Thermal decoherence of long-distance entanglement in spin- chains
The thermal entanglement is generated by weakly interacting atoms with an isotropic spin- chain. The decoherence of the entanglement is mainly investigated. The effective Hamiltonian is analytically obtained by the approximation method of perturbation. The scaling behavior of the effective coupling is numerically illustrated by the exact diagonalization. It is found out that the decay of the entanglement is slow in the case of non interacting spins. The long-distance thermal entangled states can be used as the noisy channel for the achievement of the quantum teleportation.
PACS: 03.67.Mn, 03.65.Ud, 75.10.Jm, 75.10.Pq
The entanglement plays a key role in the main tasks of quantum informationNielsen00 (); Amico07 (). In practice, entangled qubits need be accessed individually for measurements. Consequently, they are well separated in space. Recently, the long-distance entanglement Venuti06 (); Hartmann06 () has been attractive in the field of quantum information processing. A selected pair of distant qubits can retain a sizable amount of entanglement at zero temperature if they are weakly coupled to some spin models. Because spin chains can serve as an efficient communication channel for quantum teleportation Bowen01 () and state transfer Bose03 (), these models are extensively studied. In many schemes Ferreira08 (); Zhu08 (), spin- Heisenberg chains acted as the medium for the generation of quantum entanglement when the chain is kept at the ground state. It is found out that the long-distance entanglement decreases and vanishes with the length of the gapless spin chains Venuti07 (). As an appealing spin model, the spin- chain exhibits the massive and gapped ground state, which can be realized through confining an spinor condensate Demler02 (); Yip03 () in optical lattices. The quantum communication in the spin- chain has been investigated by Sanpera07 (). Here we expect that the long-distance entanglement can also be generated by the spin- chains and show the scaling property which is different from spin- chains. In the realistic optical lattices, the thermal decoherence from the temperature is unavoidable Hofstetter06 (). Therefore, it is of fundamental importance to study the impacts of the thermal noise on the long-distance entanglement. Using the long-distant entangled state as the channel, we also suggest the standard scheme of quantum teleportation.
In this report, the thermal entanglement between a pair of distant qubits is present when they are weakly coupled to the general isotropic spin- chain with bilinear-biquadratic interactions at finite low temperatures. To study the decoherence, the effective Hamiltonian between two distant sites is analytically obtained by the Fröhlich transformation Frohlich1952 (); Nakajima1953 () in Sec. II. The scaling property of the effective coupling is also given by the exact diagonalization method. The effects of the temperature and the relative strength of biquadratic interactions are considered. In Sec. III, we draw on the master equation to investigate the decay of the long-distance entanglement. The protocol of the quantum teleportation is put forward. Finally, a short discussion concludes the paper.
Ii The effective Hamiltonian at finite low temperatures
In the optical lattices, a selected pair of two-level atoms and can weakly interact with two open ends of a spin- chain. At finite low temperatures, the whole quantum system exhibits the thermal equilibrium state. To study the time evolution of quantum states, the total Hamiltonian can be expressed by
describes the intrinsic Hamiltonian of two distant atoms,
is the Hamiltonian of general isotropic spin- chain with the even length and
denotes the weak interaction between two distant atoms and open ends of the chain. Here and refer to the spin operators of distant atoms and the th site of the chain respectively. The parameter describes the transition energy from the ground state to the excited one for each atom, and gives the strength of the bilinear(biquadratic) coupling. As is well known, the energy property of the spin model is determined by the angle Affleck87 (). In the context, the biquadratic coupling for need be so weak that the general ground state of is a total singlet with the energy and the first excited ones are the degenerate triplet states with . Here the energy gap is the famous Haldane gap. In general, the thermal equilibrium state where is the -th eigenvalue of and is the corresponding eigenstate. When the low temperatures satisfy , the components of the ground state and first excited ones become dominant in the thermal equilibrium state. For lower temperatures, this assumption of considering just these states in the thermal fluctuations is more reliable. The approximate expression of the thermal state can be given by where is the partition function. For the convenience, the Plank constant and the Boltzman constant are assumed to be one.
In general, the Fröhlich transformation Frohlich1952 (); Nakajima1953 () is widely used in condensed matter physics. Recently, this method has been applied to the regime of quantum information processing Li05 (). As a second-order perturbation Frohlich1952 (); Nakajima1953 (), the effective Hamiltonian of the whole system is where the anti-Hermitian operator satisfies the relation of and the elements of this matrix are given by and the diagonal ones are zero for Li05 (). Here is the energy state of with the corresponding energy and is the degree of degeneracy. In the case with at lower temperature, the spin- chain is at the state of and then the effective Hamiltonian between two distant atoms is obtained by
where denotes the trace over the complete energy space of . To simplify the calculation, we assume that the parameter where the spin operator . Due to the invariant symmetry, it is found out that for . Here the sum is always zero if and the values are real. As a consequence, the effective Hamiltonian can be simplified by the isotropic Heisenberg one
Here the constant of is irrelevant to the long distant entanglement. The effective Heisenberg coupling is closely dependent on the energy property of . By means of the exact diagonalization method, the scaling property of at finite low temperatures is demonstrated in Fig. 1. It is shown that the values increase almost exponentially and arrive at a steady one with the length of the chain. According to Venuti07 (), the effective coupling is mainly determined by the singlet-triplet gap of the whole system . From the numerical results of White93 (), the gap of for is almost the steady one. Therefore, the values of saturate rapidly with the increase of the length. This means that the effective coupling can be obtained at finite low temperatures when distant sites are taken infinitely far away. Notice that the parameter must be calculated by all of eigenvectors of and is approximately expressed in the singlet-triplet subspace. For a simplest example of , the Hamiltonian of can be expanded by
where the projectors and . For very small , the energy spectrum is given by the ground energy , the first excited one and the second . Thus the effective coupling is analytically written by
It is necessary to consider the effects of the temperatures and relative strength of biquadratic coupling on the effective coupling. From Fig. 2, it is seen that the values are increased by the slight increase of . For the even length of the chain, the parameters are negative. In accordance with Eq.(8), the values of can be enhanced slightly because the bigger angle leads to the smaller energy gap . For the low temperature, the effective coupling is mainly determined by the first item of which is decreased with increasing the temperature.
