Distillability sudden death in two-qutrit systems with external magnetic field and Dzyaloshinskii-Moriya interaction due to decoherence

Quantum entanglement not only plays a basic role in quantum theory, but also is a key resource in quantum communication and quantum information processing [1–4]. However, on the one hand, due to the quantum systems unavoidably interacting with its environments, this leads to quantum decoherence which will degrade the entanglement of systems. Hence, studying the entanglement properties under the influence of the decoherence becomes important and necessary. Initially, the phenomenon of finite time disentanglement, also was named entanglement sudden death (ESD) which was proposed by Yu and Eberly [5,6] had been proven to occur in a quantum optics experiment.

On the other hand, as is well known, there is no simple necessary and sufficient condition of entanglement characterization except for entangled pure states. Bipartite entanglement of systems in lower dimensional Hilbert space, e.g. $2\times2$ and $2\times3$, can be characterized by the Peres Horodecki separability criterion [7,8] which implies, if a quantum state of qubit-qubit and qubit-qutrit systems has a positive partial transpose (PPT), then it is a separable state, otherwise it is a inseparable state. However, for higher dimensional bipartite systems, this criterion is in vain, there can exist entangled states that are positive after this operation, but such states are not distillable [9]. Recently, a transformation in the type of its entanglement from bound (free) to free (bound) has been demonstrated [10–12]. In the bipartite systems case, both unitary and nonunitary evolution of bound entangled states may give rise to the birth of free entanglement [13,14]. In particular, under certain type of environment interaction, certain free entangled states may be converted into bound entangled states in a finite time. For qutrit-qutrit systems, analogous to the definition of ESD, if an initial free entangled state becomes nondistillable in a finite time under the influence of local decoherence, then we say that it undergoes distillability sudden death (DSD) [15]. This discovery has attracted so many interests. Very recently, free entangled states may be converted into bound entangled states in the presence of either multilocal decoherence [16] or combination of collective and local dephasing processes [17] have been investigated.

In the present paper, we have studied this phenomenon of DSD for certain qutrit-qutrit systems which are in the presence of the external magnetic field and Dzyaloshinskii-Moriya (DM) interaction under decoherence, where the two qutrits are initially prepared in a specific family of entanglement states. Compared to previous work in refs. [15–17] where the phenomenons of DSD for certain qutrit-qutrit systems under amplitude damping, we focus here on certain initial prepared free entangled states become bound entangled in a finite time with the external magnetic field and DM interaction under decoherence. The DM interaction which is arisen from spin-orbit coupling [18], plays the role to quantum entanglement dynamics [19–21]. Besides, possessing external magnetic field B₀, a parameter b which controls the inhomogeneity of B₀, and the bilinear interaction between qubits effect on the bipartite entanglement dynamics in isotropic Heisenberg spin chains have been formulated and studied in detail [22,23]. Here, the effects of the external magnetic field strength and the DM interaction parameter, as well as the intrinsic decoherence parameter, on the possibility of DSD in our model also have been discussed in detail.

The paper is organized as follows. In the second section, we illustrate the physical model of two-qutrit systems under the intrinsic decoherence and give the basic equation of motion along with its solution. We briefly discuss the idea of DSD and demonstrate the possibility of DSD in two-qutrit systems with external magnetic field and DM interaction under decoherence in the third section. Finally, we give the conclusion in the fourth section.

