Well-protected quantum state transfer in a dissipative spin chain

In this work, a mechanism is investigated for improving the quantum state transfer efficiency in a spin chain, which is in contact with a dissipative structured reservoir. The efficiency of the method is based on the addition of similar non-interacting auxiliary chains into the reservoir. In this way, we obtain the exact solution for the master equation of the spin chain in the presence of dissipation. It is found out that entering more auxiliary chains into the reservoir causes, in general, the better improvement of the fidelity of state transfer along the mentioned chain. Furthermore, it is reveal that the protocol has better efficiency for a chain with longer length. Therefore, by this method, quantum state transfer along a linear chain with an arbitrary number of qubits, can be well-protected against the dissipative noises.

The high-fidelity transmission of quantum states from one location to another in a quantum network through a quantum channel is an important task in quantum information processing. This is so because any performance of a quantum information processing task inside a quantum computer needs to exchange quantum information between distant nodes. Among the various physical systems, quantum spin chains are the best-known ones that can serve as quantum channels. After the pioneer work of Bose 1 , in which an unmodulated ferromagnetic spin chain with nearest neighbor Heisenberg interaction was proposed as a channel for short range quantum communication, various theoretical frameworks were proposed to increase the transmission fidelity in quantum state transfer (QST) [2][3][4] and even to achieve perfect state transfer (PST) in spin chains [5][6][7][8][9][10] . Since PST in spin chains with uniform nearest neighbor couplings is possible only for the chains with two and three spins, therefore, PST in longer chains is achievable by properly engineering and modulating of these couplings 5,6,9 . On the other hand, the necessity of engineering the coupling strengths in order to achieve PST in spin networks, which in turns leads to the increment of the complexity of the system, can be removed by taking phase modulated uniform couplings 11,12 . Also, exploiting partial collapsing measurements can improve the QST in spin chains with uniform nearest neighbor couplings 13,14 . In addition, PST has been recently investigated using discrete-time quantum walk approach in refs 15,16 .
On the other hand, since any real system is inevitably subjected to its surrounding environment, achieving QST with high fidelity in the presence of noise and dissipation effects is an outstanding challenge in quantum channels. So it would be important to consider possible methods to minimize or eliminate these unwanted effects on the QST efficiency, as considered recently in 4,[17][18][19][20][21][22][23][24][25][26][27][28][29][30] .
In this paper, we propose a theoretical approach to achieve high fidelity transmission of a quantum state in a linear spin chain which is in contact with a dissipative structured reservoir. It is assumed that the PST is achievable for the isolated spin chain due to the same pre-engineered nearest-neighbor couplings discussed in the refs 9,31 . The performance of the method is based on the enterance of other similar auxiliary spin chains, without direct interaction with each other, into the reservoir. In this direction, we provide the analytical solution for the dynamics of the chains immersed in the reservoir. It is found out that increasing the number of auxiliary chains leads to access to a high fidelity state transfer. Furthermore, it is figured out that for a chain with more qubits we have a better decoupling of the unitary dynamics of the chain from the dissipation, which means that the protocol has better efficiency for the chains with longer length.
In the following sections, we first review the PST in a spin chain according to the refs 9,31 . In the next step, the exact dynamics of the system in the presence of dissipative noises is obtained and consequently, the mechanism for protection of QST process against the dissipative noises, in the spin chain, is investigated. Finally, the paper is ended by a brief conclusion.
quantum system in which, the efficiency of state transfer process is degraded due to the existence of interaction between the chain and a dissipative structured reservoir. In other words, all of the qubits in the chain are contained in a common reservoir. We introduce the protection process by considering other N − 1 auxiliary similar chains with M spin, such that each of these chains is also involved in the above mentioned reservoir (see Fig. 1). It is assumed that there is no direct interaction between the chains. The Hamiltonian of the whole system reads as where ω 0 is the transition frequency, b k ( † b k ) is the annihilation (creation) operator for the kth field mode with frequency ω k . In the above equation, we have introduced the site-dependent coupling strength g k,j as the coupling constant between the kth field mode and the qubit located at site j of the each chain, defined as ∼ is defined as a Krawtchouk function for l = 0, in Eq. (38) (see Methods). As a further illustration, it is clear from end part of the Eq. (1) that the environment has identical interactions with all of the chains. It can depend on the configuration of the chains inside the environment. In fact, it can be found a configuration for the chains in such a way that the coupling of the environment to the chains are occurred homogeneously in a similar way. Roughly speaking, the environment could be considered as electromagnetic radiations inside an imperfect cavity formed by two identical spherical mirrors 32 . Obviously, the cavity modes have a Lorentzian spectral density 33 . Since the cavity has cylindrical symmetry so all of the chains in the cavity which are parallel to the cavity axis with equal radius distance and also with equal distances from the mirrors, are coupled to the cavity modes in the same way. Therefore, by this reason, a common Lorentzian environment which interacts with a spin chain 34,35 , can be coupled to the arbitrary number of the other similar chains in the same way. Taking the site-dependent coupling strength g k,j as Eq.
