Exploring entropic uncertainty relation and dense coding capacity in a two-qubit X-state

Historically, the uncertainty principle which was introduced originally by Heisenberg [1] is an inherent feature of quantum mechanics limiting our capability to predict the measurement outcomes of two incompatible observables concurrently. Afterward, Robertson [2] and then Schrödinger [3] presented an inequality (so-called Robertson-Schrödinger uncertainty relation) for two arbitrary incompatible observables U and V as the following simple form

$$\begin{equation} \Delta U \Delta V\geq\frac{1}{2}|\langle [U, V]\rangle|, \end{equation} \tag{ 1 }$$

where $\Delta \mathcal{O} = \sqrt{\langle \mathcal{O}^2\rangle-\langle \mathcal{O}\rangle^2}$ with $\mathcal{O}\in \{U,V\}$ is the standard deviation, $\langle \mathcal{O}\rangle$ denotes the expectation value of operator $\mathcal{O},$ and [U, V] = UV − VU stands for the commutator. Unfortunately, the uncertainty relation (1) has a significant shortcoming, since the lower bound of this uncertainty relation depends on the state of the system, which leads to a trivial bound if the expectation value of the commutator on the state is zero, i.e. $|\langle [U, V]\rangle|$ = 0. To overcome this deficiency, Deutsch [4] suggested a novel expression including the Shannon entropy (so-called entropic uncertainty relation) instead of the standard deviation as

$$\begin{equation} H(U)+H(V)\geq \log_{2} \left(\frac{2}{1+\sqrt{c}}\right)^{2}, \end{equation} \tag{ 2 }$$

where $H(\mathcal{O}) = -\sum_{i}p_{i}\log_{2} p_{i}$ is the Shannon entropy, p_i represents the probability of obtaining ith outcome, and $c = \max_{j,\,k}\{|\langle u_{j}|v_{k}\rangle|^{2}\}$ where $|u_{j}\rangle$ and $|v_{k}\rangle$ are the eigenvectors of the Hermitian observables U and V, respectively. A few years later, Kraus proposed a clear formula [5], and later it was exactly demonstrated by Maassen and Uffink [6], whose form can be formulated as

$$\begin{equation} H(U)+H(V)\geq -\log_{2} c. \end{equation} \tag{ 3 }$$

Furthermore, Maccone and Pati [7] obtained the stronger uncertainty relations (so-called Maccone–Pati uncertainty relations) related to the sum of the variances, that its lower bound is non-trivial for two incompatible observables. The most initial works on uncertainty relations formulated as the sum of variances include (e.g. see [8]). Next, Wang et al [9] demonstrated these optimal outcomes for a fully optical setup.

Newly, an entropic uncertainty relation in the presence of quantum memory (EUR-QM) has been proposed [10–13] which has more efficiency than equation (3) (see [12] and [13] for detailed reviews on EUR-QM). To study EUR-QM, let us represent it from the viewpoint of uncertainty game between two legitimate players (so-called Alice and Bob). First, Bob prepares a correlated two-qubit quantum state ϱ_AB and transmits qubit A to Alice, with considering qubit B as a quantum memory during the game. Next, she measures one of the two incompatible observables U or V and sends to Bob her measurement outcomes. Bob can minimize the uncertainty from Alice's measurement outcomes. When two qubits A and B are fully entangled, Bob can precisely predict Alice's measurement results as stated in EPR paper. However, if there is no entanglement between them, the relation can be simplified to Deutsch's outcome [4]. Mathematically, this mysterious game is expressed as follows

$$\begin{equation} S(\varrho_{U|B})+S(\varrho_{V|B})\geq S(\varrho_{A|B})-\log_{2} c, \end{equation} \tag{ 4 }$$

