# Quantum steering borders in three-qubit systems

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## Abstract

We discuss a family of states describing three-qubit systems in a context of quantum steering phenomena. We show that symmetric steering cannot appear between two qubits—only asymmetric steering can appear in such systems. The main aim of this paper is to discuss the possible relations between the entanglement measures and steering parameter for two-mode mixed state corresponding to the qubit–qubit subsystem. We have derived the conditions determining boundary values of the negativity parametrized by concurrence. We show that two-qubit mixed state cannot be steerable when the negativity of such state is smaller than, or equal to, its boundary value. Finally, we have found ranges of the values of the mixedness measure, parametrized by concurrence and negativity for steerable and unsteerable two-qubit mixed states.

## Keywords

Quantum correlations Quantum steering Quantum entanglement Negativity Concurrence Three-qubit system## 1 Introduction

Quantum steering belongs to a group of correlations which appears in quantum world and does not have its counterpart in classical physics. The concept of steering was introduced by Schrödinger in 1935 [1] as a generalization of the EPR paradox formulated by Einstein, Podolsky and Rosen [2]. In 2007 Wiseman et al. [3, 4] showed that not every nonseparable states are steerable and not every nonsteerable states are nonlocal in a Bell sense. Thus, one can say that the steering represents some form of nonlocality which is, from one side, weaker than the usual Bell nonlocality, and from the other—stronger than nonseparability. Next, Oppenheimer and Weher showed that steering with uncertainty principle determine the degree of nonlocality [5]. Nowadays, steering phenomena are studied not only in space’s context, but also in time’s one (for such situations we call it *temporal steering*). Such phenomena were considered not only from theoretical point of view [6, 7] but also first experiment studies of temporal steering were proposed and performed [8]. Steering phenomena seem to be especially relevant from the point of view of quantum information processing [9], including quantum key distribution [10].

The first experimental observation of the EPR paradox was done by Ou et al. [11], whereas experimental realization of steering was described in [8, 12, 13, 14, 15, 16, 17, 18]. It should be emphasized that generation of EPR steering was considered in a context of various systems. For instance, they were three-mode optomechanical system composed of an atomic ensemble located in a cavity with an oscillating mirror [19], double-cavity optomechanical system with two separate electromagnetic fields mediated by a mechanical oscillator [20], system comprising optical cavity filled by an electromagnetic field and the mechanical oscillator [21], model involving light beams generated in time-modulated nondegenerate optical parametric oscillator [22] and many others.

To discuss steering phenomena, the criterion which was proposed by Reid [23],then developed by Cavalcanti and Reid [24], and by Walborn et al. [25] is usually applied. However, at this point one should mention very latest work by Rutkowski et al. [26] in which an experimentally feasible unbounded violation of a steering inequality was proven. The universal form of such inequality was derived there with application of the DeutschMaassenUffink entropic uncertainty relation.

In this work, we discuss steering properties of the states corresponding to a three-qubits model. In particular, we consider the relations between steering in a two-qubit subsystem and the entanglement described by the negativity and concurrence. The paper consists of two parts. In the first one, we analyze the situation when in the three-qubit system only one excitation is present, whereas in the second part we study double excited system. For the both cases, we discuss the possibility of generation symmetric steering between two qubits. We show what kind of steering (symmetric or asymmetric) can be observed in our three-qubit systems, and which type of steering will never appear for the given case. Moreover, applying entanglement measures such as concurrence and negativity, we find the conditions (upper bound) determining when unsteerable mixed states appear for qubit–qubit subsystem. In addition, we check how the mixedness of the states characterized by the linear entropy is related to the steering effect for the models discussed here. In particular, we identify ranges of the values of the entropy for steerable states described by various values of the concurrence and negativity. In this paper, we only shortly discuss cases when zero or three excitations are present—such cases lead to not interesting conclusions and correspond to trivial physical models.

## 2 Three-qubit system with one excitation

In this paper, we focus on the family of the states describing three-qubit systems in a context of finding conditions determining which states are steerable. We do not discuss here the influence of coupling among them leading to some particular time evolution.

In this article, we mostly concentrate on two cases. The first of them corresponds to the situation when single excitation is present in a model, whereas the second one concerns two excitations’ case. Obviously, there are two other situations—when we have no excitations at all and when all three qubits are excited. However, if we assume that the discussed model of three qubits does not interact with external systems, such cases are trivial and can be neglected.

