Exact dynamics of concurrence-based entanglement in a system of four spin-1/2 particles on a triangular ladder structure

Quantum state transfer is one of the crucial demands to perform quantum information processing. Depending on the accessible technology, different methods exist to accomplish the transfer of quantum states and consequently the propagation of quantum entanglement. As an illustration, in optical systems, this task is attributed to photons [1, 2]. Moreover, in the systems of trapped atoms, phonons play the role of information carriers [3].

The desire to achieve greater processing power in quantum computation makes quantum bits (qubits) get closer enough to each other for more feasible transferring quantum states. Then, such a necessity induces an additional size limitation on quantum communication devices. Short range quantum communication may be a potential device for realizing quantum state transfer from one location to another in short distances. Meanwhile, focusing on the time evolution of quantum spin chains has switched into high gear for such a goal, after Bose's idea [4, 5].

Bose suggested that the quantum state propagation can be fulfilled by the dynamic evolution of an array of permanently coupled spins. In his suggested scenario various aspects come into play: decoding a quantum state at the sth spin of a chain by Alice, traveling the state through the chain and retrieving some fidelity of the mentioned state from rth spin after a while by Bob. It was found that the amount of fidelity is equal to unity for a four-spin ring, denoting that such a ring can provide a perfect quantum communication [4]. In addition, magnetic spin chains meet many other requirements in several inspiring technologies for the processing of quantum information [6–9].

It is noticed in [10] that during the evolution of a type of spin chain over time, the unique quantum states of W type can be dynamically produced. Such states can be classified among the well-known multipartite quantum states that play a significant role in the quantum information tasks [11]. Tracing out one qubit of the system results in another W sate. In other phrase, such states are not sensitive to the loss of particles. The general form of quantum W states can be represented by

$$\begin{eqnarray}&&| W{\rangle }^{N}=\displaystyle \frac{1}{\sqrt{N}}\displaystyle \sum _{j}^{N}{{\rm{e}}}^{{\rm{i}}{\theta }_{j}}| {\bf{j}}\rangle ,\end{eqnarray} \tag{ 1 }$$

where $| {\bf{j}}\rangle =| 00...010....0\rangle$ presents a state with a flipped spin at the jth site to the $| 1\rangle$ state. While $| 0\rangle$ indicates the spin down state (i.e. spin oriented along −z orientation), $| 1\rangle$ denotes the reversed direction. Also, θ_j is an arbitrary phase factor and N is the number of system's particles.

Factually, such states are essential types of physical resources for lots of notable applications such as quantum key distribution [12, 13], quantum telecloning [14], quantum teleportation [15–17], just to name a few. Above all, the generation of W states is the primary condition of such captivating applications. From dynamical point of view, the quantum W states can be generated via evolution of an initial quantum state either in a spin chain [10] or in a spin star network [18]. It may be interesting that, opposing to spin chains through which W states can be dynamically created by four qubits at the most [10], star networks have shown that such states can be generated by the same number of qubits at least [18].

In what follows, we show how such states can be generated during the evolution of our considered system i.e. a four-qubit triangular nanomagnetic ladder. It is known that two spin chains being arranged parallel (or quasi-parallel) to each other in a two-dimensional plane, result in a system called a two-leg ladder and can be used to implement as a quantum channel. A deep insight into the physics of such quantum channels is of particular importance for the technologies associated with the entanglement and quantum state transfer on them. These technologies covers a range from the miniaturized structures such as quantum cellular automata [19] to the larger-sized components as quantum computer's data buses [20].

We assumed that in the system under consideration, the spins are coupled by the Heisenberg interaction of strength J along the ladder's rungs, together with the spin–orbit coupling over the legs of the ladder. This proposed model is of profound importance for the generation of quantum W states, because the two types of magnetic interactions coexist in such a spin arrangement beyond the spin chains or star structures that are customarily well known in the generation of W states.

There are several suggestions for engineering spin–orbit coupling for atoms in optical lattices, one of the most significant of which is based upon the superexchange mechanism of Dzyaloshinskii–Moriya (DM) effect [21, 22]. Such an interaction leads to emergence of exotic supperfluidity [23] and gives rise to a broad class of spin patterns in quantum phases [24–27] and entails a dominant form of anisotropy in spin liquids [28], just to name a few. The anisotropy parameter is important because it can broaden the line-with of electron spin resonance in kagome and honeycomb lattices as well as triangular ones with spin–orbit coupling that are the most promising candidates for spin liquids to emerge [29].