in which $S(\varrho) = -\mathrm{{Tr}}[\varrho \log_{2} \varrho]$ is the von Neumann entropy, $S(\varrho_{A|B}) = S(\varrho_{AB})-S(\varrho_{B})$ is the conditional von Neumann entropy with $\varrho_{B} = \textmd{\textrm{Tr}}_{A}(\varrho_{AB})$ which is the reduced density matrix of quantum memory, and $S(\varrho_{\mathcal{O}|B}) = S(\varrho_{\mathcal{O}B})-S(\varrho_{B})$ gives the conditional von Neumann entropy of the post-measurement state. After measuring qubit A by operator U or V, the post-measurement states are

$$\begin{align} &\varrho_{UB} = \sum_{j}\left(|u_{j}\rangle_{A} \langle u_{j}|\otimes \mathbb{I}_{B}\right)\varrho_{AB}\left(|u_{j}\rangle_{A} \langle u_{j}|\otimes \mathbb{I}_{B}\right),\nonumber\\ &\varrho_{VB} = \sum_{k}\left(|v_{k}\rangle_{A} \langle v_{k}|\otimes \mathbb{I}_{B}\right)\varrho_{AB}\left(|v_{k}\rangle_{A} \langle v_{k}|\otimes \mathbb{I}_{B}\right), \end{align} \tag{ 5 }$$

where $\mathbb{I}$ is the identity matrix.

As usual, we consider U = σ_z and V = σ_x where σ_z and σ_x are the Pauli matrices, and so we obtain c = 1/2. Herein, we use notations U_L (so-called uncertainty) and U_R (so-called uncertainty lower bound) for the left-hand side (LHS) and the right-hand side (RHS) of inequality (4) respectively.

So far, many efforts have been made to improve EUR-QM [14–23], since it has various applications in quantum information processing, e.g. quantum cryptography [24], quantum metrology [25, 26], quantum key distribution [27, 28], entanglement witness [29, 30], and so on.

In recent years, many investigations have risen regarding quantum correlations and EUR-QM, and the relation between them in various quantum systems has been studied by [31–66]. Therefore, we are insisting that there is some intrinsic relationship between the uncertainty and quantum correlations. For example, Hu and Fan [32] determined a tight upper bound for quantum discord using EUR-QM, as well as the other works in this field [33, 34]. Impressively, Wang et al [35–56] investigated solid-state systems such as Heisenberg models and open quantum systems [57], and they also examined the relationship between quantum correlations and uncertainty. Recently, they experimentally investigated the linear-entropy-based uncertainty relations in all-optical systems [58] and they certificated the steering criterion based on a general entropic uncertainty relation [59]. They also experimentally explored the entropic uncertainty relations and coherence uncertainty relations [60]. Furthermore, Pourkarimi [61, 62] and Huang [63] investigated EUR-QM in the spin chain models with Dzyaloshinskii–Moriya interaction. Fang et al [64] studied the dynamic behavior of EUR-QM and quantum coherence in the structured reservoir, and they have shown that the EUR-QM is smallest in the disappearing limit of noise regardless of the forms of the initial state, while the quantum coherence keeps the maximum value. More recently, we investigated thermal quantum correlations and EUR-QM under two kinds of spin squeezing models and got a simple relationship between the quantum discord and entropic uncertainty [65]. Next, we examined EUR-QM, quantum discord, and fidelity of teleportation under decohering environments and found an explicit relationship between them [66].

The dense coding protocol initially proposed by Bennett and Wiesner [67], where the sender can send two bits of classical information to the receiver by transmitting a single-qubit, if they share a two-qubit maximally entangled state. After that, several works on dense coding have been presented theoretically and experimentally [68–80]. The dense coding capacity (DCC) is a vital parameter to measure the quality of the chosen quantum channel, i.e. when the sender (Bob) transmits a single-qubit to the receiver (Alice), how much classical information might be transmitted. The amount of information that can be moved is known to be limited by the Holevo quantity [81]. Since the Holevo quantity is asymptotically obtainable [82], we put forward DCC as

$$\begin{equation} \chi = S(\bar{\varrho})-S(\varrho), \end{equation} \tag{ 6 }$$