We consider our system composed of two Heisenberg qutrits A and B in the presence of an inhomogeneous magnetic field and DM interaction. The Hamiltonian of the system is given as $(\hbar=1)$ [24]

$$\begin{eqnarray} H&=J\hat{S}_{A}\cdot\hat{S}_{B}+(B_{0}+b)\hat{S}_{A}+(B_{0}-b)\hat{S}_{B} \nonumber \\ &\quad +\,\vec{D}\cdot(\hat{S}_{A}\times\hat{S}_{B}), \end{eqnarray} \tag{ 1 }$$

where the coupling coefficients J is the isotropic bilinear spin-spin interaction between two Heisenberg qutrits, B₀ is the external magnetic field, the inhomogeneity of B₀ which is controlled by the parameter b, $\vec{D}=D \vec{k}$ is the DM vector, D is the strength of the DM interaction which is arisen from spin-orbit coupling. When two isotropic Heiseberg qutrits are prepared in high excited states with the JJ coupling, the isotropic bilinear spin-spin interaction between two Heisenberg qutrits is weaker enough than the spin-orbit coupling, then it is reasonable to neglect isotropic term and remain only DM interaction. In this paper, we only consider two qutrits A and B in the presence of an inhomogeneous magnetic field and DM interaction between qutrits A and B, choosing DM interaction vector $\vec{D}=D\vec{z}$, and the magnetic fields are assumed to be along the z-direction. The Hamiltonian can be expressed as

$$\begin{eqnarray} &&H = (B_{0}+b)\hat{S}_{A}^{z}+(B_{0}-b)\hat{S}_{B}^{z}+D (\hat{S}_{A}^{x} \hat{S}_{B}^{y}-\hat{S}_{A}^{y}\hat{S}_{B}^{x}).\quad \end{eqnarray} \tag{ 2 }$$

Here $\hat{S}_{i}=(S^{x},S^{y},S^{z}),i=A,B$ denotes the spin-1 operators

$$\begin{eqnarray} &S^{x}=\frac{1}{\sqrt{2}}\left(\begin{array}{c@{\quad}c@{\quad}c} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0 \end{array} \right)\!,\qquad S^{y}=\frac{1}{\sqrt{2}}\left(\begin{array}{c@{\quad}c@{\quad}c} 0 & -i & 0\\ i & 0 & -i\\ 0 & i & 0 \end{array} \right),\nonumber S^{z}=\left(\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -1 \end{array} \right).\end{eqnarray} \tag{ 3 }$$

Denoting $|0\rangle$, $|1\rangle$ and $|2\rangle$ as the ground, first excited and second excited state of a qutrit, respectively. We choose the basis $\{|00\rangle, |01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle, |22\rangle\}$. The Hamiltonian eq. (2) has the following matrix form:

$$\begin{equation} H=\left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 2B_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & B_{0}+b & 0 & iD & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2b & 0 & iD & 0 & 0 & 0 & 0\\ 0 & -iD & 0 & B_{0}-b & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -iD & 0 & 0 & 0 & iD & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & b-B_{0} & 0 & iD & 0\\ 0 & 0 & 0& 0 & -iD & 0 & -2b & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -iD & 0 & -b-B_{0} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2B_{0} \end{array} \right).\end{equation} \tag{ 4 }$$

According to the Milburn's decoherence model, the master equation $(\hbar=1)$ describing the dynamics of two-qutrit systems with external magnetic field and DM interaction under the Markovian approximation is [24,25]

$$\begin{eqnarray} &&\frac{\text{d}}{\text{d}t}\rho(t)=-i[H,\rho(t)]-\frac{\gamma}{2}\left[H,[H,\rho(t)]\right], \end{eqnarray} \tag{ 5 }$$

where γ is the intrinsic decoherence parameter. For the model studied here, it is possible to solve the dynamics of the total closed system exactly. The formal solution of Milburn equation can be written as

$$\begin{eqnarray} &&\rho(t)=\sum_{k=0}^{\infty}\frac{(\gamma t)^{k}}{k!}U_{k}(t) \rho(0) U_{k}^{\dagger}(t), \end{eqnarray} \tag{ 6 }$$

with $U_{k}(t)=H^{k}e^{-iHt}e^{-(\gamma t/2)H^2}$ and $\rho(0)$ being the initial state of the system. Further calculations allow us to obtain the time evolution of the density matrix

$$\begin{eqnarray} \rho(t)&=\sum_{mn}\exp \left[\frac{-\gamma t}{2}(E_{m}-E_{n})^2-i(E_{m}-E_{n}) t\right]\nonumber \\ &\quad \times\,\langle \psi_{m}|\rho(0)|\psi_{n}\rangle |\psi_{m}\rangle \langle \psi_{n}|, \end{eqnarray} \tag{ 7 }$$

where E_n and $\psi_{n}$ are the eigenvalues and eigenvectors of H, respectively, given in the appendix.

