**4. The First-Order Correlation Function and Linear Entropy**

In Section 3, we discussed the relationship between the internal coherence of subsystems (quantified by the degree of coherence *D*<sup>2</sup> *ij*), linear entropy and concurrence. Here, we shall consider the relationships among the mutual coherence quantified by the first-order correlation function and linear entropy and concurrence. Such first-order cross-correlation function for subsystems *i* and *j* can be written as [78,79]:

$$\log\_{ij}^{(1)} = \frac{|\langle \hat{a}\_i^\dagger \hat{a}\_j \rangle|}{\sqrt{\langle \hat{a}\_i^\dagger \hat{a}\_i \rangle \langle \hat{a}\_j^\dagger \hat{a}\_j \rangle}} \cdot \tag{34}$$

The function *g* (1) *ij* can take values from zero to unity. For maximally coherent states, it equals 1, whereas, when we do not observe coherence between subsystems *i* and *j*, *g* (1) *ij* = 0. All states corresponding to the single excitation's case, described by the wave func-

tion (1), are fully coherent and thus *g* (1) *ij* = 1. In contrast, if we assume the presence of two excitations (see, the wave function (3), the first-order correlation function can take various values from 0 to 1. Therefore, in further analysis, we focus only on the relations between linear entropy and first-order coherence for double excited systems.

In Figure 3, we present the results of numerical analysis concerning the ensemble of randomly generated states describing double excited systems. For such states, the blue area shows possible values of linear entropy for given values of the first-order correlation function. The boundary values of linear entropy are represented by black lines: solid and dashed ones.

To derive the maximal values of linear entropy parametrized by the first-order correlation function, we find the formulas describing *g* (1) *ij* function expressed by probabilities:

$$\begin{array}{rcl} \mathcal{S}\_{12}^{(1)} &=& \frac{\mathsf{C}\_{011}^{\*} \mathsf{C}\_{101}}{\sqrt{\left(P\_{101} + P\_{110}\right) \left(P\_{011} + P\_{110}\right)}}, \\ \mathcal{S}\_{13}^{(1)} &=& \frac{\mathsf{C}\_{011}^{\*} \mathsf{C}\_{110}}{\sqrt{\left(P\_{110} + P\_{101}\right) \left(P\_{011} + P\_{101}\right)}}, \\ \mathcal{S}\_{23}^{(1)} &=& \frac{\mathsf{C}\_{101}^{\*} \mathsf{C}\_{110}}{\sqrt{\left(P\_{110} + P\_{011}\right) \left(P\_{101} + P\_{011}\right)}}. \end{array} \tag{35}$$

In further analysis, we will consider real probability amplitudes *C*∗ *ijk* <sup>=</sup> *Cijk* <sup>=</sup> *Pijk*.

From Figure 3, we see that, for *g* (1) *ij* ≤ 1/3, the maximal value of *Eij* does not depend on the value of the first-order correlation function. For such a case, the two-qubit matrix is expressed by Equation (19), and the corresponding first-order correlation function is given as

$$\mathbf{g}\_{ij}^{(1)} = \frac{\sqrt{(1/2 - \alpha)\alpha}}{\sqrt{(1 - \alpha)(\alpha + 1/2)}} \,\mathrm{}.\tag{36}$$

Thus, for *g* (1) *ij* ≤ 1/3, the maximal value of *Eij* is equal to 2/3 and does not depend on the values of the parameter *α*.

From the other side, when *g* (1) *ij* > 1/3, the maximal possible value of linear entropy decreases with the increasing value of the first-order correlation function (see the dashed line in Figure 3). In such a case, the density matrix describing the system is:

$$
\begin{array}{rcl}
\rho\_{ij} &=& \begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & a & \sqrt{a\beta} & 0 \\
0 & \sqrt{a\beta} & \beta & 0 \\
0 & 0 & 0 & 1 - a - \beta
\end{bmatrix}
\end{array}
\tag{37}
$$

where the probabilities *α* and *β* have to be equal to

$$\alpha = \beta = \frac{\mathcal{g}\_{ij}^{(1)}}{1 + \mathcal{g}\_{ij}^{(1)}},\tag{38}$$

and the probabilities *α* and *β* can take values within the range 1/4, 1/2. When *α* = *β* = 1/4, the first-order correlation function is 1/3, and *Eij* = 2/3. However, if *α* = *β* = 1/2, the linear entropy reaches zero, and function *g* (1) *ij* is equal to unity.