*I*) denotes that we study the system with a single excitation.

*i*and

*j*label the qubits. Qubit

*j*steers qubit

*i*when the parameter \(S_{ij}\) takes values greater than zero. If at the same time \(S_{ji}>0\), then symmetric steering can be observed. The second type of steering is present when only one of two parameters (\(S_{ij}\) or \(S_{ji}\)) is greater than zero. For such a case, we are dealing with asymmetric steering.

Therefore, to get symmetric steering (\(S_{12}^{(I)}>0\) and \(S_{21}^{(I)}>0\)) two probabilities must take values greater than 0.5 what is not possible to occur. A similar situation we get for other pairs of qubits. Then, we see that symmetric steering is impossible to obtain in the system discussed here and the generated steering is always asymmetric.

If we carefully look at Eq. (4), we see that when we assume that \(P_{100}>0.5\) two steering parameters are greater than zero—\(S_{21}^{(I)}\) and \(S_{31}^{(I)}\). For such situation, qubit 1 steers simultaneously the remaining two qubits, labeled by 2 and 3. Analogously, when \(P_{010}>0.5\) qubits 1 and 3 are steerable by qubit 2 (\(S_{12}^{(I)}>0, \,S_{32}^{(I)}>0\)),whereas when \(P_{001}>0.5\) qubit 3 steers those labeled by 1 and 2 (\(S_{13}^{(I)}>0, \,S_{23}^{(I)}>0\)). From the other side, if we look for the situation when two qubits steer the third one (for example, when qubits 1 and 2 steer qubit 3) two steering parameters ( \(S_{31}^{(I)}\) and \(S_{32}^{(I)}\)) should be positive, and hence, two probabilities: \(P_{100}\) and \(P_{010}\) must exceed 0.5. Obviously such situation is impossible. Therefore, for our three-qubit model, one qubit can steer two qubits, whereas two qubits cannot steer the third qubit at the same time. This fact is consistent with the monogamy results of Reid [28].

The problems related to the nonlocality seems to be one of the most interesting topics of the quantum theory and are discussed extensively in numerous papers (for instance, see [3, 4, 29, 30] *and the references quoted therein*). Steering and entanglement phenomena are different types of quantum nonlocality. It is known that steerable states are the subset of the set of entangled states [3, 4]. Therefore, we shall find relations among entanglement measures and the steering parameter present in our model. As the measures of two-qubit entanglement, we will apply the negativity *N* and concurrence *C*. In 2001 Verstraete et al. [31] checked the relations between the negativity and concurrence and found upper and lower bounds of the negativity’s value corresponding to a given concurrence. They proved that for two-qubit mixed states the negativity takes values smaller or equal to the corresponding concurrence *C*. Moreover, they have found that such negativity is always greater than or equal to \(\sqrt{(1-C)^2 + C^2}-(1-C)\). Obviously, if we are dealing with pure states and for the Bell diagonal states, the negativity always is equal to the concurrence. In general, the negativity takes its minimal value for very special mixed states. The degree of entanglement of such states cannot be increased by any global unitary operation. The example of such states are two-qubit mixed states defined as a mixtures of Bell and separable states [32, 33], named Werner states [34].

*R*obtained from the relation \(R=\rho \tilde{\rho }\), where \(\tilde{\rho }\) is defined as \(\tilde{\rho }=\sigma _{y}\otimes \sigma _{y}\rho ^{*}\sigma _{y}\otimes \sigma _{y}\) and \(\sigma _{y}\) is \(2\times 2\) Pauli matrix. For the qubits 1 and 2, we have calculated eigenvalues of the matrix \(R_{12}=\rho _{12}\tilde{\rho }_{12}\) and found three of four eigenvalues are zero, whereas the fourth one takes the following form

*D*denotes dimension of \(\rho \). The linear entropy takes values from zero to unity. It is equal to zero for a pure states, whereas it reaches unity for maximally mixed states. For two-qubit states, dimension \(D=4\) and linear entropy takes the following form:

The range of linear entropy for given values of the concurrence for steerable and unsteerable states