In this paper, we consider a particular initial state given as

$$\begin{eqnarray} &&\rho_{\alpha}(0)=\frac{2}{7}|\psi_{+}\rangle \langle \psi_{+}|+\frac{\alpha} {7}\sigma_{+}+\frac{5-\alpha}{7}\sigma_{-}, \end{eqnarray} \tag{ 8 }$$

where $2 \leq\alpha \leq5$. In eq. (8) the maximally entangled state $|\psi_{+}\rangle=\frac{1}{\sqrt{3}}(|01\rangle+|10\rangle+|22\rangle)$ is mixed with separable state $\sigma_{+}=\frac{1}{3}|00\rangle \langle 00|+|12\rangle \langle 12|+|21\rangle \langle 21|$ and $\sigma_{-}=\frac{1}{3}|11\rangle \langle 11|+|20\rangle \langle 20|+|02\rangle \langle 02|$. It was shown [26] that $\rho_{\alpha}(0)$ is separable for $2 \leq\alpha \leq3$, bound entangled for $3 < \alpha \leq4$, and free entangled for $4 < \alpha \leq5$.

First of all, we review the realignment criterion [27] which can detect certain bound entangled states as well as the quantification of entanglement for qutrit-qutrit systems. The realignment criterion for a given density matrix ρ is defined as

$$\begin{eqnarray} &&RC=||\rho_{}^{R}||-1, \end{eqnarray} \tag{ 9 }$$

where $\rho_{ij,kl}^{R}=\rho_{ik,jl}$. For a separable state ρ, the realignment criterion implies that $RC\leq 0$. For a PPT state, the positive value of the quantity RC can prove the bound entangled state.

In order to determine the time evolution of entanglement for qutrit-qutrit systems, we adopt negativity to measure the entanglement, its definition is given as [28]

$$\begin{eqnarray} &&N(\rho)=||\rho_{}^{T}||-1, \end{eqnarray} \tag{ 10 }$$

where T represents the partial transpose of ρ, and $||.||$ takes its trace norm $\rho_{}^{T}$. It has a non-negative value of this measure, if a given state is a negative partial transpose (NPT) then it is an entangled state. For a PPT state which has zero negativity, we cannot conclude its entanglement or separability until some other measures or steps reveal its status. Note that the realignment criterion also cannot detect all bound entangled states. Once the negativity becomes zero, we can study the time evolution of a realignment criterion to detect the possibility of bound entangled states. In this paper, we adopt the realignment criterion to investigate the possibility of DSD in the presence of the external magnetic field and DM interaction due to decoherence.

Now let us investigate that the time-evolved density matrix $\rho_{\alpha}(0)$ does undergo DSD. Taking these states with $4 <\alpha \leq5$, by numerical results, the partial transpose of time-evolved density matrix $\rho(t)$ given in eq. (A.10) can have three possible negative eigenvalues $\lambda =\lambda_{1}\leq \lambda_{2} \leq \lambda_{3}$. In fig. 1, we plot λ as a function of t and α with external magnetic field parameter $B_{0}=1$, b = 1, DM interaction parameter D = 1 and the intrinsic decoherence parameter $\gamma=1$. We can see that the eigenvalues of partial transposition of the states $\rho(t)$ will always arrive at a positive value in a finite time. It is worth pointing out that, our results are in agreement with the outcomes of previous works [15–17]. In fig. 2, we plot the negativity and realignment criterion against the time t and a specific choice of the single parameter $\alpha=4.3$, with external magnetic field parameter $B_{0}=1$, b = 1, DM interaction parameter D = 1 and the intrinsic decoherence parameter $\gamma=1$. It clearly shows that for two-qutrit systems with external magnetic field and DM interaction under decoherence, an initial free entangled state becomes bound entangled at a time $t\approx 0.0494$. The entanglement of the PPT state is verified by the positive value of $||\rho_{}^{R}||- 1$ in the range $0.0494\leq t\leq 0.107$. However, this realignment criterion fails to detect the possible entanglement after time $t \approx0.107$. Hence, we have demonstrated that free entangled states exhibit DSD for two-qutrit systems with external magnetic field and DM interaction under decoherence.