In general, for the two-qubit states represented by Equation (37), *Eij* fulfills the following relation:

$$E\_{ij} = -\frac{16\left(\mathcal{g}\_{ij}^{(1)} - 1\right)\mathcal{g}\_{ij}^{(1)}}{3\left(1 + \mathcal{g}\_{ij}^{(1)}\right)^2} \,. \tag{39}$$

In the next step, we discuss the case when the states are strongly entangled ones, i.e., the concurrence is assumed to be equal to or higher than 0.9. For such a situation, the minimal value of *Eij* parametrized by *g* (1) *ij* is defined by the condition represented by the dash-dotted line in Figure 3b. The red area corresponds to the values linear entropy and first-order correlation function for states presenting strong entanglement. From Figure 3b, we see that the states with *Cij* ≥ 0.9 exhibit a high level of the first-order correlation function *g* (1) *ij* ∈ 9/11; 1. Moreover, for such the case, the linear entropy is limited to values determined by:

$$E\_{ij} \ge -\frac{27\left(\mathcal{g}\_{ij}^{(1)^2} - 1\right)\left(481\mathcal{g}\_{ij}^{(1)^2} - 81\right)}{20000\mathcal{g}\_{ij}^{(1)^4}}.\tag{40}$$

We derived that condition using the definitions (12), (28) and (36) and assuming that *Cij* = 0.9.

Thus, one can state that the strongly entangled two-qubit states are simultaneously characterized by low levels of mixedness and high values of the first-order coherence function.

**Figure 3.** (**a**) Linear entropy *Eij* versus first-order correlation function *g* (1) *ij* for two-qubit states described by the density matrix *ρij*, calculated numerically (blue area). Black lines are plotted according to the analytical formulas derived here determining the borders between various regions of the states. (**b**) The same as in (**a**). Additionally, the red area presents the possible values of linear entropy and the first-order correlation function for two-qubit states with concurrence *Cij* > 0.9 (red area).

#### **5. The Second-Order Correlation Function and Linear Entropy**

Analogously, as in the previous section, we will analyze at this point relations between the degree of mixedness and second-order coherence function *g* (2) *ij* . This function quantifies the correlations between intensities of field, contrary to *g* (1) *ij* considered earlier that described the correlations between the amplitudes of two fields. *g* (2) *ij* is defined here for two subsystems *i* and *j* and can be expressed as [78,79]:

$$\mathbf{g}\_{ij}^{(2)} = \frac{\langle \mathbf{a}\_i^\dagger \mathbf{a}\_j^\dagger \mathbf{a}\_i \mathbf{a}\_j \rangle}{\langle \mathbf{a}\_i^\dagger \mathbf{a}\_i \rangle \langle \mathbf{a}\_j^\dagger \mathbf{a}\_j \rangle} \,. \tag{41}$$

Applying the procedure described in the previous section, we shall concentrate here on the case of double excited systems described by the density matrix (4). For such a situation, the second-order correlation function expressed by probabilities for each qubit– qubit subsystem can be written as:

$$\begin{array}{rcl} \mathcal{S}\_{12}^{(2)} &=& \frac{P\_{110}}{(P\_{101} + P\_{110})(P\_{011} + P\_{110})} \\ \mathcal{S}\_{13}^{(2)} &=& \frac{P\_{011}}{(P\_{110} + P\_{101})(P\_{011} + P\_{101})} \\ \mathcal{S}\_{23}^{(2)} &=& \frac{P\_{101}}{(P\_{110} + P\_{011})(P\_{101} + P\_{011})} \end{array} \tag{42}$$

