Robust and efficient transport of two-qubit entanglement via disordered spin chains
Rafael Vieira Gustavo Rigolin







Robust and efficient transport of entanglement via disordered spin chains





Rafael Vieira Departamento de Física,
	   Universidade Federal de São Carlos, São Carlos, SP 13565-905,
	   Brazil          
          Gustavo Rigolin Departamento de Física,
	   Universidade Federal de São Carlos, São Carlos, SP 13565-905,
	   Brazil 
  
           rigolin@ufscar.br  






Robust and efficient transport of two-qubit entanglement via disordered spin chains
    Rafael Vieira  Gustavo Rigolin
    Received: date / Accepted: date
===================================================================================




We investigate how robust is the modified XX spin-1/2 chain of 
[R. Vieira and G. Rigolin, Phys. Lett. A 382, 2586 (2018)]
in transmitting entanglement when several types of disorder 
and noise are present. 
First, we consider how deviations 
about the optimal settings that lead to almost perfect transmission of a maximally entangled two-qubit state
affect the entanglement reaching the other side of the chain. 
Those deviations are modeled by static, dynamic, and fluctuating disorder. 
We then study how spurious or undesired interactions and external magnetic fields diminish 
the entanglement transmitted through the chain. For chains of the order of hundreds of qubits,
we show for all types of disorder and
noise here studied that the system is not appreciably affected when we have weak disorder
(deviations of less than 1% about the optimal settings) and that for moderate disorder  
it still beats the standard and ordered XX model when deployed to accomplish
the same task.











§ INTRODUCTION
 

An important tool in the practical implementation of quantum computation and communication tasks is
the reliable transmission of quantum states from one location to another <cit.>.  
Quantum information, or equivalently a quantum state, can be sent from one party (Alice) to another (Bob)
in at least three ways, namely, via direct transmission, via the quantum teleportation protocol <cit.>,
and via
spin chains <cit.>. This last strategy, where Alice and Bob are connected by a spin chain through which 
they can send quantum states to each other is the main focus of this work.
This task is accomplished by properly tuning the interaction among the qubits in the chain such that after 
the system evolves a certain time t>0, a state prepared by one party at t=0 reaches 
the other one with high fidelity. Note that here as well as in the quantum teleportation protocol no physical
system is transmitted from Alice to Bob. 

One important characteristic of a spin chain when employed to transmit quantum states 
is that the interaction strength among the qubits, once set up to give a high fidelity transmission,
is fixed along the time evolution of the system. Also, by using spin chains as the mean through which 
quantum information is sent back and forth within a silicon-based quantum chip, we will deal with the same
physical
system in order to process and transmit quantum information. In this way we avoid the technologically difficult 
task of integrating different physical platforms, one for processing and other for
transmitting information <cit.>.

The several works investigating the usefulness of spin chains to perform 
quantum communication tasks have centered their focus on one of the 
three following scenarios, namely, the transmission of a single qubit 
<cit.>, 
the generation of entanglement between the two ends or two specific sites of the chain 
<cit.>, and the transmission of 
quantum states composed of two or more qubits <cit.>.
We should also mention important contributions on the usefulness of chains of continuous variable systems in transmitting quantum
states <cit.>. Most of the aforementioned works studied strictly one dimensional chains
with open or periodic boundary conditions and extensive investigations on the performance of those systems in the presence of disorder and
noise can be found in <cit.>. 

In this work our main goal is to investigate the performance of the quantum state transfer
protocol presented in <cit.> when subjected to several types of noise and disorder. 
The protocol of <cit.> was specifically built to 
transmit maximally entangled two-qubit states from Alice to Bob and to be simple in 
its construction and operation, namely, we avoided any modulation in the coupling constants between the 
qubits along the chain <cit.> and we excluded the use of external magnetic fields to drive the transmission of 
the quantum state from Alice to Bob <cit.>.
With such stringent requirements, we could achieve almost perfect entanglement transmission by going 
slightly beyond a one dimensional spin chain. In Fig. <ref> we show the optimal configurations for the model
proposed in <cit.> and for the standard strictly one dimensional system usually seen in the literature.
We also point out a related geometry given in the work of Chen et al. <cit.> aiming at sending
a single qubit from 1 to N. 






