EPR steering and entanglement are two fundamental characteristics of quantum mechanics and that are inextricably linked. For the moment, the researchers believe that EPR steering stems from entanglement, but entanglement does not imply EPR steering [1, 2]. EPR entanglement characterizes quantum nonlocal correlations among remote parties that are totally forbidden within the classical regime. Moreover, multipartite entangled states have important applications in the field of quantum information. Utilizing and characterizing such quantum resources stemming from multipartite nonlocal correlations [3] are rather crucial for the applications of the information theory [4–10] and from foundational perspectives. A multipartite state is deemed to be genuinely multipartite entangled [11] if and only if the state may not be written as a convex linear combination of states, each of which is disentangled with reference to some partition.

One the other hand, the concept of EPR steering was first introduced by Schrödinger [12, 13] in the context of the EPR argument [14]. Conceptually, EPR steering describes a nontrivial trait of quantum mechanics that an observer can immediately “steer” a distant party by employing the local quantum measures. EPR steering can be detected by utilizing EPR steering inequalities [15–22]; the violation of EPR steering inequalities can indicate that EPR steering occurs. At first, Reid [23] derived an inequality for EPR steering based on Heisenberg uncertainty relation in 1989. Then, EPR steering was formally defined in 2007 [24]. Numerous EPR steering inequalities have since been given; however, most respect was given to detecting bipartite EPR steering [25]. Additionally, multipartite steerable states have vital applications in the future quantum information field. Consequently, the detection and investigation of multipartite EPR steering is more important and challenging. The concept of multipartite EPR steering was first introduced by He and Reid [26] and developed for Gaussian states by Kogias et al. [27]. Experiments were followed [28–32], which motivated research studies of the monogamy relationship of EPR steering [33, 34]. Moreover, Wang et al. [35] have optimized the collective EPR steering for the tripartite state within a particular optics-based system in 2014.

In a realistic world, a quantum system ineluctably suffers from the influence of the decoherence attributed to the mutual effect between the system and its external noises. Typically, noisy environments usually can be classified into two species, namely, non-Markovian and Markovian environments [36–40]. In detail, the Markovian noisy environment is featured by leading to the degeneration of quantum nonlocal correlations [40]. By contrast, as a normative non-Markovian noisy environment [41, 42], a dynamic characteristic of quantum nonlocal correlations can be discovered, which is the renewal of quantum nonlocal correlations after a finite time period of the entire disappearance [43]. As a consequence, in the course of quantum information processing, considering the external noisy environments is indispensable and significant under a realistic regime. However, in the past years, there have been only a few authors to examine the steerability of multipartite states in the local noisy environments [44–46]. Hence, we will concentrate on exploring the genuine multipartite steering (GMS) and genuine multipartite entanglement (GME) under the noise channels. We here mainly probed the dynamic characteristics of GME and GMS and the relationship between them under the noise channels.

The remainder of this article is organized as follows. In Section II, we introduced the measuring method of GME and GMS within the multi-body systems, respectively. Then, we investigated the dynamic behaviors of GMS for the initial tripartite state under two kinds of different noises in Section 3. In Section 4, we probed the characteristic of GME and compared it with GMS as the tripartite state under the two kinds of different noises. Finally, we ended up our article with a brief conclusion.

In this section, we assume that there are three parties and they share an initial three-qubit state in the form of [48, 49]. ρ = Q ( | c G H Z 〉 〈 c G H Z | ) + 1 − Q 8 I 8 , 0 ≤ Q ≤ 1 , (5) where | c G H Z 〉 = α | 000 〉 + 1 − α 2 | 111 〉 ,   0 ≤ α ≤ 1 , and I 8 is the 8 × 8 identity matrix. Based on Eqs 2, 4, we can obtain the three-qubit states of GME 2 α Q ( 1 − α 2 ) − 3 / 4 ( 1 − Q ) and GMS inequality 39 / 16 − 3 / 2 Q ( 1 + α 1 − α 2 ) ≥ 1 , respectively. In Figure 1, the red dashed line is below the black dashed line, which means the tripartite state is a genuine tripartite steerable state. On the contrary, if the red dashed line is above the black dashed line, which means the tripartite state is not a genuine tripartite steerable state. Thus, when α is equal to 2 / 2 , one can obtain that the tripartite state is a genuine steerable state in the case of 23 / 36 < Q ≤ 1 , while it is a genuine unsteerable state for 0 ≤ Q ≤ 23 / 36 in Figure 1. Moreover, the tripartite state is entangled for 1 ≥ Q > 3 / 7 and is separable for 3 / 7 ≥ Q ≥ 0 . The maximally entangled state ( Q = 1 , α = 2 / 2 ) is a maximally genuine tripartite steerable state. Hence, we can draw a conclusion that for the whole set of the three-qubit states, it holds that G M S ⇒ G M E , suggesting a hierarchy according to which all GMS’s states are genuinely entangled, while GME does not imply GMS, which means that the set of genuine tripartite steerable states is a strict subset of the set of genuine tripartite entangled states.