In this work, we demonstrate how tuning DM interaction in a four-qubit spin ladder influences on the dynamical behavior of entanglement shared between any pairs of the system. This objective is fulfilled by means of concurrence [30–32] to monitor the dynamical procedure. Recently, many researches focused on developing the role of concurrence in many body systems as well as in the processing of quantum information. For instance, in solid state physics, characterizing some phases of several materials is devoted to this quantity [33]. In addition to its interest in analytical approaches, several numerically effective methods for calculating this quantity have been developed as well [34]. As it is known, a situation in which the concurrences being shared on any pairs of a system are all equal is related to the quantum W states [35, 36].

For dealing with the above situations in details, our paper is organized as follows: in section 2 we introduce the considered model and describe temporal behavior of the quantum state of the system. Section 3 begins with recalling concurrence as the considered measure tool for qubit-qubit entanglement. The analytical results presented in this section have provided the exhaustive descriptions of the dynamical behavior of entanglement between any pairs. Section 4 addresses the two-point correlation functions and their relations with entanglement. In the concluding section we give a summary of our results.

Let us consider a ladder in which spin 1/2 particles interact via XX Heisenberg Hamiltonian on the rungs. In addition, DM interaction links spins over the legs. The Hamiltonian that describes such a system with assuming a periodic boundary condition on the rungs, reads:

$$\begin{eqnarray}&&\hat{H}=J\displaystyle \sum _{\langle i,j\rangle }({\hat{S}}_{i}^{x}{\hat{S}}_{j}^{x}+{\hat{S}}_{i}^{y}{\hat{S}}_{j}^{y})+\vec{D}\,{\boldsymbol{\cdot }}\,\displaystyle \sum _{ \langle\langle i^{\prime} ,j^{\prime} \rangle\rangle }{\hat{\vec{S}}}_{i^{\prime} }\times {\hat{\vec{S}}}_{j^{\prime} },\end{eqnarray} \tag{ 2 }$$

where $\langle i,j\rangle$ $\left( \langle\langle i^{\prime} ,j^{\prime} \rangle\rangle \right)$ denotes sites on the rungs (legs). In this equation, the first term in the right-hand side stands for XX Heisenberg interaction with the parameter J showing its strength on the rungs. The second term takes into account the DM interaction (as depicted in figure 1) over the legs. Also, $\vec{D}=D\hat{z}$ plays the role of control parameter in the system dynamics.

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In order to study the dynamics of the system, we make use of the time evolution operator approach for which the initial quantum state of the system is required. As long as the system undergoes a unitary dynamics generated by the system's Hamiltonian due to the time evolution operator $\hat{U}(t)=\exp (-{\rm{i}}\hat{H}t/{\hslash })$ we wish to examine its application on the initial state which can be written as

$$\begin{eqnarray}\begin{array}{ccl}\hat{U}(t)| \psi (0) \rangle & = & \exp \left\{\displaystyle \frac{-{\rm{i}}Jt}{\hslash }\left[\displaystyle \sum _{ \langle i,j \rangle }\left({\hat{S}}_{i}^{x}{\hat{S}}_{j}^{x}+{\hat{S}}_{i}^{y}{\hat{S}}_{j}^{y}\right)\right.\right.\\ & & \left.\left.+\displaystyle \frac{\overrightarrow{D}}{J}\cdot \,\displaystyle \sum _{ \langle \langle {i}^{{\rm{{\prime} }}},{j}^{{\rm{{\prime} }}} \rangle \rangle }{\hat{\overrightarrow{S}}}_{{i}^{{\rm{{\prime} }}}}\times {\hat{\overrightarrow{S}}}_{{j}^{{\rm{{\prime} }}}}\right]\right\}| \psi (0) \rangle .\end{array}\end{eqnarray} \tag{ 3 }$$

In what follows, t and D respectively are normalized to ℏ/J and J. Without loss of generality, we take ℏ ≡ 1 and J = 1 here. Therefore, t and D become dimensionless parameters in the following computations.

Let us consider the case in which the entanglement is initially encoded in the entangled Bell state $\tfrac{1}{\sqrt{2}}(| 10{\rangle }_{12}\,+| 01{\rangle }_{12})$ over the first two qubits (1, 2), while the last two qubits (3, 4) are in the disentangled state $| 00{\rangle }_{34}$. Hence, the initial state of the whole system can be shown by a tensor product state, $| \psi (0)\rangle =\tfrac{1}{\sqrt{2}}(| 10{\rangle }_{12}+| 01{\rangle }_{12})| 00{\rangle }_{34}$.