§ THE MODEL




The ordered system is described by the following slightly modified XX Hamiltonian,  


    H = H_A + H_M + H_B,


where


    H_A     =    J (σ_1^xσ_2^x+σ_1^yσ_2^y) +J (σ_A^xσ_2^x+σ_A^yσ_2^y), 
    
    H_M     =   ∑_j=2^N-2J_m(σ_j^xσ_j+1^x+σ_j^yσ_j+1^y), 
    
    H_B     =    J (σ_N-1^xσ_N^x+σ_N-1^yσ_N^y) + 
    J (σ_N-1^xσ_B^x+σ_N-1^yσ_B^y).



The notation σ_i^ασ_j^α means σ_i^α⊗σ_j^α, the superscript
indicates a specific Pauli matrix, and the subscript denotes the qubit acted by it. Here   
σ^z|0⟩=|0⟩,σ^z|1⟩=-|1⟩,
σ^x|0⟩=|1⟩,σ^x|1⟩=|0⟩,σ^y|0⟩=i|1⟩,σ^y|1⟩=-i|0⟩,  and i=√(-1). 
We can get the standard XX model setting J=0 and letting the summation for H_M
run from 1 to N-1. In Ref. <cit.> it was shown that for a system of N+2 qubits, with N=100, Bob gets
about 99% of the entanglement created by Alice if J_m/J=49.98 in the proposed model.
And for small values of J_m/J, namely, J_m/J≤ 5, we get 82% of the entanglement created by Alice 
arriving at Bob if J_m/J=2.86. The strictly linear chain (standard model with N=100 qubits) transmits only about 
40% of the entanglement with Alice in the best scenario (when we set J_m/J=2.32).

In order to study how disorder and noise affect the proposed model, we work with the following Hamiltonian,


    H̃ = H̃_A + H̃_M + H̃_B +H̃_z+H̃_zz,


where


    H̃_A     =    J_1,2(t) (σ_1^xσ_2^x+σ_1^yσ_2^y) +J_A,2(t)(σ_A^xσ_2^x+σ_A^yσ_2^y), 
    H̃_M     =   ∑_j=2^N-2J_j,j+1(t)(σ_j^xσ_j+1^x+σ_j^yσ_j+1^y), 
    H̃_B     =    J_N-1,N(t) (σ_N-1^xσ_N^x+σ_N-1^yσ_N^y) + 
    J_N-1,B(t) (σ_N-1^xσ_B^x+σ_N-1^yσ_B^y). 
    H̃_z     =     h_A(t)(1-σ_A^z) + ∑_j=1^Nh_j(t)(1-σ_j^z) + 
    h_B(t)(1-σ_B^z), 
    H̃_zz   =   Δ_A,2(t)σ_A^zσ_2^z + ∑_j=1^NΔ_j,j+1(t)σ_j^zσ_j+1^z 
    +  Δ_N-1,B(t)σ_N-1^zσ_B^z.


Here H̃_A,H̃_M, and H̃_B extend the proposed model such that 
the coupling constants J_i,j(t) may now change with time and with position (qubit's lattice location), allowing
us to model several types of disorder as later explained.
H̃_z and H̃_zz represent, respectively, external magnetic fields and 
the σ_j^zσ_j+1^z interaction, which are two types of noise that may act on the proposed model.
When J_A,2(t)=J_1,2(t)=J_N-1,N(t)=J_N-1,B(t)=J, J_j,j+1(t)=J_m, for j=2,…,N-2,
and all h_j(t) and Δ_i,j(t) are zero, we recover the proposed model of Ref. <cit.>.