Next, we considered that the tripartite states each independently and locally interacts with a zero-temperature reservoir. Herein, the two kinds of different noisy channels were considered: the bit flip (BF) channel and phase damping (PD) channel, respectively. In this context, the system–environment interaction via the operator-sum representation formalism is utilized. Following the approach of the Kraus operators, the time-evolution of the initial three-qubit states under the local noisy environment can be expressed by the trace-preserving quantum operation ξ ( ρ ) , which is ξ ( ρ ) = ∑ i K i ρ K i † with the Kraus operators satisfying the trace-preserving condition ∑ i K i K i † = I . The influence of the flip noises is to damage the correlations contained in the phase relations without the exchange of energy. The Kraus operators for the BF noise channel can be given by K 0 = d I , K 1 = 1 − d σ x , (6) where one can call that d is the channel-noise factor and 0 ≤ d ≤ 1 , and I is the 2 × 2 unit density matrix. The set is interpreted as corresponding to a probability d of remaining in the same state and a probability 1 − d of having an error 0 ↔ 1 . The factor K 1 in Eq. 6 ensures that at d = 1 / 2 has maximal ignorance about the occurrence of an error and thereby has minimum information about the state [50].Furthermore, the PD noise channel depicts the losing correlations without the loss of energy. It leads to decoherence without relaxation. The Kraus operators can be given as K 2 = ( 1 0 0 1 − d ) , K 3 = ( 0 0 0 d ) , (7) where d is the decoherence strength, and 0 ≤ d ≤ 1 . For convenience, here, we collectively call that d is the channel-noise factor in the BF and PD noise channels.

As a consequence, when three parties (all subsystem) of the three-qubit states suffer from the two different noisy environments, we then can obtain the non-zero elements of two kinds of the different final states, ρ B F and ρ P D , respectively.

To be precise, as three parties of the three-qubit states undergo the BF channel, the final state can be written as ρ B F = K 0 ⊗ K 0 . ρ . ( K 0 ⊗ K 0 ) † + K 1 ⊗ K 1 . ρ . ( K 1 ⊗ K 1 ) †                         + K 0 ⊗ K 1 . ρ . ( K 0 ⊗ K 1 ) † + K 1 ⊗ K 0 . ρ . ( K 1 ⊗ K 0 ) † , (8) Hence, we can obtain the non-zero elements of the final states ρ B F as follows: ρ B F 18 = ρ B F 81 = α 1 − α 2 [ 1 + 3 ( d − 1 ) d ] Q , ρ B F 27 = ρ B F 72 = ρ B F 36 = ρ B F 63 = ρ B F 45 = ρ B F 54 = α 1 − α 2 ( 1 − d ) d Q , ρ B F 11 = 1 8 { 1 + ( 2 d − 1 ) [ 2 ( 5 − 2 d ) d − 7 + 8 α 2 ( 1 + ( d − 1 ) d ) ] Q } , ρ B F 22 = ρ B F 33 = ρ B F 55 = 1 8 − 1 8 ( 2 d − 1 ) { − 1 + 2 [ 3 + 4 α 2 ( d − 1 ) − 2 d ] d } Q , ρ B F 44 = ρ B F 66 = ρ B F 77 = 1 8 { 1 + ( 2 d − 1 ) [ 1 + 2 ( 1 + 4 α 2 ( d − 1 ) − 2 d ) d ] Q } , ρ B F 88 = 1 8 + { d 3 − 1 8 + α 2 [ 1 + d ( ( 3 − 2 d ) d − 3 ) ] } Q . (9) Then, as three parties of the three-qubit states, Eq. 5 suffers from the PD channel; the final state can be written as ρ P D = K 2 ⊗ K 2 . ρ . ( K 2 ⊗ K 2 ) † + K 3 ⊗ K 3 . ρ . ( K 3 ⊗ K 3 ) †                         + K 3 ⊗ K 2 . ρ . ( K 3 ⊗ K 2 ) † + K 2 ⊗ K 3 . ρ . ( K 2 ⊗ K 3 ) † , (10) and the non-zero elements of final states ρ P D are ρ P D 11 = 1 8 + ( α 2 − 1 8 ) Q , ρ P D 88 = 1 8 [ 1 + ( 7 − 8 α 2 ) Q ] , ρ P D 18 = ρ P D 81 = α 1 − α 2 ( 1 − d ) 3 / 2 Q , ρ P D 27 = ρ P D 72 = ρ P D 36 = ρ P D 63 = ρ P D 45 = ρ P D 54 = 0 , ρ P D 22 = ρ P D 33 = ρ P D 44 = ρ P D 55 = ρ P D 66 = ρ P D 77 = 1 − Q 8 . (11)

