1. IntroductionQuantum entanglement as an important physical resource has received much attention.[1] For an entangled system, there is the nonlocal correlation between its subsystems. It has been proved that the nonlocal correlation can be used to accomplish various quantum information processing tasks such as quantum teleportation,[2,3] quantum cryptographic key distribution,[4,5] quantum secret sharing,[6,7] and superdense coding.[8] Quantum teleportation was first proposed by Bennett et al. in 1993,[2] and they presented the standard teleportation protocol, in which a sender teleported an unknown quantum state to a distant receiver by sharing a pair of maximal entangled states with the receiver in advance. Since then, quantum teleportation has been extensively investigated both experimentally and theoretically.[3,9–15]
Earlier studies focused mainly on the teleportation of quantum states.[2,16–18] For contributing to entanglement manipulation, the entanglement teleportation gradually attracts researchers’ attention. Lee and Kim investigated the entanglement teleportation of a two-body unknown entangled state via two identical kinds of noisy but independent quantum channels, and they found that the quantum channel must have the critical value of minimum entanglement in order to guarantee the entanglement teleportation of the initial state.[19] Due to the decoherence effect, the resource of maximally entangled state is hard to hold in real circumstance during teleportation. Thus, researchers pay more attention to the mixed entangled state as a resource. Popescu has shown that the mixed state as a quantum channel also has better fidelity than the classical one.[9] Furthermore, the standard teleportation protocol of an arbitrary mixed state as quantum channel has been proposed by Bowen and Bose in 2001.[10] Considering that the thermal fluctuation can destroy the quantum state, Arnesen et al. firstly investigated the thermal entanglement in a one-dimensional Heisenberg model.[20] Since then, a number of studies on the thermal entanglement have been conducted in various condensed matter systems.[21–24] Moreover, teleportation via the thermally entangled state has also attracted much attention.[25–33]
The Heisenberg spin chain as one of the simplest solid state systems has been studied extensively in the quantum teleportation, including the two-qubit XX,[25] XY,[26] XXX,[27] XXZ Heisenberg models,[30] the three-qubit Heisenberg model with the Dzyaloshinsky–Moriya interaction,[32] and so on. In view of practical applications, it is necessary to study the model describing real materials. For example, Cu3(CO3)2(OH)2, known as natural azurite, can be described by the generalized diamond chain.[34] Rojas et al. used the standard teleporatation protocol to study an arbitrary entangled state teleportation through a couple of Heisenberg dimers in an infinite Ising–Heisenberg diamond chain.[33] Due to a high energy barrier for the spin relaxation, the single-chain magnets (SCMs) have been deemed to be a class of advanced materials which can be applied to quantum computing.[35–39] The Ising–Heisenberg diamond chain is one of representative of SCMs, and the magnetic frustration, the field induced phase transitions and the entanglement have been predicted in this heterometallic chains.[40–43] Moreover, it is suggested that heterotrimetallic complexes will have more intriguing magnetic properties than those in heterobimetallic complexes. Wang et al. used a stepwise method to synthesize the first heterotrimetallic coordination compounds CuIIMnII(L1)][FeIII(bpb)(CN)2]·ClO4 ·H2O, abbreviated as Fe–Mn–Cu.[44] Two different magnetic metal ions Fe3+ and Mn2+ regularly alternate along the main chain axis of the polymeric coordination compounds, and Cu2+ ion is laterally attached to each Mn2+ ion. Recently, Souza et al. introduced a simple exactly solvable model to characterize the most essential features of the heterotrimetallic coordination compounds Fe–Mn–Cu.[45] The exchange coupling between Fe3+ and Mn2+ ions is assumed to be Ising-like, while the exchange coupling between Cu2+ and Mn2+ ions is assumed to be an anisotropic XXZ Heisenberg interaction. They have investigated the magnetization processes and the quantum entanglement in the model.[45]
Inspired by the work of Souza et al.,[45] we study the entangled state teleportation through a couple of quantum channels composed of the Heisenberg dimers in the Ising–Heisenberg chain of the heterotrimetallic coordination compounds Fe–Mn–Cu. The magnetic structure of the considered compound and the bipartite quantum coupling which serves as a quantum channel are shown in Fig. 1(a), and the schematic drawing of teleportation is shown in Fig. 1(b).
The rest of the paper is organized as follows. In Section 2, we introduce the Hamiltonian of the infinite Ising–Heisenberg chain, and use the transfer-matrix approach to obtain the density operator of the quantum channel. Using the standard teleportation protocol, we obtain the density matrix expressions of the output state. We study the concurrence of the output state and the fidelity of the entanglement teleportation in Section 3. The conclusions will be presented in Section 4.