Figure 4 depicts numerical results of analysis of randomly generated states for the system with double excitation. The same as previously, colored areas correspond to the possible achievable states characterized by various pairs of the values of the linear entropy and *g* (2) *ij* . The black lines appearing there denote the boundary values of the entropy for the particular *g* (2) *ij* . When *g* (2) *ij* < 8/9, the maximal possible value of *Eij* monotonously increases with the increasing value of the second-order correlation function (see the dashed line in Figure 4). In such a case, using Equations (12) and (43), we find that the maximal value of *Eij* fulfills the relation:

$$E\_{ij} = \frac{16\left(\sqrt{1 - \mathcal{g}\_{ij}^{(2)}} - 1\right)^2 \left(\sqrt{1 - \mathcal{g}\_{ij}^{(2)}} + \mathcal{g}\_{ij}^{(2)} - 1\right)}{3\mathcal{g}\_{ij}^{(2)^2}}.\tag{43}$$

The entropy *Eij* given by (43) reaches its maximal values when the system is described by the density matrix (37) with the probabilities *α* and *β* equal to:

$$\alpha = \beta = \frac{g\_{ij}^{(2)} + \sqrt{1 - g\_{ij}^{(2)}} - 1}{g\_{ij}^{(2)}} \tag{44}$$

where *α* and *β* can take values in the range 0; 1/2. When both *α* and *β* are simultaneously equal to 0 or 1/2, the second-order correlation function and entropy become equal to zero. However, if *α* = *β* = 1/4, the linear entropy *Eij* = 2/3, and *g* (2) *ij* reaches = 8/9.

However, when *g* (2) *ij* ≥ 8/9, the maximal possible value of linear entropy stops being dependent on the second-order correlation function and remains equal to 2/3 (see the black solid line in Figure 4. For such a case, the two-qubit density matrix is described by Equation (19).

In Figure 4b, the red area corresponds to the strongly entangled states with concurrence *Cij* ≥ 0.9. The dash-dotted line appearing there represents the condition for the minimal values of *Eij* parametrized by *g* (2) *ij* . Simultaneous analysis of Equations (12), (28) and (43), describing the entropy, second-order correlation function, concurrence, respectively, and assuming that concurrence is equal to 0.9 gives us the minimal achievable entropy for strongly entangled states:

$$E\_{ij} = \frac{27 \left( 400 - 481g\_{ij}^{(2)} \right) g\_{ij}^{(2)}}{20000 \left( g\_{ij}^{(2)} - 1 \right)^2},\tag{45}$$

where *g* (2) *ij* ∈ 0; 40/121. It is seen that the strongly entangled two-qubit states are characterized by simultaneously low levels of both mixedness and second-order coherence function.

**Figure 4.** (**a**) Linear entropy *Eij* versus second-order correlation function *g* (2) *ij* for two-qubit states described by the density matrix *ρij*, calculated numerically (yellow area). Black lines are plotted according to the analytical formulas derived here and determining the borders between various regions of the states. (**b**) The same as in (**a**). Additionally, the red area presents the possible values of linear entropy and the second-order correlation function for two-qubit states with concurrence *Cij* > 0.9 (red area).

#### **6. Conclusions**

In this work, we have analyzed two families of three-qubit states in the context of the appearance of coherence and entanglement as quantum resources, and the mixedness of discussed states. In particular, we have focused on the characteristics of possible achievable states describing the two-qubit subspace of the system. Applying the tracing out procedure, we have analyzed the degree of mixedness of such two-qubit states, the bipartite coherences, and entanglement. We have compared the degree of mixedness and the parameters describing coherences, such as the degree of coherence, the first- and second-order correlation function, and have shown the relations among them. Based on such performed analysis, we have derived boundary conditions for possible achievable strongly entangled two-qubit states. We have shown that the strongly entangled states can be characterized by low levels of mixedness and degree of coherence. On the other hand, analyzing the correlation functions *g* (1) *ij* and *g* (2) *ij* , it turned out that highly entangled states are states with high and low levels of the first and second-order correlation function, respectively.

**Author Contributions:** Conceptualization, J.K.K. and W.L.; methodology, J.K.K. and J.P.J.; software, J.K.K.; validation, J.K.K. and R.S.; formal analysis, J.K.K. and W.L.; investigation, J.K.K., W.L., R.S. and J.P.J.; writing—original draft preparation, J.K.K. and W.L.; writing—review and editing, J.K.K., R.S. and J.P.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** J.K.K. and W.L. acknowledge the support of the program of the Polish Minister of Science and Higher Education under the name "Regional Initiative of Excellence" in 2019-2022, Project No. 003/RID/2018/19, funding amount 11 936 596.10 PLN.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