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In fig. 3, we show the time evolution of the negativity and realignment criterion for an initial quantum state $\rho_{\alpha}(0)$ with $\alpha=4.8$, external magnetic field parameter $B_{0}=1$, b = 1, DM interaction parameter D = 1 and the intrinsic decoherence parameter $\gamma=1$. Clearly, the negativity becomes zero at $t\approx 0.50$, and the $||\rho_{}^{R}||- 1$ becomes zero at $t\approx 0.188$. As mentioned above, the realignment criterion also fails to detect the entangled states after time $t\approx 0.188$. In order to understand the initial state α effects on the possibility of DSD, we shows the sensitivity of the realignment criterion on the parameter α (see fig. 4). For this particular case $\alpha=4.8$ in fig. 3, we can see that although the initial NPT states (free entangled) become PPT after a finite time, we can not conclude their separability or entanglement immediately. Whatever, we show the PPT states might be entangled suffering DSD followed by ESD for two-qutrit systems with external magnetic field and DM interaction under decoherence.

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Following, we will investigate the influence of some other parameters on the possibility of DSD. First, the influence of the external magnetic field on the possibility of DSD is shown in fig. 5 and fig. 6. Figure 5 presents the time dependence of the realignment criterion $||\rho_{}^{R}||- 1$ for different magnetic field parameters B₀ with $\alpha=4.2$, b = 1, D = 1 and $\gamma=1$, and fig. 6 shows the time dependence of the realignment criterion $||\rho_{}^{R}||- 1$ for different parameters b which is controlled magnetic field $B_{0}=1$ and the other parameters are same as in fig. 5. It can be seen that, the stronger the external magnetic field B₀, and the larger the parameter b which controls the inhomogeneity of B₀, the quicker the initial NPT states may become PPT states after a finite time. This indicates the PPT states might be entangled suffering DSD for two-qutrit systems with external magnetic field and DM interaction under decoherence.

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Next, we will consider the DM interaction effect on the possibility of DSD. In fig. 7, we plot the time evolution of the realignment criterion $||\rho_{}^{R}||- 1$ for an initial quantum state $\rho_{\alpha}(0)$ for different D with $\alpha=4.2$, $B_{0}=1$, b = 1 and $\gamma=1$. It is observed that, the stronger the DM interaction, the quicker the initial NPT states may become PPT states after a finite time. This indicates the PPT states might be entangled suffering DSD for two-qutrit systems with external magnetic field and DM interaction under decoherence. In fig. 8, we plot the time evolution of the realignment criterion $||\rho_{}^{R}||- 1$ for an initial quantum state $\rho_{\alpha}(0)$ for different decoherence parameters γ with $\alpha=4.2$, $B_{0}=1$, b = 1 and D = 1. It is clear that, increasing the intrinsic decoherence parameter can accelerate the initial NPT states to be PPT states after a finite time. This indicates the PPT states might be entangled suffering DSD for two-qutrit systems with external magnetic field and DM interaction under decoherence. According to Milburn's decoherence model, it is worth noting the effect of the decoherence parameter γ on the evolution of a two-outrit system. When decoherence parameter $\gamma=0$, the evolution of density matrix $\rho(t)$ is unitary. This means the entanglement dynamics exhibits periodic oscillations in time. Based on our model, if the decoherence parameter γ is very small within a finite time t, environmentally induced intrinsic decoherence scheme may inhibit applying unitary operations. Whence this implies that, controlling specific parameter γ, (e.g., $\gamma=0.001$), can avoid DSD in this type of open system.