The columns of the matrix Û are the eigenvectors of Ĥ and related to the Krawtchouk polynomials as follows From the orthogonality of Û , the inverse relation follows as   Now, we consider the dynamics of the system by noting to the point that at initial time t = 0, there exists only a single excitation in one of the chains and the other N − 1 chains along with the reservoir are in their respective ground states. Let us assume that the initial state can be written, in general, as follows Since the Hamiltonian conserves the number of excitations in the system, the time-evolved state |ψ(t)〉 is where |1 k 〉 E denotes the state of the reservoir with only one excitation in the kth mode. The time-dependent coef-  (10) and (11), we can obtain the following differential equations Integrating Eq. (14) and substituting it into Eq. (13) gives the integro-differential equation where the correlation function f(t − t′) is related to the spectral density J(ω) of the reservoir by Here, the structure of the common reservoir can be described by an effective Lorentzian spectral density of the form. where λ is the spectral width, γ 0 is the coupling strength, and ω 0 is the central frequency of the reservoir, which is equal to the transition frequency of qubits. Using the Laplace transformation and its inverse, we can obtain a formal solution for  C s ( )  = = Therefore, the probability amplitude for finding the initial excitation, at time t, in the qubit located at site j of the 1th chain is given by   . Therefore, it is concluded that the fidelity of state transfer for an excitation between two ends of the dissipative spin chain in the presence of other N − 1 similar auxiliary chains contained in the reservoir is written as , as shown in Fig. 2. On the other hand, in the presence of additional chains (for example N = 50), a considerable improvement is observed in the efficiency of state transfer (see Fig. 2(a) and (b)). Evidently, whatever N becomes larger, the state transfer process in the mentioned chain is better protected against the dissipative noises. , so the interaction of the three-qubit spin chain with the reservoir is possible only through the eigenstate |Φ 0 〉. Therefore, by entering the corresponding three-qubit auxiliary chains (N = 45), the QST for the three-qubit spin chain can be well-protected against the noises. This procedure can be repeated for the four-qubit chain by considering the four-qubit auxiliary chains (N = 40), as depicted in Fig. 4(a) and (b).
To explain why the presence of auxiliary chains in the reservoir leads to the protection of QST in the considered chain, let us remember from refs [36][37][38] that the protection of entanglement or coherence in a qubit (2-dimensional) system or in a qutrit (3-dimensional) system, is achieved by entering auxiliary qubit or qutrits, into the related reservoir respectively. In fact, entering auxiliary systems into the respective reservoirs leads to more separation of system-reservoir bound state, as an isolated eigenstate of the whole system, from the remainder spectrum. This approach can be extended for protection of an open quantum system with M-dimensional Hilbert space, which could be considered as a spin chain with M spin in the single excitation subspace. Therefore, entering the other similar auxiliary chains into the reservoir leads to improvement of the formed bound state, i.e, a better separation of it from the remainder spectrum. Consequently, this situation gives the protection process for the mentioned spin chain with length of M.
It should be noted that a common reservoir with Lorentzian spectral density makes it possible to solve the dynamics of the open system (system with N identical chains) analytically. While, if we take for example Ohmic spectral density, it is not possible to obtain an exact master equation for the dynamics of the considered system. Also, it is impossible to obtain an exact dynamics for the system by taking local environments interacting individually with the chains. In addition, we note that the protection process introduced in this paper has similar performance for Markovian and Non-Markovian noises (see, for example 36 ).

Conclusion
In summery, we investigated a mechanism for the protection of the intrinsic PST of a pre-engineered linear spin chain in the presence of dissipative noises. By obtaining the exact dynamics, it was shown that the protection process can be well-controlled through the entering non-interacting auxiliary chains into the structured reservoir and therefore, high fidelity state transmission is achievable in the considered spin chain. Furthermore, it was illustrated that the protocol has better efficiency for the chains with more qubits.

Methods
Perfect State transfer for an isolated spin chain. We consider a set of M identical qubits on a linear chain with nearest-neighbor XY coupling. The Hamiltonian of the system is given by