Iii Decoherence of entanglement in thermal noise
The state of two distant atoms can be gained by tracing out the variables of the chain from the thermal state of the whole system. However, if the temperature , we do not expect real excitations of the spin chain to be present Ferreira08 (). Only the subspace of the states described by will be populated and then we can calculate the correlations between two atoms using where . When two distant sites are simultaneously coupled to the chain, the thermal state can be generated. In accordance with Wang02 (); Bayat05 (); Nielsen (); Arnesen01 (); Wang01 (), the concurrence of can be written by . Therefore, thermal entanglement exists if the effective coupling satisfies . From the point of view of practice, the local operations concerning two distant entangled atoms are needed. It is reasonable to assume that two atoms are coupled with its local thermal reservoirs . According to Yu04 (), the two independent reservoirs can lead to the local decoherence of entanglement. Suppose that the initial state at is where denotes the vacuum state of the two local reservoirs. The evolution of quantum state between atoms and is given by the master equation
where the Lindbald operator
Here is the mean number of the thermal reservoir and signifies the rate of spontaneous emission for each atom.
If one of two weak couplings is turned off after the preparation of the long-distance entanglement, the effective Hamiltonian of two atoms is obtained by which means there is no mutual interaction between atoms. In this case, the evolution of can be described by a completely positive trace-preserving map Aolita08 (). For a general two-qubit mixed state , the evolved state in time can be written by where the Kraus operators , , and . Here is the ground(excited) state of atoms and means the probability of the atom exchanging a quantum with the reservoir. The density matrix of the quantum state at any time is expanded in the Hilbert space of
On the other hand, it is assumed that the two atoms directly interact with each other in the form of the Hamiltonian given by Eq.(6). In this case, the analytical solution of the master equation is tedious. The expression of the density matrix of quantum states is also similar to that of Eq.(11). The decoherence of the thermal entanglement in two cases can be illustrated by Fig. 3(a). It is seen that the entanglement of two qubits without mutual interactions is decreased much more slowly than that of two directly interacting qubits. This point demonstrates that the decoherence time for long distant entanglement is so long as to be useful for the implementation of solid-state quantum computation.
The standard teleportation through the mixed states can be regarded as a general depolarising channel Bowen01 (). An arbitrary unknown quantum state is destroyed and its replica state appears at the remote place after applying the Bell measurement and the corresponding local operations. When single-qubit state is teleported via the noisy channel of like Eq.(12), the output state is written by
In the above equation, and the projectors where a Bell state . The average fidelity of this teleportation is given by
According to Nielsen00 (), the fidelity for a pure input state . The effect of the thermal noise on the average fidelity of the standard teleportation is illustrated by Fig. 3(b). It is shown that the average fidelity of quantum teleportation with thermal decoherence is larger than before a certain time. This means that the quantum teleportation via the channel of long-distance entangled state is better than the classic communication in the range of finite time. In the condition of the thermal noise, the quantum teleportation as the channel of the long-distant thermal entangled state is better than that of the thermal entangled state between two qubits interacting directly.
The long-distance thermal entanglement can be obtained when two atoms are weakly interacting with the isotropic spin- chain at finite low temperatures. For the massively gapped quantum systems, the scaling law for the effective coupling shows the exponential increase with the length of the spin chain. Under the influence of thermal noise, the entanglement of two distant qubits without mutual interactions is decreased much more slowly. It is demonstrated that the resource of long-distance entanglement can be used for quantum information processing. We suggest the efficient scheme of the standard teleportation via the channel of long-distance entanglement.
X.H. was supported by the Initial Project of Research in SUST and the National Natural Science Foundation of China No. 10774108.
The finite-size scaling behavior of the effective coupling is demonstrated at the low temperature . The square data were obtained with the exact diagonalization method when the couplings and . The solid line was the exponential fit curve for the data.
The effective coupling is plotted as functions of the temperature and the relative strength of the biquadratic coupling when the couplings .
(a). The decoherence of the entanglement is shown. (b). The average fidelity of the standard teleportation is illustrated. The dash line denotes the case that two qubits interact directly. The solid one refers to the case that one of the weak couplings between the distant atoms and the chain is turned off. The length of the chain is infinite and , and . The mean number of the thermal reservoir is chosen to be and the rate of spontaneous emission is .
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