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In summary, we have studied this phenomenon of DSD for certain qutrit-qutrit systems which are in the presence of the external magnetic field and DM interaction under decoherence. We show that certain initial prepared free entangled states may become bound entangled in a finite time due to the external magnetic field and the DM interaction, as well as the intrinsic decoherence. Besides, we have investigated in detail the influence of the external magnetic field, the DM interaction and the intrinsic decoherence on the possibility of DSD. We conclude the following: Firstly, if given initial states, after the negativity becomes zero in a finite time, and then the initial NPT states become PPT after a finite time, they undergo DSD. Secondly, controlling parameters of the external magnetic field, DM interaction and the intrinsic decoherence, can lead to the initial NPT states to be PPT more quickly, this phenomenon of DSD may occur.

We would like to thank Jiang Huang for interesting discussions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11374096 and 11074072) and Hunan Provincial Innovation Foundation for Postgraduate (CX2014B194).

In this appendix, we give the explicit forms of the time-evolved state of two qutirts. First we calculate the eigenvalues and eigenvectors of H given in eq. (4): $E_{1}=0$:

$$\begin{eqnarray} \psi_{1}&=\pm\frac{D}{\sqrt{2D^2+4b^2}}|02\rangle\pm\frac{i2b}{\sqrt{2D^2+4b^2}}|11 \rangle\nonumber\\ &\quad \pm\,\frac{D}{\sqrt{2D^2+4b^2}}|20\rangle, \end{eqnarray} \tag{ A.1 }$$

 $E_{2}=-2B_{0}$:

$$\begin{eqnarray} &&\psi_{2}=|22\rangle, \end{eqnarray} \tag{ A.2 }$$

 $E_{3}=2B_{0}$:

$$\begin{eqnarray} &&\psi_{3}=|00\rangle, \end{eqnarray} \tag{ A.3 }$$

 $E_{4}=-\sqrt{4b^2+2D^2}$:

$$\begin{eqnarray} \psi_{4}&=\mp\frac{\mu_{-}}{4b^2+2D^2}|02\rangle\pm\frac{iD(2b-\sqrt{4b^2+2D^2})} {2\mu_{-}}|11\rangle \nonumber\\ &\quad \pm\,\frac{D^2}{2\mu_{-}}|20\rangle, \end{eqnarray} \tag{ A.4 }$$

 $E_{5}=\sqrt{4b^2+2D^2}$:

$$\begin{eqnarray} \psi_{5}&=\mp\frac{\mu_{+}}{4b^2+2D^2}|02\rangle\pm\frac{iD(2b+\sqrt{4b^2+2D^2})} {2\mu_{+}}|11\rangle \nonumber\\ &\quad \pm\,\frac{D^2}{2\mu_{+}}|20\rangle, \end{eqnarray} \tag{ A.5 }$$

 $E_{6}=-B_{0}-\sqrt{b^2+D^2}$:

$$\begin{eqnarray} &&\psi_{6}=\pm\frac{i(-b+\sqrt{b^2+D^2})}{\nu_{-}}|12\rangle\mp\frac{D}{\nu_{-}} |21\rangle, \end{eqnarray} \tag{ A.6 }$$

 $E_{7}=B_{0}-\sqrt{b^2+D^2}$:

$$\begin{eqnarray} &&\psi_{7}=\pm\frac{i(-b+\sqrt{b^2+D^2})}{\nu_{-}}|01\rangle\mp\frac{D}{\nu_{-}}|10 \rangle, \end{eqnarray} \tag{ A.7 }$$