where $S(\bar{\varrho})$ is the von Neumann entropy of the signal ensemble average state $\bar{\varrho}$ and S(ϱ) is the von Neumann entropy of original state ϱ shared between Alice and Bob. The average quantum state of the signal ensemble for two-qubit can be expressed as

$$\begin{equation} \bar{\varrho} = \frac{1}{4}\sum_{i = 0}^{3}(\mathcal{U}_{i}\otimes\mathbb{I}_{2})\varrho(\mathcal{U}_{i}^{\dagger}\otimes\mathbb{I}_{2}), \end{equation} \tag{ 7 }$$

where the set of the mutually orthogonal unitary transformations are $\mathcal{U}_{0} = \mathbb{I}_{2}$, $\mathcal{U}_{1} = \sigma_{x}$, $\mathcal{U}_{2} = i\sigma_{y}$, and $\mathcal{U}_{3} = \sigma_{z}$. What physically happens is that Bob performs the set of mutually orthogonal unitary transformations on his quantum system to put the initially shared entangled state ϱ in $\varrho_{i} = (\mathcal{U}_{i}\otimes\mathbb{I}_{2})\varrho(\mathcal{U}_{i}^{\dagger}\otimes\mathbb{I}_{2})$, and sends off his quantum system to Alice, then Alice makes appropriate measurements on the quantum system ϱ_i to extract the signal. Note that when χ > 1, we have the valid dense coding, and for the optimal dense coding, χ must be the maximum ($\chi_{\max} = 2)$.

As discussed above, lots of investigations have been devoted to reviewing the entropic uncertainty relations and quantum correlations. Still, no one has examined EUR-QM and DCC together in any study of quantum systems. That is why we are motivated to study EUR-QM and DCC in arbitrary two-qubit X-states to be looking for a connection between them. Hence, the framework of this work is structured as follows. In the next section, we obtain the uncertainty and its bound as well as DCC for arbitrary two-qubit X-states. In section 3, we consider a typical solid-state system (known as two-qubit spin squeezing model) as an example and then we try to analyze the relationship between EUR-QM and DCC analytically and numerically in section 4. Finally, we close the paper with a brief conclusion in section 5.

Suppose that the density matrix of arbitrary two-qubit X-states based on the standard basis of $|00\rangle_{1}, |01\rangle_{2}, |10\rangle_{3}$ and $|11\rangle_{4}$ is as follows

$$\begin{equation} \varrho_{AB} = \left( \begin{array}{cccc} a & 0 & 0 & e \\ 0 & b & f & 0 \\ 0 & g & c & 0 \\ h & 0 & 0 & d \\ \end{array} \right), \end{equation} \tag{ 8 }$$

where the unit trace ($a+b+c+d = 1$) and positivity conditions ($bc\geq|f|^{2}$ and $ad\geq|e|^{2}$) are fulfilled.

For this density matrix (8), from (5), the post-measurement states are

$$\begin{align*} &\varrho_{\sigma_{x}B} = \frac{a+c}{2}(|00\rangle\langle00|+|10\rangle\langle10|)+\frac{b+d}{2}(|01\rangle\langle01|\\ &\qquad+|11\rangle\langle11|) +\frac{e+g}{2}(|01\rangle\langle10|+|11\rangle\langle00|)\\ &\qquad+\frac{f+h}{2}(|10\rangle\langle01|+|00\rangle\langle11|),\\ &\varrho_{\sigma_{z}B} = a|00\rangle\langle00|+b|01\rangle\langle01|+c|10\rangle\langle10|+d|11\rangle\langle11|, \end{align*}$$

therefore, the eigenvalues of $\varrho_{\sigma_{z}B}$ are as {a, b, c, d} and the eigenvalues of $\varrho_{\sigma_{x}B}$ are as $\{(1-t)/4, {(1-t)/4,}{(1+t)/4,} {(1+t)/4\}}$ in which $t = \sqrt{[1-2(b+d)]^{2}+4|e+g|^{2}}$. Besides, the von Neumann entropy of reduced density matrix is given by $S(\varrho_{B}) = -(a+c)\log_{2}(a+c)-(b+d)\log_{2}(b+d)$. Then, the von Neumann entropies of the post-measurement states read