Herein, by using Eqs 3, 4, one can gain an analytical expression of the GMS inequality for the initial state within the two kinds of different noisy channels, respectively. In accordance with the abovementioned analysis, one can draw the GMS inequality of the states ρ B F and ρ P D as a function of the state parameters α in terms of the different channel-noise factor d for Q = 1 in Figure 2. From these figures, one can see that the overall trend of the GMS inequality first decreases and then increases with the increase in the state parameter α for a fixed d , whatever the initial state is under the BF channel or PD channel. The value of α is equal to 2 / 2 , which corresponds to the position of the maximal genuine steerability for the tripartite state. As the channel-noise factor grows, it does not change. It turns out that the noisy environments cannot destroy the symmetry of GMS for the inertial state. Moreover, we observed that GMS will rapidly disappear with the increasing channel-noise factor d in the BF channel. However, GMS will not fleetly disappear with the increasing channel-noise factor d in the PD channel. It means that the BF and PD noises can seriously influence and damage the GMS. However, the impact of the PD noise on GMS is weaker than that of the BF noise.

Then, in order to explore the influence of the state parameters Q on the GMS inequality in terms of different channel-noise factors d for α = 2 / 2 , Figure 3 is drawn. As shown in Figure 3, one can see that GMS inequality rapidly decreases to zero with the increase in the state parameters Q , when there is no effect of the decoherence noise, namely, d = 0 . This demonstrates that the steerability of the state is stronger. We also found that the GMS occurs only when the state parameters Q increases to a fixed value. However, the properties of the GMS are different in the BF and PD noises, when the channel-noise factor is nonzero. In the BF channel, when the channel-noise factor is equal to 0.2, 0.4, and 0.5, respectively, GMS disappears whatever the state parameter Q is. Particularly, for the channel-noise factor d = 0.5 , the tripartite state has minimum information. In addition, GMS can appear with the increase in the state parameter Q , while the channel-noise factor is equal to 0.2, 0.4, and 0.5 in Figure 3 (2), respectively.

Next, we considered the effects of the state parameters Q and the channel-noise factor d on the GMS inequality, for which Figures 4,5 were drawn. As shown in Figures 4, 5, it can be concluded that the GMS inequality first increases and then decreases with the increase in the channel-noise factor d within the BF channel, whatever the value of the state parameter Q is; however, the GMS inequality increases with the increase in the channel-noise factor d in the PD channel. We discovered that the GMS can be detected if and only if the channel-noise factor d is larger than ( 6 − 23 ) / 12 and less than ( 6 + 23 ) / 12 under the BF channel in Figure 4(i) and Figure 5A. Moreover, when the channel-noise factor d is equal to 0.5, the values of the GMS inequality are invariable in the BF channel. At the moment, the tripartite state has minimum information and no quantum correlation.

Hence, we can conclude that the decoherence effect can destroy the steerability of quantum states or even completely disable the steerability. In order to more intuitively observe the influence of the three parameters (the channel-noise factor d and the state parameters Q and α ) on GMS, we drew a three-dimensional contour map of the GMS inequality in Figure 6. We can draw the same conclusions as mentioned earlier, and we will not go into them here.