In order to proceed equation (3), $| \psi (0)\rangle$ is spanned over energy eigenstates. Factually, we can write $| \psi (0)\rangle$ in term of energy eigenvectors ($| {E}_{i}\rangle$) as

$$\begin{eqnarray}&&| \psi (0)\rangle =\displaystyle \sum _{i=1}^{16}{c}_{i}| {E}_{i}\rangle .\end{eqnarray} \tag{ 4 }$$

To reach $| \psi (t)\rangle$ we apply the time evolution operator as

$$\begin{eqnarray}&&| \psi (t)\rangle ={{\rm{e}}}^{\tfrac{-{\rm{i}}{H}{t}}{{\hslash }}}| \psi (0)\rangle =\displaystyle \sum _{i=1}^{16}{c}_{i}{{\rm{e}}}^{\tfrac{-{{\rm{i}}{E}}_{i}t}{{\hslash }}}| {E}_{i}\rangle .\end{eqnarray} \tag{ 5 }$$

Moreover, we know that it is possible to extend energy states onto standard bases as $| {E}_{i}\rangle ={\sum }_{j=1}^{16}{a}_{j}| {\phi }_{j}\rangle$ which can be inserted in equation (5). Here $| {\phi }_{j}\rangle$ are included in the set of standard bases $\{| 1111\rangle ,| 1110\rangle ,\ldots | 0000\rangle \}$. Then we have

$$\begin{eqnarray}&&| \psi (t)\rangle =\displaystyle \sum _{i,j=1}^{16}{c}_{i}{a}_{j}{{\rm{e}}}^{\tfrac{-{{\rm{i}}{E}}_{i}t}{{\hslash }}}| {\phi }_{j}\rangle .\end{eqnarray} \tag{ 6 }$$

After straightforward calculation, it is found that all coefficients emerge in equation (6) are zero except those relating to one-particle state [10]. In such states, one particle is up and the rest are down in the z direction of spin space. Consequently, the time evolution of the quantum state reads:

$$\begin{eqnarray}\begin{array}{rcl}| \psi (t)\rangle & = & \displaystyle \frac{\eta (t,D)}{2\sqrt{2}}(| 1000\rangle +| 0100\rangle )\\ & & +\displaystyle \frac{\xi (t,D)}{2\sqrt{2}}(| 0010\rangle +| 0001\rangle ),\end{array}\end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}\eta (t,D) & = & \cos \displaystyle \frac{\mu t}{2}+\cos \displaystyle \frac{\nu t}{2}\\ & & -\displaystyle \frac{{\rm{i}}}{\omega }\left[\displaystyle \frac{\omega +{\omega }^{2}}{\mu }\sin \displaystyle \frac{\mu t}{2}+\displaystyle \frac{\omega -{\omega }^{2}}{\nu }\sin \displaystyle \frac{\nu t}{2}\right],\\ \xi (t,D) & = & \displaystyle \frac{-({\rm{i}}+D)}{\omega {D}^{2}}\left\{{{\rm{i}}{D}}^{2}\left[\cos \displaystyle \frac{\mu t}{2}-\cos \displaystyle \frac{\nu t}{2}\right]\right.\\ & & \left.+(\omega +1)\nu \sin \displaystyle \frac{\mu t}{2}+(\omega -1)\mu \sin \displaystyle \frac{\nu t}{2}\right\},\end{array}\end{eqnarray} \tag{ 8 }$$

in which $\omega =\sqrt{1\,+\,{D}^{2}}$, and μ and ν are positive parameters depending on D as:

$$\begin{eqnarray}\begin{array}{rcl}\mu & = & \sqrt{2+{D}^{2}+2\sqrt{1+{D}^{2}}},\\ \nu & = & \sqrt{2+{D}^{2}-2\sqrt{1+{D}^{2}}}.\end{array}\end{eqnarray} \tag{ 9 }$$

As we can see in equation (7), the quantum state for an arbitrary time instant is a linear combination of the states in which the qubits pair (1, 2) is in an entangled Bell-state while the pair (3, 4) is in the ground state, and the other way around. In what follows, we make use of equation (7) to calculate the value of entanglement between different parts of the considered system at different time instants.