$$\begin{equation} S(\varrho_{\sigma_{x}B}) = -\frac{1-t}{2}\log_{2}\frac{1-t}{4}-\frac{1+t}{2}\log_{2}\frac{1+t}{4}, \end{equation} \tag{ 9 }$$

$$\begin{equation} S(\varrho_{\sigma_{z}B}) = -a\log_{2}a-b\log_{2}b-c\log_{2}c-d\log_{2}d, \end{equation} \tag{ 10 }$$

so we get the uncertainty (LHS) as

$$\begin{align} &U_{L} = -a\log_{2}a-b\log_{2}b-c\log_{2}c-d\log_{2}d\nonumber\\ &\qquad+2((a+c)\log_{2}(a+c)+(b+d)\log_{2}(b+d))\nonumber\\ &\qquad-\frac{1-t}{2}\log_{2}\frac{1-t}{4}-\frac{1+t}{2}\log_{2}\frac{1+t}{4}, \end{align} \tag{ 11 }$$

and for the lower bound of uncertainty (RHS) we find

$$\begin{align} U_{R} =& -M^{+}\log_{2}M^{+}-M^{-}\log_{2}M^{-}-Q^{+}\log_{2}Q^{+}\nonumber\\ &-Q^{-}\log_{2}Q^{-} +(a+c)\log_{2}(a+c)\nonumber\\ &+(b+d)\log_{2}(b+d)+1, \end{align} \tag{ 12 }$$

where $M^{\pm} = ((a+d)\pm\sqrt{(a-d)^{2}+4|e|^{2}})/2$ and $Q^{\pm} = ((b+c)\pm\sqrt{(b-c)^{2}+4|f|^{2}})/2$. On the other hand, the average quantum state of the signal ensemble for two-qubit ϱ_AB can be expressed as

$$\begin{align} \bar{\varrho}_{AB} = &\frac{1}{2}[(a+c)(|00\rangle\langle00|+|10\rangle\langle10|)+(b+d)(|01\rangle\langle01|\nonumber\\ &+|11\rangle\langle11|)], \end{align} \tag{ 13 }$$

hence, the analytic expression of DCC (6) can be obtained as

$$\begin{align} \chi =&\, M^{+}\log_{2}M^{+}+M^{-}\log_{2}M^{-}+Q^{+}\log_{2}Q^{+}\nonumber\\ &+Q^{-}\log_{2}Q^{-}-(a+c)\log_{2}\left(\frac{a+c}{2}\right)\nonumber\\ &-(b+d)\log_{2}\left(\frac{b+d}{2}\right). \end{align} \tag{ 14 }$$

Significantly, we obtain a simple relationship between the lower bound of uncertainty and DCC after algebraic computations as

$$\begin{equation} U_{R} = 2-\chi. \end{equation} \tag{ 15 }$$

Due to their suitable scalability and integrability, solid-state systems attract much attention nowadays [83–95]. In this section, we consider a typical solid-state system with the Hamiltonian H for the two-qubit spin squeezing model under the external magnetic field B as (ћ = 1) [83, 84]

$$\begin{equation} H = \mu S_{x}^{2}+\varsigma S_{y}^{2}+\gamma (S_{x}S_{y}+S_{y}S_{x})+BS_{z}, \end{equation} \tag{ 16 }$$

where µ(≥ 0) and ς(≥ 0) are the strength of the spin squeezing interaction in x and y directions respectively, γ(≥ 0) describes the strength of the spin squeezing interaction in the x − y plan, and B(≥ 0) explains the strength of the external magnetic field in z-direction. Herein, $S_{\alpha} = \frac{1}{2}\Sigma_{i = 1}^{2}\sigma_{\alpha}^{i}$ is the collective spin operators, and $\sigma_{\alpha}^{i} (\alpha\in x, y, z)$ is the Pauli matrix of the ith spin.