It is generally acknowledged that quantum steering originates from quantum entanglement; however, entanglement does not imply steering, which means that the set of steerable states is a strict subset of the set of entangled states. In this section, we probed the dynamic characteristics of GME and then discussed the relationship between GMS and the GME under the two kinds of the noisy channels.

By employing Eq. 2, we can give the expressions of the GME as G M E ( B F ) = 2 max [ 0 , | ρ B F 18 | − 3 ρ B F 22 ⋅ ρ B F 77 ,                                                         | ρ B F 27 | − ( ρ B F 11 ⋅ ρ B F 88 + 2 ρ B F 33 ⋅ ρ B F 66 ) ] , (12) and G M E ( P D ) = 2 max [ 0 , α 1 − α 2 ( 1 − d ) 3 / 2 Q − 3 8 ( 1 − Q ) ] , (13) under the BF and PD channels, respectively.

To begin with, we considered the influence of the state parameters Q and α on GME, when d is a constant value. As shown in Figure 7, the GME first increases and then reduces with the increasing state parameter α , as Q is a constant value. Additionally, the tripartite state is a product state with no GME, when the state parameter α is equal to zero or one. We also obtained that GME increases with the increase in the state parameter Q . Thus, we think that Q is a purity parameter for the tripartite state. The bigger the Q , the bigger the GME is. The tripartite state is a maximal entangled state, when α = 2 / 2 ,   Q = 1 , and there is no decoherence.

Next, for comparing GME with GMS and the relationship between GMS and GME, we investigated the influence between the state parameters α and the channel-noise factor d on the GME and the GMS for Q = 0.9 . In the BF channel, both GME and GMS first rapidly decay to death with the increasing channel-noise factor d and then come back to life (see Figure 8A). However, both GME and GMS tardily decay to death with the increasing channel-noise factor d within the PD channel. Meanwhile, as shown in Figure 8B, when GMS and GME just disappear, the channel-noise factor d has a critical value, and the critical values are d ≈ 0.744 and d ≈ 0.809 , respectively. In other words, as the channel-noise factor is approximately smaller than 0.744, the tripartite state is both genuine steerable and entangled. If the channel-noise factor is larger than 0.744 but less than 0.809, the tripartite state is unsteerable and only genuinely entangled. The channel-noise factor is larger than 0.809, and the tripartite state is both unsteerable and disentangled. It is indicated that GMS originates from GME, but GME does not imply to GMS, which means that the set of genuine multipartite steerable states is a strict subset of the set of genuine multipartite entangled states. This result is also true in the BF channel (see Figure 8A). These conclusions may be useful for analyzing the relationship of quantum nonlocal correlations (GME and GMS) in the decoherence noise.

In this article, we mainly investigated the physical characteristic of GME and GMS within the two kinds of the different noisy channels. In contrast with our previous work [49], we used different initial states, and this state (see Eq. 5) is more general. In addition, here, we utilized different measurement methods for the multipartite quantum nonlocal correlation (GME) in this work. The anti-decoherence ability of GME is stronger than that of GMN. In the next place, a tripartite state is subjected to different decoherence noisy environments, but one is under curved spacetime (non-inertial frame) and one is without (this work). Consequently, in this study, we first discussed that the dynamic properties of GMS and GME for the initial tripartite state and the conditions for entangled and steerable states can be given. Then, the effect of BF and PD noises on the GMS is discussed, respectively. The results indicated that GMS is very flimsy under the influence of the decoherence. Specifically, GMS will perish with the increase in the channel-noise factor under the PD channel. However, GMS rapidly decays to death with the increase in the channel-noise factor and then come back to life soon in the BF channel. At the end, we studied the dynamic characteristics of GME and discussed the relationship between GME and GMS under decoherence noises. The decoherence noises can also degrade the GME and even induce its death. In addition, we can draw a conclusion that GMS originates from GME, but the GME does not imply GMS, which means that the set of genuine multipartite steerable states is a strict subset of the set of genuine multipartite entangled states. These conclusions may be useful for analyzing the relationship of quantum nonlocal correlations in the decoherence noises.