Hamiltonian (<ref>) is such that [H,Z]=0, i.e., it commutes with the operator Z=σ_A^z+ ∑_j=1^Nσ_j^z+σ_B^z. 
This implies that the number of spins up and down (excitations) does not change in time, which restricts considerably the size of 
the Hilbert space needed to describe the system. In this scenario, the Schrödinger equation can 
be efficiently solved when the system has just a few excitations <cit.>.   

Following Ref. <cit.>, we aim at sending from Alice to Bob entangled states containing just one excitation. 
In this case any system of N+2 qubits, as depicted in Fig. <ref>,
is described by the linear combination of N+2 single-excitation states, 


    |Ψ(t)⟩ = ∑_j c__j(t)|1_j⟩,


where j=A,1,2, …, N-1,N,B, and






    |1_j⟩ = σ_j^x|00⋯0_j-th qubit⋯ 00⟩ 
     = |00⋯1_j-th qubit⋯ 00⟩.


We assume that when t=0 Alice's qubits 1 and A are the maximally entangled 
Bell state |Ψ^+⟩=(|01⟩+|10⟩)/√(2), leading to the following 
initial state for the whole system, |Ψ(0)⟩=(|010⋯ 0⟩+|10⋯ 0⟩)/√(2). 
In the notation of Eq. (<ref>) we have at t=0 


    c__A(0)=c_ _1(0)=1/√(2)      c__j(0)=0,  j ≠ A,1.



After substituting Eq. (<ref>) into the Schrödinger equation 

    iħd|Ψ(t)⟩/dt = H̃(t) |Ψ(t)⟩
 
and left multiplying it by the bra ⟨ 1_k| we get


    iħdc__k(t)/dt = ∑_j c__j(t) ⟨ 1_k|H̃(t)|1_j⟩.


Using H̃(t), Eq. (<ref>), we can compute ⟨ 1_k|H̃(t)|1_j⟩ 
via the same strategy presented in Ref. <cit.>. This allows us to rewrite the 
Schrödinger equation as 


    d𝐜(t)/dt = 𝐌̃(t)   𝐜(t),


where 


    𝐜(t) = ( c__A(t), c__1(t), c__2(t), …, c__N-1(t), 
    c__N(t), c__B(t) )^T


is a column vector (T denotes transposition) and  






𝐌̃(t) is the matrix of dimension (N+2) × (N+2)
shown in Eq. (<ref>): 


    𝐌̃(t) = -i2/ħ([    D_A(t)         0 J__A,2(t)         0         0         0         ⋯;         0    D_1(t) J__1,2(t)         0         0         0         ⋯; J__A,2(t) J__1,2(t)    D_2(t) J__2,3(t)         0         0         ⋯;         0         0 J__2,3(t)    D_3(t) J__3,4(t)         0         ⋯;         0         0         0 J__3,4(t)    D_4(t) J__4,5(t)         ⋯;         0         0         0         0 J__4,5(t)    D_5(t)         ⋯;         ⋮         ⋮         ⋮         ⋮         ⋮         ⋮         ⋱;           ]).












































The diagonal elements of Eq. (<ref>) are 

    D_A(t)    =    h_A(t) + Δ(t)/2 - Δ_A,2(t), 
    
    D_1(t)    =    h_1(t) + Δ(t)/2 - Δ_1,2(t), 
    
    D_2(t)    =    h_2(t) + Δ(t)/2 - Δ_A,2(t)  -Δ_1,2(t)-Δ_2,3(t), 
    
    D_j(t)    =    h_j(t) + Δ(t)/2 - Δ_j-1,j(t) 
    - Δ_j,j+1(t),  3≤ j≤ N-2,  
    
    D_N-1(t)    =    h_N-1(t) +Δ(t)/2 -Δ_N-2,N-1(t) 
    -Δ_N-1,N(t)-Δ_N-1,B(t), 
    
    D_N(t)    =    h_N(t) + Δ(t)/2 - Δ_N-1,N(t), 
    
    D_B(t)    =    h_B(t) + Δ(t)/2 - Δ_N-1,B(t),


with


    Δ(t) = Δ_A,2(t)+∑_j=1^N-1Δ_j,j+1(t) + Δ_N-1,B(t).