Case 1. If ς = γ = 0, the Hamiltonian is reduced to the one-axis twisting model (OATM) with a transverse field

$$\begin{equation} H_{1} = \mu S_{x}^{2}+BS_{z}. \end{equation} \tag{ 17 }$$

Case 2. If ς = µ = 0, the Hamiltonian is reduced to the two-axis counter-twisting model (TACM) with a transverse field

$$\begin{equation} H_{2} = \gamma (S_{x}S_{y}+S_{y}S_{x})+BS_{z}. \end{equation} \tag{ 18 }$$

The density operator when the typical two-qubit system is in a thermal equilibrium at temperature T can be described as

$$\begin{equation} \varrho_{AB} = Z^{-1}e^{-\beta H} = Z^{-1}\sum_{i = 1}^{4}e^{-\beta E_{i}}|\psi_{i}\rangle \langle\psi_{i}|, \end{equation} \tag{ 19 }$$

where $Z = \mathrm{{Tr}}[\exp( -\beta H)]$ is the partition function of the system, β = 1/κ_BT for which κ_B denotes the Boltzmann constant (considered as κ_B = 1 for simplicity), and E_i and $|\psi_{i}\rangle$ are respectively eigenvalues and eigenvectors of the Hamiltonian.

4.1. EUR-QM and DCC in OATM

From equation (17), the eigenvalues and the corresponding eigenvectors of the Hamiltonian for case 1 are obtained as

$$\begin{align} &E_{1} = \frac{\mu-\kappa}{2},\quad|\psi_{1}\rangle = \frac{1}{A_{-}}((2B-\kappa)|00\rangle+\mu|11\rangle),\nonumber\\ &E_{2} = 0,\quad\quad\quad\,|\psi_{2}\rangle = \frac{1}{\sqrt{2}}(|10\rangle-|01\rangle),\nonumber\\ &E_{3} = \mu,\quad\quad\quad\,|\psi_{3}\rangle = \frac{1}{\sqrt{2}}(|10\rangle+|01\rangle),\nonumber\\ &E_{4} = \frac{\mu+\kappa}{2},\quad\,|\psi_{4}\rangle = \frac{1}{A_{+}}((2B+\kappa)|00\rangle+\mu|11\rangle), \end{align} \tag{ 20 }$$

where $\kappa = \sqrt{\mu^{2}+4B^{2}}$ and $A_{\pm} = \sqrt{\mu^{2}+(2B\pm\kappa)^{2}}$. Thus, by using equation (19) one obtains the density matrix of the bipartite system ϱ_AB as

$$\begin{equation} \varrho_{AB} = \left( \begin{array}{cccc} \nu^{-} & 0 & 0 & \omega \\ 0 & \eta^{+} & \eta^{-} & 0 \\ 0 & \eta^{-} & \eta^{+} & 0 \\ \omega & 0 & 0 & \nu^{+} \\ \end{array} \right), \end{equation} \tag{ 21 }$$

where

$$\begin{align} &\nu^{\pm} = e^{-\beta \mu/2}\left(\cosh\left(\beta\kappa/2\right)\pm 2B\kappa^{-1}\sinh\left(\beta\kappa/2\right)\right)/Z,\nonumber\\ &\omega = -\mu e^{-\beta \mu/2} \sinh\left(\beta\kappa/2\right)/Z\kappa,\nonumber\\ &\eta^{\pm} = \left(\pm1+e^{-\beta\mu}\right)/2Z,\nonumber\\ &Z = 2e^{-\beta \mu/2}\left(\cosh\left(\beta\kappa/2\right)+\cosh\left(\beta\mu/2\right)\right). \end{align} \tag{ 22 }$$