 $E_{8}=-B_{0}+\sqrt{b^2+D^2}$:

$$\begin{eqnarray} &&\psi_{8}=\pm\frac{i(b+\sqrt{b^2+D^2})}{\nu_{+}}|12\rangle\pm\frac{D} {\nu_{+}}|21\rangle, \end{eqnarray} \tag{ A.8 }$$

 $E_{9}=B_{0}+\sqrt{b^2+D^2}$:

$$\begin{eqnarray} &&\psi_{9}=\pm\frac{i(b+\sqrt{b^2+D^2})}{\nu_{+}}|01\rangle\pm\frac{D}{\nu_{+}}|10 \rangle \end{eqnarray} \tag{ A.9 }$$

with $\mu_{\pm}=\sqrt{(2b^2+D^2)(4b^2+D^2\pm2b\sqrt{4b^2+2D^2})}$, $\nu_{\pm}=\sqrt{2D^2+2b^2\pm2b\sqrt{b^2+D^2}}$.

Now let us focus on the time evolution of the system initial prepared in the state $\rho_{\alpha}(0)$ given by eq. (9). According to Milburn's decoherence model, the formal solution of Milburn equation is given by eq. (7):

$$\begin{equation} \hspace{-7pc} \rho(t)=\left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \displaystyle\frac{\alpha}{21} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \displaystyle\frac{1}{21}(2-iD \eta_{-}) & 0 & \displaystyle\frac{1}{21}(\Delta \eta_{+}-b\eta_{-}) & 0 & 0 & 0 & 0 & \displaystyle\frac{1}{21}(\epsilon_{-}^{\ast} \chi_{-}^{\ast}+\epsilon_{+}^{\ast} \chi_{+}^{\ast})\\ 0 & 0 & \displaystyle\frac{5-\alpha}{21} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \displaystyle\frac{1}{21}(\Delta \eta_{+}+b\eta_{-}) & 0 & \displaystyle\frac{1}{21}(2+iD \eta_{-}) & 0 & 0 & 0 & 0 & \displaystyle\frac{D}{21}(\varepsilon_{-}^{\ast}\chi_{-}^{\ast}+ \varepsilon_{+}^{\ast} \chi_{+}^{\ast}) \\ 0 & 0 & 0 & 0 & \displaystyle\frac{5-\alpha}{21} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \displaystyle\frac{\alpha}{21} & 0 & 0 & 0\\ 0 & 0 & 0& 0 & 0 & 0 & \displaystyle\frac{5-\alpha}{21} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \displaystyle\frac{\alpha}{21} & 0\\ 0 & \displaystyle\frac{1}{21}(\epsilon_{-} \chi_{-}+\epsilon_{+} \chi_{+}) & 0 & \displaystyle\frac{D}{21}(\varepsilon_{-} \chi_{-}+\varepsilon_{+} \chi_{+}) & 0 & 0 & 0 & 0 & \displaystyle\frac{2}{21} \end{array} \right), \end{equation} \tag{ A.10 }$$

where $\eta_{\pm}=\frac{e^{-2i\Delta t-2\Delta^2\gamma t}(e^{4i\Delta t}\pm 1)}{\Delta}$, $\epsilon_{\pm}=(\Delta\pm b)(\Delta \pm b \mp iD)$, $\varepsilon_{\pm}=(ib+D\pm i \Delta)$, $\chi_{\pm}=\frac{e^{\frac{-1}{2}(\Delta \pm 3B_{0})}[\mp2i t+(\Delta \pm 3B_{0}) \gamma t]}{\Delta^2\pm b \Delta}$, with $\Delta=\sqrt{b^2+D^2}$ and $\epsilon_{\pm}^{\ast}, \varepsilon_{\pm}^{\ast}, \chi_{\pm}^{\ast}$ is conjugate of $\epsilon_{\pm}, \varepsilon_{\pm}, \chi_{\pm}$.