2. Model and methodsWe consider the Heisenberg dimer of the Ising–Heisenberg spin chain of the heterotrimetallic coordination compounds Fe–Mn–Cu as the quantum channel (see the dashed box in Fig. 1(a)). The Hamiltonian of the Ising–Heisenberg spin chain can be expressed as follows:[45]
where
Here
Si,1 represents the Heisenberg spin reflecting the Mn
2+ magnetic ion after renormalization of the respective magnetic moment,
Si,2 corresponds to the Heisenberg spin related to the Cu
2+ magnetic ion,
denote the raising and lowering spin operators of spin-1/2, and
stands for the Ising spin approximating the highly anisotropic Fe
3+ magnetic ion.
J1 and
J2 denote the Ising interaction between Fe
3+–Mn
2+ magnetic ions and the Heisenberg interaction between Cu
2+-Mn
2+ magnetic ions, respectively.
Δ is the anisotropy parameter. Zeeman’s terms (
h1 =
g1μ0h,
h2 =
g2μ0h,
h3 =
g3μ0h) are involved in the Hamiltonian with
h being the external magnetic field, and we will consider the specific values (
g1 = 2.4,
g2 = 10.0,
g3 = 2.2) of the gyromagnetic factor according to the individual magnetic ions. Here
μ0 is the Bohr magneton which will be set to unity (
μ0 = 1) in the calculation. According to the experimental result, Souza
et al. studied the quantum entanglement in this model for the antiferromagnetic coupling
J2 < 0.
[45] In addition, the periodic boundary condition (
μN + 1 =
μ1) is considered here for convenience.
Based on the expression of
, it is easy to find
. Thus we can adopt the transfer-matrix approach to gain the entanglement measurement of this infinite chain. The calculation process of the transfer-matrix approach has been presented detailedly in the literature.[40,42] Firstly, we obtain the eigenvalues of
by diagonalization of the two-qubit Heisenberg spin dimer and by assuming fixed values for μi and μi + 1 as follows:
where
a =
h2 −
h3 +
J1(
μi +
μi + 1). The corresponding eigenstates based on the standard basis {|00⟩, |01⟩, |10⟩, |11⟩} are given by
Then, we obtain the two-qubit operator ρ by fixing μi and μi + 1 as follows:
The two-qubit operator in the natural basis becomes
The expressions of the elements of the two-qubit operator (
6) are too complex, thus we do not present them here.
The element of the reduced density operator for the quantum channel can be expressed as
where
ZN is the partition function,
w(
μi,
μi + 1) is the element of the transfer matrix, and
κi,j(
μi,
μi + 1) is the two-qubit operator about
μi and
μi + 1. Considering all
i and
j, one can obtain the reduced density matrix of the quantum channel as follows:
Now we apply the standard teleportation protocol to study the quantum teleportation in the Ising–Heisenberg chain considered above. Supposing that these two quantum channels are far from each other, we can ignore any possible coupling between them. According to the work by Bowen and Bose,[10] the standard teleportation through the mixed states can be regarded as a general depolarizing channel. When a two-qubit state ρin is teleported via the thermal states of two independent Ising–Heisenberg chains, the output state ρout is given as[10,25]
where
and |
Ψ±⟩, |
Φ±⟩ are the Bell states. Here
σi (
i = 0,
x,
y,
z) are the unit matrix and three components of the Pauli matrices.
We consider an initial state of two qubits expressed as
where
θ ∈ [0,
π],
ϕ∈[0,2
π],
θ and
ϕ are the polar and phase parameters. Then the initial state will be obtained by
ρin = |
ψin⟩⟨
ψin|. The input concurrence, as the measurement of entanglement, is
According to Eq. (9), we can obtain the output state ρout, which is expressed as
where
Next, we use the concurrence to measure the entanglement of the output state. The definition of the concurrence has been proposed by Wootters et al.,[46,47] which is expressed in terms of a matrix R as follows:
Here
is the complex conjugate of the matrix
ρout, and
σy denotes the
y component of the Pauli operator. Then the concurrence of the output state can be gained in terms of the eigenvalues of the matrix
R,
where
ei are the eigenvalues of the matrix
R (assuming
e1 ≥
e2 ≥
e3 ≥
e4).
When the input state is a pure state, the teleportation performance is usually measured by the fidelity defined as[48]
Since the teleported state is an unknown pure state, the efficiency of the quantum communication is characterized by the average fidelity
[49]
In the following section, we specifically analyze the entangled state teleportation via a couple of quantum channels composed of a spin-1/2 Ising–Heisenberg chain.