It is straightforward to obtain analytical expressions of U_L, U_R, and DCC by substituting (21) into (11), (12), and (14) respectively. So we obtain the uncertainty and its lower bound as what follows

$$\begin{align*} U_{L} =& -\sum_{i = 1}^{4}\Omega_{i}\log_{2}\Omega_{i}-(\nu^{-}\log_{2}\nu^{-}+2(\eta^{+}\log_{2}\eta^{+})\\ &+\nu^{+}\log_{2}\nu^{+})+2(\epsilon^{-}\log_{2}\epsilon^{-}+\epsilon^{+}\log_{2}\epsilon^{+}), \end{align*}$$

and

$$\begin{align*} U_{R} =& 1-(\lambda^{+}\log_{2}\lambda^{+}+\lambda^{-}\log_{2}\lambda^{-}+\xi^{+}\log_{2}\xi^{+}\\ &+\xi^{-}\log_{2}\xi^{-})+\epsilon^{-}\log_{2}\epsilon^{-}+\epsilon^{+}\log_{2}\epsilon^{+}. \end{align*}$$

Also for DCC we get

$$\begin{align*} &\chi = \lambda^{+}\log_{2}\lambda^{+}+\lambda^{-}\log_{2}\lambda^{-}+\xi^{+}\log_{2}\xi^{+}+\xi^{-}\log_{2}\xi^{-}\\ &\quad-\epsilon^{-}\log_{2}\frac{\epsilon^{-}}{2}-\epsilon^{+}\log_{2}\frac{\epsilon^{+}}{2}, \end{align*}$$

where $\Omega_{1} = \Omega_{2} = (1-\tau)/4$, $\Omega_{3} = \Omega_{4} = (1+\tau)/4$, $\tau = \sqrt{4(\omega+\eta^{-})^{2}+(\nu^{-}-\nu^{+})^{2}}$, $\lambda^{\pm} = ((\nu^{-}+\nu^{+})\pm\sqrt{4|\omega|^{2}+(\nu^{-}-\nu^{+})^{2}} )/2$, $\xi^{\pm} = \eta^{+}\pm\eta^{-}$, and $\epsilon^{\pm} = \nu^{\pm}+\eta^{+}$.

In the ground state $T\rightarrow0$, by analyzing the eigensystem (20), we know that $E_{1} = (\mu-\kappa)/2$ has minimal energy, and $|\psi_{1}\rangle = ((2B-\kappa)|00\rangle+\mu|11\rangle)/A_{-}$ is the corresponding eigenstate. Therefore, we obtain the simplified expressions of U_L, U_R, and DCC in the ground state as

$$\begin{align*} U_{L}&\equiv U_{R} \\ &=\frac{1}{A_{-}^{2}}\left(\mu^{2}\log_{2}\frac{\mu^{2}}{A_{-}^{2}}+(2B-\kappa)^{2}\log_{2}\frac{(2B-\kappa)^{2}}{A_{-}^{2}}\right)+1, \end{align*}$$

and

$$\begin{equation*} \chi = -\frac{1}{A_{-}^{2}}\left(\mu^{2}\log_{2}\frac{\mu^{2}}{2A_{-}^{2}}+(2B-\kappa)^{2}\log_{2}\frac{(2B-\kappa)^{2}}{2A_{-}^{2}}\right). \end{equation*}$$

As a result, $U_{L}\equiv U_{R} = 2-\chi$ which implies a fully anti-correlated relationship between EUR-QM and DCC.

4.2. EUR-QM and DCC in TACM

As described by the Hamiltonian (18), the eigenvalues and the corresponding eigenstates for case 2 are achieved as

$$\begin{align} &E_{1} = -v,\quad\,|\psi_{1}\rangle = \frac{1}{F_{-}}(i(v-B)|00\rangle+\gamma|11\rangle),\nonumber\\ &E_{2} = 0,\quad\quad|\psi_{2}\rangle = |01\rangle,\nonumber\\ &E_{3} = 0,\quad\quad|\psi_{3}\rangle = |10\rangle,\nonumber\\ &E_{4} = v,\quad\:\:\;\:|\psi_{4}\rangle = \frac{1}{F_{+}}(-i(v+B)|00\rangle+\gamma|11\rangle), \end{align} \tag{ 23 }$$