As suggested by Wootters et al [30–32], the concurrence is an appropriate quantifier for the pairwise entanglement between qubits p and q in a quantum system. The starting point is tracing of the density matrix ρ over two qubits p and q, which results in the reduced density matrix $\rho (p,q)={\mathrm{Tr}}_{p,q}(\rho )$. Then, the concurrence between pth qubit and qth one is given by ${C}_{p,q}=\max \{2{\lambda }_{1}-{\sum }_{i=1}{\lambda }_{i},0\}$ where λ_is are square roots of eigenvalues of the matrix $R\,=\rho (p,q)\left({\sigma }_{y}\otimes {\sigma }_{y}\right){\rho }^{* }(p,q)\left({\sigma }_{y}\otimes {\sigma }_{y}\right)$. Moreover, σ_y denotes the Pauli-y-matrix and ρ*(p, q) is complex conjugate of ρ(p, q). In definition, $0\lt {C}_{p,q}\lt 1$ indicates that pth qubit is partially entangled with qth one. While, ${C}_{p,q}=1$ corresponds to a maximally entangled state, ${C}_{p,q}=0$ stands for a completely disentangled state between two considered qubits. While, for a thorough discussion and comparison between concurrence and other quantum correlations we refer for instance to [37] and bibliography quoted therein.

The concurrence, for a quantum state that includes one-particle states as equation (7), can be simplified by ${C}_{p,q}(t)=2| {b}_{p}(t){b}_{q}(t)|$, where b_i(t)s are the coefficients [10, 38]. For the considered system, our calculation shows that the concurrences are given by ${C}_{\mathrm{1,2}}={\cos }^{2}\left(\tfrac{\mu +\nu }{4}t\right)$ and ${C}_{\mathrm{3,4}}={\sin }^{2}\left(\tfrac{\mu +\nu }{4}t\right)$ for the first and the last rung respectively. Moreover, for every leg we have ${C}_{\mathrm{1,3}}=\tfrac{1}{2}| \sin \left(\tfrac{\mu +\nu }{2}t\right)|$. It might be worthy to mention that C_2,3 and C_1,4 behave as C_1,3. Figures 2(a) and (b) represent the dynamical behavior of the concurrence, respectively, between the first and last leg in terms of the DM interaction parameter D. Sharafullin et al demonstrated in details how DM Hamiltonian relates to the angle between spins and the value of nonmagnetic ions (for example oxygen) displacement in materials [39]. However, other possible schemes might come into play for determining DM interaction. For instance, temperature and voltage bias applied to the system and spin polarization can affect the DM interaction [40]. In [41], it is shown that the interfacial DM interaction is not a given material parameter. Instead, it can be controlled and managed by a spin current injected externally to the system. In any way, without concerning about the experimental method to control DM interaction, we consider them theoretically in a range of values.

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As it can be seen from the plots (a) and (b) in figure 2, at specific time instants, which decrease in a monotonic manner with increasing D, C_1,2 vanishes and C_3,4 equates with unity. At the same time instants C_1,3 vanishes (see figure 2(c)) as expected from concurrence relations. Hence, one can certainly infer that the entanglement completely transfers from the first initially entangled rung to the last initially disentangled one. The time instants of transition are

$$\begin{eqnarray}&&{t}_{{tr}}=\displaystyle \frac{(2n+1)2\pi }{(\mu +\nu )},\ \ (n=0,1,2,...).\end{eqnarray} \tag{ 10 }$$

It turns out, when C_3,4 is equal to unity, the concurrence between the other components are zero, as can be seen in figure 3(a), which is a cross-sectional cut of figure 2 for a specific value of D. In other words, as the entanglement turns off at one end of the spin ladder, the other end is found in the entangled Bell state. In addition, it is observed that at $t=\tfrac{(2n+1)\pi }{(\mu +\nu )}$ (n = 0, 1, 2, ...), all concurrences are equal. These time instants, which henceforth are denoted by t_w, indicate the emerge of quantum W states (see figure 4). Since the concurrences of all qubit pairs in the states of W type are equal to 2/N, then for N = 4 (as the number of particles), one can expect C_1,2 = C_3,4 = C_1,3 = 1/2 at t = t_w, as shown in figure 3(a).

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The dependence of entanglement on two-point quantum correlations [42] makes us explore a signature of such correlations to detect entanglement transition and W states generation. Two-point quantum correlations (${\chi }_{p,q}^{\alpha \alpha }$) between αth component of spin pairs (${\hat{\vec{S}}}_{p},{\hat{\vec{S}}}_{q}$) can be defined as

$$\begin{eqnarray}&&{\chi }_{p,q}^{\alpha \alpha }(t)=\langle \psi (t)| {\hat{S}}_{p}^{\alpha }{\hat{S}}_{q}^{\alpha }| \psi (t)\rangle ,\end{eqnarray} \tag{ 11 }$$

that is a substantial quantity. With the quantum state being given by equation (7) the correlation functions have been calculated and written in table 1.

Table 1.  The two-point correlations in different directions.