The solution to Eq. (<ref>) can be formally written as
a time-ordered matrix exponential with the aid of the time-ordering operator 
𝒯̂,


    𝐜(t) = 𝒯̂exp(∫_0^t𝐌̃(t')dt')𝐜(0),


where 𝐜(0) is the column vector containing the initial conditions listed 
in Eq. (<ref>).





§ QUANTIFYING ENTANGLEMENT



We use the Entanglement of Formation (EoF) <cit.> to quantify 
the entanglement shared between two qubits. Specifically, we want to determine 
as a function of time the EoF between qubits N and B with Bob. Tracing 
out the other N qubits from ρ(t)=|Ψ(t)⟩⟨Ψ(t)|, 
the density matrix describing the whole system, we get 
<cit.>


    ρ__NB(t) 
       =   ([ 1-|c__N(t)|^2-|c__B(t)|^2                         0                         0                         0;                         0               |c__B(t)|^2          c__B(t)c__N^*(t)                         0;                         0          c__N(t)c__B^*(t)               |c__N(t)|^2                         0;                         0                         0                         0                         0 ]),


where * means complex conjugation. Eq. (<ref>) is the density matrix describing the two qubits 
N and B with Bob written in the basis {|00⟩,|01⟩,|10⟩,|11⟩}. 

With the aid of ρ__NB(t) the EoF can be computed following the recipe in <cit.>,


    (ρ__NB) = -f(C)log_2f(C) - [1-f(C)]log_2[1-f(C)],


with f(C)=(1 + √(1 - C^2))/2 and concurrence C given by  
C(t) = 2|c__N(t)c__B(t)|.
Note that a maximally entangled two-qubit state has EoF = 1 and 
when the pair of qubits shares no entanglement EoF = 0.




§ MODELING DISORDER AND NOISE



We deal with the three major types of disorder that can affect the system's coupling
constants J_i,j, namely, static, dynamic, and fluctuating disorders <cit.>, and assume 
that any fluctuation in those coupling constants is about the optimal settings
for the ordered case with N=100 +2 qubits.  Specifically, we either employ J_m/J=2.86, the optimal setting
for J_m/J≤ 5.0 and whose transmitted entanglement is 
EoF = 0.816, or J_m/J=49.98, the best case when J_m/J≤ 50.0 and with transmitted
entanglement given by EoF = 0.986 <cit.> (see the Appendix for calculations related to chains of N=1000 + 2
qubits).
In the rest of this work we set J=1.0 and ħ = 1.0, which
defines J as our unit of energy, and work with the two values of J_m 
given above.
Following Ref. <cit.>, we also assume that when dealing with time dependent disorder,
the Hamiltonian changes with time only a finite number of times and after the same period τ.  


In this scenario, we can model disorder if we set 


    J_i,j(t_k) = J_i,j(t_k-1)[ 1 + δ J_i,j(k) ],


with

J_A,2(0) = J_1,2(0) = J_N-1,N(0) = J_N-1,B(0) = J, 
∑_j=2^N-2J_j,j+1(0) = J_m, and  
J_i,j(t) = 0, any t, for other values of i,j.







Here 

    (k-1)τ < t_k ≤ kτ,

where k is an integer such that 1≤ k ≤ n.
We also have t_0=0 and we define T_max=max{t_n}=nτ. Sometimes we call T_max by Jt/ħ, both notations meaning
the time when in the optimal ordered model the transmitted entanglement from Alice 
to Bob is maximal.

The quantity δ J_i,j(k) is responsible for introducing
disorder in our system. Its behavior depends on which type of disorder we have 
but all cases boil down to random continuous uniform distributions 
centered in zero and whose ranges lie between -p and p, with 
p the maximal percentage fluctuation 
of J_i,j about its optimal value. Note that δ J_i,j(k)
is a function of the index k, i.e., whenever k changes we have a new uniform 
distribution from which the random values of δ J_i,j are drawn. 