where $v = \sqrt{\gamma^{2}+B^{2}}$ and $F_{\pm} = \sqrt{\gamma^{2}+(v\pm B)^{2}}$. By substituting (23) into (19), one can obtain the density operator as follows

$$\begin{equation} \varrho_{AB} = \left( \begin{array}{cccc} \vartheta^{-} & 0 & 0 & \zeta \\ 0 & q & 0 & 0 \\ 0 & 0 & q & 0 \\ -\zeta & 0 & 0 & \vartheta^{+} \\ \end{array} \right), \end{equation} \tag{ 24 }$$

where

$$\begin{align} &\vartheta^{\pm} = \left(v\cosh(v\beta)\pm B\sinh(v\beta)\right)/Zv,\nonumber\\ &\zeta = i\gamma\sinh(v\beta)/Zv,\nonumber\\ &q = 1/Z,\nonumber\\ &Z = 2\left(1+\cosh(v\beta)\right). \end{align} \tag{ 25 }$$

Having known the density matrix (24) and by using the definition of EUR-QM and DCC, we obtain the analytical expressions of U_L, U_R, and DCC in TACM. First, we obtain the uncertainty (11) as

$$\begin{align*} &U_{L} = -\sum_{i = 1}^{4}\delta_{i}\log_{2}\delta_{i}-(\vartheta^{-}\log_{2}\vartheta^{-}+2(q\log_{2}q)+\vartheta^{+}\log_{2}\vartheta^{+})\\ &\quad\quad+2(\varepsilon^{-}\log_{2}\varepsilon^{-}+\varepsilon^{+}\log_{2}\varepsilon^{+}), \end{align*}$$

then, for the lower bound of uncertainty (12) we get

$$\begin{align*} &U_{R} = 1-(\Lambda^{+}\log_{2}\Lambda^{+}+\Lambda^{-}\log_{2}\Lambda^{-}+2q\log_{2}q)\\ &\quad\quad+\varepsilon^{-}\log_{2}\varepsilon^{-}+\varepsilon^{+}\log_{2}\varepsilon^{+}, \end{align*}$$

on the other hand, we find DCC (14) as

$$\begin{align*} &\chi = \Lambda^{+}\log_{2}\Lambda^{+}+\Lambda^{-}\log_{2}\Lambda^{-}+2q\log_{2}q\\ &\quad-\varepsilon^{-}\log_{2}\frac{\varepsilon^{-}}{2}-\varepsilon^{+}\log_{2}\frac{\varepsilon^{+}}{2}, \end{align*}$$

where $\varepsilon^{\pm} = \vartheta^{\pm}+q$, $\delta_{1} = \delta_{2} = (1-\Delta)/4$, $\delta_{3} = \delta_{4} = (1+\Delta)/4$, $\Lambda^{\pm} = ((\vartheta^{-}+\vartheta^{+})\pm\Delta)/2$, and $\Delta = \sqrt{(\vartheta^{-}-\vartheta^{+})^{2}-4|\zeta|^{2}}$.

As explained in the previous section, we find the simplified expressions of U_L, U_R, and DCC in the ground state as

$$\begin{equation*} U_{L}\equiv U_{R} = \frac{1}{F_{-}^{2}}\left(\gamma^{2}\log_{2}\frac{\gamma^{2}}{F_{-}^{2}}+(v-B)^{2}\log_{2}\frac{(v-B)^{2}}{F_{-}^{2}}\right)+1, \end{equation*}$$

and

$$\begin{equation*} \chi = -\frac{1}{F_{-}^{2}}\left(\gamma^{2}\log_{2}\frac{\gamma^{2}}{2F_{-}^{2}}+(v-B)^{2}\log_{2}\frac{(v-B)^{2}}{2F_{-}^{2}}\right). \end{equation*}$$

Consequently, $U_{L}\equiv U_{R} = 2-\chi$ implies that the uncertainty and its bound have a fully anti-correlated relationship with DCC in the ground state.