Concurrence | Linear entropy | Generated states |
---|---|---|

\(C<\sqrt{(}2)-1\) | \(E\left( \rho _{ij}^{(I)}\right) <\frac{8}{3}\left( C-C^2 \right) \) | Only steerable states |

\(E\left( \rho _{ij}^{(I)}\right) \ge \frac{8}{3}\left( C-C^2 \right) \) | Steerable or unsteerable states | |

and \(E\left( \rho _{ij}^{(I)}\right) <\frac{2}{3}-\frac{2}{3}C^2\) | ||

\(E\left( \rho _{ij}^{(I)}\right) \ge \frac{2}{3}-\frac{2}{3}C^2\) | Only unsteerable states | |

\(C\ge \sqrt{(}2)-1\) | \(E\left( \rho _{ij}^{(I)}\right) < \frac{2}{3}-\frac{2}{3}C^2\) | Only steerable states |

\(E\left( \rho _{ij}^{(I)}\right) \ge \frac{2}{3}-\frac{2}{3}C^2\) | Only unsteerable states |

Figure 4 shows the same as Fig. 3, but the concurrence was replaced there by the negativity. Again, the red region represents steerable states (Fig. 4a), whereas the turquoise area corresponds to unsteerable states (Fig. 4b). When we compare Fig. 4a, b we see that for the values of linear entropy corresponding to the area confined by solid and dashed lines, the both steerable and unsteerable states can appear.

The ranges of the values of linear entropy and negativity for steerable and unsteerable states

Negativity | Linear entropy | Generated states |
---|---|---|

\(N<\frac{1}{7}\) | \(E\left( \rho _{ij}^{(I)}\right) <-\frac{8}{3}\left( -1-N+\sqrt{2(N+N^2)}\right) \left( -N+\sqrt{2(N+N^2)}\right) \) | Only steerable states |

\(E\left( \rho _{ij}^{(I)}\right) \ge -\frac{8}{3}\left( -1-N+\sqrt{2(N+N^2)}\right) \left( -N+\sqrt{2(N+N^2)}\right) \) | Steerable or unsteerable states | |

and \(E\left( \rho _{ij}^{(I)}\right) <-\frac{2}{3}\left( N^2-1\right) \) | ||

\(E\left( \rho _{ij}^{(I)}\right) \ge -\frac{2}{3}\left( N^2-1\right) \) | Only unsteerable states | |

\(N\ge \frac{1}{7}\) | \(E\left( \rho _{ij}^{(I)}\right) < -\frac{2}{3}\left( N^2-1\right) \) | Only steerable states |

\(E\left( \rho _{ij}^{(I)}\right) \ge -\frac{2}{3}\left( N^2-1\right) \) | Only unsteerable states |

## 3 Three-qubit system with two excitations

As we have shown in the previous section, for the case of single excited system one qubit can steer two others simultaneously. For double excited systems, such type of steering can not appear. Let us analyze situations when the first qubit steers two other. For such a case, two steering parameters must be greater than zero \(S_{21}^{(II)}>0, \,S_{21}^{(II)}>0\). From relations (33), we obtain that two conditions must be satisfied simultaneously: \(P_{011}P_{101}+\frac{3}{2}P_{101}+P_{011}>\frac{3}{2}\) and \(P_{011}P_{110}+\frac{3}{2}P_{110}+P_{011}>\frac{3}{2}\). Unfortunately, the system of these two inequalities for the probabilities has no solution and, in consequence, one subsystem cannot steers two others.

At this point, we would like to mention two other possible situations—when there are no excitations in the system or three excitations are present. For such cases, the system remains in the state \(\vert 000\rangle \) or \(\vert 111\rangle \), respectively. Since we assumed here that all qubits do not interact with other systems, we deal here with trivial situations for which all steering parameters remain equal to zero or are negative. In consequence, for all states describing such systems the steering phenomenon does not appear.