3. Results and discussions3.1. Entangled state teleportationFirstly, we study how the output concurrence Cout is affected by the input concurrence Cin and the temperature T. In Fig. 2(a), Cout is illustrated as a function of T with various Cin in the absence of the external magnetic field (h = 0). It is shown that Cout is nonmonotonic when the temperature increases. As the temperature increases from zero, the entanglement remains unchanged firstly, then increases to the maximum, and then gradually reduces to zero when the critical temperature is reached. Taking Cin = 1 (the black line in Fig. 2(a)), for instance, Cout = 0.8 when T = 0.001, with increasing the temperature, the output entanglement Cout gets the maximum 0.8603 (T = 0.171), and then it goes down to 0 (T = 0.851). In addition, it is easy to obtain from Fig. 2(a) that Cin affects the range of Cout. The larger the input concurrence Cin is, the larger the maximum value of Cout is, and the higher the critical temperature T is. It is noted that Cout is never as large as Cin in Fig. 2(a). This means that the entanglement of the initial state is lossy when passing through the quantum channel. In Fig. 2(b), we consider the case with the external magnetic field h = 0.1. The variation trend of the curves in Fig. 2(b) is similar to that in Fig. 2(a). However, it is worth noting that the external magnetic field h will suppress the generation of the output concurrence.
In Fig. 3, we investigate the effect of the anisotropy Δ of the quantum channel on the output concurrence Cout. Cout as a function of Cin is plotted in Fig. 3(a) for T = 0.1, J1 = J2 = −1, and the different anisotropy parameters Δ. It is shown that, in the same case of anisotropy, the relationship between Cout and Cin is linear, the larger the input concurrence is, the larger the output concurrence is. Moreover, we can obtain the fitting equation of each line in Fig. 3(a) as follows:
From Eq. (
18), it is shown that the slope of the line becomes larger and larger, finally approaches 1 with the increase of the anisotropy. The density plot of the output concurrence depending on
Δ and
T for
Cin = 1 is shown in Fig.
3(b). It also reveals that, in a certain range of anisotropy, with the increase of temperature, the tendency of the entanglement is to be promoted first and then suppressed. However, when the anisotropy exceeds this range, the temperature only inhibits entanglement. From Fig.
3, we can reach the conclusion that the stronger the anisotropy of the Heisenberg dimer as a quantum channel is, the higher the entanglement of the output state is.
In Fig. 4, we plot the output concurrence Cout as a function of the temperature T and the Heisenberg coupling parameter J2 for Cin = 1, J1 = −1, and h = 0. First, we study the case of isotropy Δ = 1 in Fig. 4(a). It is shown that the output concurrence Cout occurs only in the region of the antiferromagnetic coupling (J2 < 0). There is no entanglement in the ferromagnetic region (J2 > 0). For the ferromagnetic case (J2 < 0), when |J2| is a small value, it is found that the output concurrence Cout decreases with the temperature T increasing. Taking J2 = −0.5 as an example, Cout = 0.20 when T = 0.01, Cout vanishes when T is beyond 0.16. While |J2| is taken as a larger value, Cout varies nonmonotonically with the temperature increasing. Taking J2 = −2 as an example, Cout = 0.80 when T = 0.01. With increasing the temperature, the output entanglement Cout reaches the maximum 0.87 (T = 0.23), and then it goes down to 0 (T = 1.05). Moreover, with the increasing |J2|, the critical temperature corresponding to the vanishing of Cout increases. For example, when J2 = −1, −2, −3, and −4, the critical temperature Tc = 0.49, 1.05, 1.60, and 2.13, respectively. Figure 4(b) shows the case of the anisotropy Δ = 2. Compared with Fig. 4(a), the output concurrence occurs in both the cases of coupling region (J2 < 0 and J2 > 0) in Fig. 4(b). In the case of the antiferromagnetic coupling region, for Δ = 2 the critical temperature obviously increases (e.g., when J2 = −1, −2, −3, and −4, the critical temperature Tc = 0.86, 1.74, 2.64, and 3.52, respectively.) This indicates that the anisotropy promotes the generation of the output concurrence again. From Fig. 4, it is observed that the output concurrence Cout is dependent on the sign of J2. The physical origin of the phenomenon can be explained as follows. As the mentioned above, the coupling parameter J2 denotes the Heisenberg interaction between Cu2+–Mn2+ magnetic ions in the heterotrimetallic coordination compounds Fe–Mn–Cu. The coupling parameter J2 > 0 corresponds to the ferromagnetic case, and J2 < 0 to the antiferromagnetic case. Δ is the anisotropy parameter. When Δ = 1, the anisotropic model becomes the isotropic model. Through the straight calculating, it is found that in the isotropic case (i.e., Δ = 1) for the antiferromagnetic case (J2 < 0), the ground state of the Heisenberg interaction between Cu2+-Mn2+ magnetic ions is a maximally entangled state, while for the ferromagnetic case (J2 > 0), the ground state corresponds to the nonentangled state. On the other hand, when Δ = 2, for both the ferromagnetic case (J2 > 0) and the antiferromagnetic case (J2 < 0), the ground states are entangled, whereas the eigenvalues of the ground states in the two cases are different.