4.3. The comparison of EUR-QM and DCC for two cases

Now, let us analyze and compare the thermal EUR-QM and DCC in two cases as functions of temperature (T), spin squeezing parameters (µ and γ), and the external magnetic field (B).

The thermal evolution of the uncertainty and DCC has been displayed in figures 1 and 2 for OATM and TACM respectively with different values of the strength of the magnetic field and spin squeezing. As shown in figure 1(a), DCC collapses with increasing temperature when the strength of the magnetic field, B, and spin squeezing, µ, are weak. Conversely, the uncertainty and its bound rise from more than 0.5 to get their maximal value. Based on this figure and the other figures in this work, one observes that there is a fully anti-correlated relationship between DCC and the lower bound of the uncertainty as expected from (15). DCC remains unchanged at the beginning for low temperatures and then suddenly falls to disappear asymptotically as displayed in figure 1(a). However, as seen from figures 1(b) and (c), it is clear that the threshold value of the critical temperature T_c is enhanced by increasing µ and B. We note that T_c is the critical point for which EUR-QM and DCC suddenly change and they remain unchanged for 0 < T < T_c.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

This scenario happens similarly for TACM except for the beginning points of DCC and EUR-QM which are upper and lower than ones in OATM for the same values of the magnetic field and spin squeezing parameters. Notably, in this case, U_L, U_R, and DCC most remain unchanged for higher temperatures than the previous case, i.e. the threshold value of T_c is enhanced by increasing γ and B.

Figures 3 and 4 reveal the uncertainty and its bound, as well as DCC as functions of B for specific value of T = 0.5 and different values of µ and γ for OATM and TACM, respectively. In both the cases, DCC increases and the uncertainty decreases with increasing the strength of the external magnetic field. As mentioned before, χ must be greater than 1 to get a valid dense coding. Hence, this condition cannot be fulfilled in OATM case even for strong values of B and µ. However, the valid value can be achieved in TACM case when $\gamma \rightarrow 1$ and the strong value of B as shown in figure 4(c). In this case, the uncertainty becomes lesser than 1 and it gets the lower bound, U_L ≡ U_R.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figures 5 and 6 are devoted to studying the effect of µ and γ on the uncertainty of incompatible observables and DCC. From these figures, it is clear that µ and γ play a constructive role to increase DCC and decrease the amount of EUR-QM. Obviously, the strength of spin squeezing can lead to growing χ and reduce the uncertainty in two cases, which is more desirable in quantum information processing. Again, OATM shows its inability to obtain valid dense coding. Nevertheless, the strength of spin squeezing in TACM can cause the uncertainty and its bound to vanish, but DCC to get its maximal value, which is called optimal dense coding.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In summary, we have presented the analytical expressions for the uncertainty and its lower bound and also DCC for an arbitrary two-qubit X-state. Interestingly, we have obtained a simple relationship between the lower bound of the uncertainty and DCC as U_R = 2 − χ, which is true for all two-qubit X-state density matrices. To be specific, we have examined the thermal EUR-QM and DCC in two kinds of two-qubit spin squeezing models under an external magnetic field. Generally, the results reveal that EUR-QM and DCC have different behaviors, that means the uncertainty and its bound decrease when DCC increases and vice versa. In the ground state, we have provided the analytical expressions of EUR-QM and DCC for two cases which lead to a simple relation between them. This relation reveals a fully anti-correlated relationship between EUR-QM and dense coding. In both the cases, the uncertainties achieve the lower bounds in the limit of zero temperature. Remarkably, our analysis shows that the valid dense coding cannot be fulfilled in OATM case. However, if one properly chooses the parameters γ and B, TACM case not only carries out the valid dense coding but also the optimal dense coding easily can be achieved. Hopefully, we believe that our examinations may present a non-trivial insight into thermal dense coding and entropic uncertainty in spin squeezing systems, and be helpful to understand about quantum measurement precision in quantum information theory, especially solid-state systems.