Pairs positions	χxx(t) = χyy(t)	χzz(t)
First Rung (1, 2)	$\tfrac{1}{4}{\cos }^{2}(\tfrac{\mu +\nu }{4}t)$	$\tfrac{-1}{4}\cos (\tfrac{\mu +\nu }{2}t)$
Legs	$\tfrac{1}{8}\sin (\tfrac{\mu +\nu }{2}t)$	0
Last Rung (3, 4)	$\tfrac{1}{4}{\sin }^{2}(\tfrac{\mu +\nu }{4}t)$	$\tfrac{1}{4}\cos (\tfrac{\mu +\nu }{2}t)$

Since, we seek to find the role of correlations in entanglement transition and W state generation, the first and last pairs whose concurrence can take the value of unity are considered in the following. For this reason, xx and zz correlations between (1, 2) and (3, 4) are plotted in figure 3(b). The comparison between figures 3(a) and (b), shows that when ${C}_{p,q}=1$, zz correlation is equal to −1/4. On the other hand, if ${C}_{p,q}=0$, then such a correlation equates 1/4. This fact reflects the nature of the particular initial state that has been taken. Therefore, the correlation in z direction can easily show the transition of entanglement. In addition, such a correlation between all pairs quenches when a W state emerges as it depicted in figure 3(b).

Focusing on xx (yy) correlation also allows us to reveal their role in dynamical behavior of entanglement. At t = t_w, all pairs agree on the same value of xx (yy) correlation that is 1/8, as shown in figure 3(b) and presented in table 1. In other phrase, because of quenching zz correlation at t = t_w, only the correlations in x and y directions play the main role in W state production. In addition, it can be said that the zero value for such correlations coincides with the complete disentanglement between the considered pair. On the other hand, the maximum value of correlations in x and y direction is according to complete entanglement shared between that pairs.

The other types of two-point correlations such as ${\chi }_{p,q}^{\alpha \beta }(t)$ vanish for all pairs, as long as $\alpha \ne \beta$. Given that, the DM interaction causes xx and yy correlations as it is exhibited in table 1, however induces a term such as ${\hat{S}}^{x}{\hat{S}}^{y}$ in the system's Hamiltonian.

In addition, as can be seen in table 1, the amplitudes of xx and yy correlations between leg pairs are half of that of pairs on the rungs. The lower values for this type of amplitude, together with the lack of zz correlation between such pairs, prevent the maximum of the entanglement exceeding from 1/2, which takes place at t = t_w.

Another important quantity is ${\sum }_{p\,=\,1}^{4}{\hat{S}}_{p}^{z}={\hat{S}}_{{\rm{tot}}}^{z}$, that does not commute with the system's Hamiltonian, nevertheless, its expectation value is temporally constant ($\langle \psi (t)| {\hat{S}}_{{\rm{tot}}}^{z}| \psi (t)\rangle \,=-1$). This is a generic character of W states and, based on the uncertainty principle, one can reach $\langle \psi (t)| {\hat{S}}_{{\rm{tot}}}^{x}| \psi (t)\rangle \,=\langle \psi (t)| {\hat{S}}_{{\rm{tot}}}^{y}| \psi (t)\rangle =0$.

This work deals with the entanglement dynamics through a nanomagnetic triangular ladder with the Heisenberg interaction on the rungs in addition to the DM interaction over the legs, which has been treated by means of an analytical technique. In the system's initial state, the first rung is completely entangled via a Bell state and the other elements are in a separated state. The unitary time evolution of the system's quantum state leads to the transit of entanglement from the first leg to the last. The dependence of such transition time instants on the value of DM interaction is exactly obtained. It is found that the increasing DM interaction boosts the speed of entanglement transmission. In examining the dynamical behavior between different pairs, we arrived at the conclusion that the entanglement frequency for the legs is greater than that of the rungs. During the temporal evolution, the system passes several times through specific quantum states in which one-particle states take the equal probability. Such states whose generation is the most spectacular feature of the proposal model are known as W states.

Since the dynamical behavior of the entanglement over various pairs reflects the effect of their two-point quantum correlations, we investigated such functions in detail. Analytical results reveal that zz correlations play no role in the W state generation, while xx and yy correlations play the main character. In addition, the maximum value of such correlations (xx and yy ones) and the minimum value of the correlation in z direction have the main role as an entanglement transition detector.

On the other side, the interaction between a quantum system and its environment is unavoidable. For example, a quantum ladder (as the present model) might be affected by the host crystal's phonons carrying the thermal energy of the crystal at $T\ne 0$. It is expected that the system-environment interaction will give rise to decoherence effects that can destroy the quantum correlations and can result in other effects such as entanglement sudden death [43] and sudden birth as reported in [44].

MM is really grateful to Seyed Mohesen Rasa for drawing some figures.