It would be also desirable to mention here the problem of physical realization of the ideas presented in the last two sections. As we discuss here three-qubits states, we are looking for physical models which could be good candidates for systems described by such states. In particular, they should be two-level systems such as two-level atoms [40], two-state spin systems [41] or systems which can be treated as the so-called quantum scissors [42] (systems generally described in infinite Hilbert space but under some conditions their evolution remains closed within a finite set of the states—here we deal with two states \(\vert 0\rangle \) and \(\vert 1\rangle \)). Such scissors can lead to photon(phonon) blockade [43, 44] effects, and hence, considered system can evolve within two states. The examples of such scissors system can be found in a group of the optical Kerr or Kerr-like systems [45, 46, 47, 48]. Another interesting proposals are those related to the atoms trapped in optical lattices [49] (even in 3D lattices [50]), superconducting circuits [51, 52], quantum dot systems [53, 54], optomechanical models [19, 55] and other. What is important, the systems mentioned here were not only considered theoretically but also were applied in various experiments. Thus, measurements of qubits in molecular single-ion magnet were presented in [56], whereas three-qubit entanglement in a superconducting circuit was achieved experimentally by DiCarlo et al. [57] and by Neely et al. [58]. From other side, Neumann et al. presented the creation of (bipartite and tripartite) entangled quantum states in a tiny quantum register built from individual \(^{13}\)C nuclei located in a diamond lattice [59]. The blockade effects have also been observed successfully in experiments. In particular, one should mention at this point Santa Barbara experiment [60] in which photon blockade has been observed for a single atom trapped inside optical cavity [61]. Moreover, quantum scissors system which was based on the Kerr media effect, leading to the single-photon states generation, has been accomplished and experimentally observed by Kirchmair et al. [62].

If we look at the results concerning two models discussed here (those comprising one or two excitations), we can ask which of them is more feasible experimentally and which can lead to more interesting results. When we compare the map presented in Fig. 1 with its counterpart from Fig. 5, we see that the range of steerable states is wider for the situation when we deal with systems involving only single excitation. Thus, such systems seem to be more interesting and promising from the point of view of steerable states generation. From other side, if we look at the current progress in the field of experiments, systems for which single-photon (phonon) blockade can appear, seem to be more attainable experimentally than those where two or more photons blockade can be achieved. Additionally, for systems based on the trapped atoms, manipulation of a single atom seems to be easier than of groups of them. For the situation when we are dealing with two atoms, we should avoid the situation when both of them will be located at the same site. Since our system should be described in only two states basis (that defined by one-atom and no-atoms states), maximally one atom can be located at a given site. Therefore, models with single excitations seem to be more promising and interesting than those with two excitations.

## 4 Summary

In this work, we discussed the properties of a family of three-qubit states, especially in the context of quantum steerability. In particular, we have concentrated on two cases—(i) one qubit was in its upper state (one excitation was present in physical system) and (ii) two qubits were excited. Applying the parameters based on the Cavalcanti inequality, we analyzed the possibility of appearance of the steering between two qubits for the both situations.

We have shown that for the discussed model symmetric steering cannot appear—only asymmetric one can be observed. Moreover, we proved that for the single excited model one qubit can simultaneously steer two qubits. However, this type of steering we cannot observe for double excited cases. Additionally, it was proved that two qubits cannot steer simultaneously the remaining one.

It is known that the value of negativity for two-qubit system can not be greater than that for the corresponding concurrence. The negativity always exceeds or is equal to the bound found by Verstraete et al. [31]. In this paper, we have also compared the values of negativity, concurrence, linear entropy characterizing mixedness of the state and steering parameter. Such parameters were calculated for the states corresponding to the two-qubit subsystem. As a result, we have found the upper bound of unsteerable two-qubit mixed state for the states corresponding to the model with single and double excitations. We showed that if the negativity is greater than its boundary value, which was derived here, the discussed states are steerable. Moreover, we have found a condition when the considered here states are characterized by the strongest steering; see Eq. (20). It was also shown here that for a given value of concurrence the steering parameter reaches its maximal value when the negativity is equal to that of concurrence. In addition, maximal values of steering parameter for a given concurrence (Eq. 19) and for given negativity (Eq. 21) were found here.

Finally, we discussed how mixedness of the considered here states is related to the steering phenomenon. We have found linear entropy and conditions defining bounds between the regions corresponding to the steerable states and unsteerable ones. Those conditions were determined by formulas for the linear entropy parametrized by (i) the concurrence and (ii) the negativity. All derived here conditions were derived for the both cases—states with single and two excitations.

## Notes

### Acknowledgements

The authors would like to thank Prof. J. Peřina Jr., Dr. A. Barasiński and Dr. K. Bartkiewicz for their valuable and inspiring discussions.

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