows the entanglement and steering region parameterized by transmission efficiency and excess noise, where the purple region represents the two-way steering region, the cyan region represents the one-way steering region of G A→B , the light red region represents the one-way steering region of G B→A , and the yel-FIG. 5: The region plot parameterized by transmission efficiency and excess noise. The region below the black curve, which is the bound of entanglement sudden death, represents the existence of the entanglement. The region below the blue and red curves, which are the bound of the disappearance of G A→B and G B→A respectively, represents the existence of EPR steering.
FIG. 6: Results for distillation of Gaussian steering with a correlated noisy channel and measurement-based NLA. Black solid and dash curves are the theoretical predicted steerabilities in a noisy channel with excess noise 5 times of vacuum noise. Red solid and dash curves are the predicted steerabilities after revival of Gaussian steering with a correlated noisy channel. The orange and purple curves are the predicted steerabilities when the measurement-based NLA based on Bob's measurement results with gain of 1.2 is applied before and after the revival of Gaussian steering, respectively. The Blue solid and dash curves are the predicted steerabilities in a pure lossy but noiseless channel. low region represents the entanglement without steering region, respectively. Firstly, the direction of the EPR steering is changed with the increase of excess noise which is not the case in entanglement sudden death. The sudden death of EPR steering and entanglement happens when there is excess noise in the quantum channel. For a pure −3 dB TMSS, the direc-tion of EPR steering is changed when the excess noise is 0.25 times of vacuum noise. This result comes from the intrinsic asymmetric character of EPR steering. Secondly, it is obvious that the sudden death of Gaussian EPR steering happens earlier than that of the entanglement with same excess noise level. Thirdly, different from entanglement, the sudden death of Gaussian EPR steering in a noisy channel is directional.
Recovering EPR steering with correlated noisy channel can only remove the effect of excess noise in the transmission channel, the effect of loss in the transmission channel cannot be removed. Some other technologies such as noiseless linear amplification can be used to eliminate the effect of loss [65][66][67][68]. Recently, it has been shown that the measurement-based noiseless linear amplification can be used to distill Gaussian EPR steering in both lossy and noisy channels, which provides another feasible way to recover Gaussian EPR steering in quantum channel [69].
Comparing the present work of EPR steering revival with a correlated noisy channel and distillation of EPR steering with measurement-based NLA, the differences are as following. Firstly, the steering revival with a correlated noisy channel can only remove the effect of excess noise in the quantum channel, the effect of loss in the transmitted channel cannot be removed. But the distillation of EPR steering works for both lossy and noisy channels. Secondly, the revival of Gaussian EPR steering with a correlated noisy channel can remove the excess noise in the quantum channel totally, but distillation of EPR steering with measurement-based NLA cannot remove the excess noise in all range of transmission efficiency [orange curves in Fig. 6]. Thirdly, the revived steerability with a correlated noisy channel cannot exceed the initial steerability (that in a pure lossy channel), but the distilled steerability can exceed the initial steerability.
Combining the methods of correlated noisy channel and the NLA may provide an option to distill Gaussian EPR steering in lossy and noisy environment. As shown in Fig. 6, if the measurement-based NLA based on Bob's measurement results with gain of 1.2 is applied after the revival of Gaussian EPR steering with a correlated noisy channel, the revived steerability could be enhanced by using the measurementbased NLA. In this case, the revival of steering with a correlated noisy channel removes the effect of excess noise in quantum channel. Followed by the measurement-based NLA, the effect of loss in quantum channel can be eliminated and the distillation of Gaussian steering can be obtained.
Our work demonstrates evolution of bipartite Gaussian EPR steering of a TMSS transmitted in lossy and noisy quantum communication channels. We show the noise in quantum channel and the noise in the initial state result in different influences on EPR steerability. The impurity of the state only decrease the steerabilities and two-way steering region, but never leads to sudden death of EPR steering. While the excess noise in the quantum channel leads to sudden death of Gaussian EPR steering. We successfully revive EPR steering after the sudden death occurs in noisy channel by establishing a correlated noisy channel. The presented results provide useful references for understanding the steering properties of quantum state which interacts with different environments, also have potential applications in asymmetric quantum information processing exploiting EPR steering as a valuable resource.