Now we study the effect of the external magnetic field on the output concurrence in Fig. 5(a). We plot the dependence of Δ on Tc for h = 0, 0.1, and 0.5 with J1 = J2 = −1 and Cin = 1. The areas surrounded by the curve and the y-axis represent the occurrence of the entanglement. Obviously, as the magnitude of the external magnetic field increases, the area of the entanglement becomes smaller and smaller, and eventually disappears completely. This means that the external magnetic field will destroy the formation of the entanglement. Finally, we also study the density plot of Cout depending on J1 and J2 for T = 0.1, Δ = 2, h = 0 and Cin = 1 in Fig. 5(b). It is shown that the result of the output entanglement is symmetric with respect to J1 = 0. In addition, the range of Cout of J2 < 0 is larger than that of J2 > 0.
3.2. The average fidelity of entangled state teleportationAs we know, the average fidelity FA is to describe the quality of the process of teleportation. In order to transmit a quantum state better than the classical communication protocol, FA must be greater than 2/3, which is the best fidelity in the classical word.[49] Now we will illustrate the behavior of the average fidelity in the standard teleportation protocol. In Fig. 6, we depict the average fidelity FA as functions of temperature T and Δ for J1 = J2 = −1. The yellow region corresponds to the maximum average fidelity FA = 1, and the blue region represents the minimum FA = 0. The red curve surrounding the yellow region is the region where the quantum teleportation will become successful (FA > 2/3), whereas the outside means that the quantum teleportation fails. We consider the case without the external magnetic field (i.e., h = 0) in Fig. 6(a), and the case with h = 0.3 in Fig. 6(b). It clearly shows that the region of successful fidelity with the case of h = 0 is greater than that of h = 0.3.
In Fig. 7, we discuss the effect of the Heisenberg coupling parameter J2 on the average fidelity FA for J1 = −1, h = 0 and various T. First, we consider the case with Δ = 1 in Fig. 7(a). The dark, red, and blue curves correspond to the case with T = 0.1, 0.6, and 1, respectively. The dark dotted line denotes the best fidelity 2/3 in the classical system. It is clearly shown that the teleportation will be successful only when J2 < 0, and with increasing the temperature, it needs the larger antiferromagnetic coupling J2 to ensure FA > 2/3. Moreover, all the curves will eventually reach an uniform maximum as the antiferromagnetic coupling parameter increases. For example, when T = 0.1, J2 ≤ −0.48, the average fidelity FA will be larger than 2/3, and finally up to FA ≈ 1. The change trends of curves of T = 0.6 and 1 are similar to those with T = 0.1. We also study the case of the anisotropy (Δ = 2) in Fig. 7(b). Compared with Fig. 7(a), it is obvious that there exists the successful average fidelity when J2 > 0 in Fig. 7(b).
Finally, we investigate the effect of the external magnetic field h on the average fidelity, and plot FA as a function of h for different T in Fig. 8. We observe the case with Δ = 1 in Fig. 8(a). It is shown that when the magnitude of the external magnetic field increases, the average fidelity decreases sharply firstly, then gets the minimum. After that, it gradually increases and finally stabilizes at FA = 0.333. Taking T = 0.6 for instance, when h = 0, the average fidelity FA = 0.8074. When h = 0.75, the average fidelity gets the minimum 0.2268. Lastly when h = 2.52, the average fidelity gets the stable value 0.333. In addition, the average fidelity is always smaller than 2/3 with increasing the temperature (e.g., T = 1). The case of Δ = 2 is plotted in Fig. 8(b). It is shown that the increasing magnitude of the anisotropy will suppress the effect of the temperature on the average fidelity for a weak external magnetic field. However, when the external magnetic field is enough high, the average fidelity always keeps 0.333, that is, in this case the quantum teleportation fails.