In a similar way we can introduce noise. 
Noting that the optimal settings have no external magnetic field (h_j(t)=0) and
no σ_i^zσ_j^z interaction (Δ_i,j(t)=0), the following
prescription allows us to insert noise in our system,


    h_j(t_k)    =    h_j(t_k-1) + δ h_j(k), 
    Δ_i,j(t_k)    =   Δ_i,j(t_k-1) + δΔ_i,j(k),


where t_k and k were defined above, h_j(0)=0 for j=A,1, 2,…, N-1,N,B, Δ_A,2(0) = Δ_1,2(0)=Δ_N-1,N(0)=Δ_N-1,B(0)=0,
∑_j=2^N-2Δ_j,j+1(0)=0, and Δ_i,j(t)=0, any t, for other values
of i,j.
Analogously to δ J_i,j(k), δ h_j(k) or δΔ_i,j(k) 
depends on the kind of disorder we have and will be explained shortly. But 
as before these uniform distributions are centered in zero, ranging  
between -p and p, where p now means the greatest possible value of  
δ h_j(k) and δΔ_i,j(k). Since our unit of energy is J=1.0,
p can be read as the percentage of J 
defining the upper bound for 
δ h_j(k) or δΔ_i,j(k) in a given uniform distribution.




 §.§ Static disorder
 

In this scenario we have noise or fluctuations in
the coupling constants that are time independent but site (position) dependent,


    δ J_i,j(1) = δ J_i,j,  δ h_j(1) = δ h_j, δΔ_i,j(1) = δΔ_i,j, 
          δ J_i,j(k)=δ h_j(k)=δΔ_i,j(k)=0,  k≠ 1.


This means that 
J_i,j(t) =  J_i,j(0)( 1 + δ J_i,j), h_j(t) = δ h_j, 
Δ_i,j(t) = δΔ_i,j, where 0≤ t ≤ nτ. 
In other words, each quantity above is chosen from independent continuous uniform
distributions at t = 0 and fixed during the time evolution. 

In this case 
Eq. (<ref>) can be integrated and we get


    𝐜(t) = exp(𝐌̃   t )𝐜(0).


By numerically computing the matrix exponential we obtain 𝐜(t)
and then, using Eq. (<ref>), the entanglement at any t 
between the qubits N and B with Bob.





 §.§ Dynamic disorder
 

In contrast to the case of static disorder, we have time dependent but position independent noise
or fluctuations in the coupling constants,


    δ J_i,j(k) = δ J(k),    δ h_j(k) = δ h(k),    δΔ_i,j(k) = δΔ(k).


Now, each coupling constant changes equally by δ J(k) at the time (k-1)τ, remaining fixed from (k-1)τ to
kτ. After each period of time τ a new random number is drawn from a continuous uniform distribution as previously explained
and a new value of the coupling constant is obtained: J_i,j(t_k) = J_i,j(t_k-1)[ 1 + δ J(k) ]. 
Similarly, all qubits are subjected to the
same external field from (k-1)τ to kτ and the σ_i^zσ_j^z interaction strength 
is the same for any
pair of nearest neighbors during this time span. After a period of time τ new values for the field and for the 
σ_i^zσ_j^z interaction strengths are obtained from two independent continuous uniform distributions
δ h(k) and δΔ(k) according to the prescription given in Eq. (<ref>) and (<ref>).
We should note that it is convenient sometimes to express τ as a percentage
of T_max=Jt/ħ. Thus τ = (Jt/ħ)/n, with 1/n denoting the percentage of the total time of evolution
T_max giving the period of changes. 

Noting that between time (k-1)τ and kτ, where k is an integer 
between 1 and n, the matrix 𝐌̃(t) is time independent, the time evolution
of the system all the way down to T_max=nτ can be written as


    𝐜(nτ) = exp(𝐌̃_𝐧  τ )exp(𝐌̃_𝐧-1  τ )
    …exp(𝐌̃_1  τ )𝐜(0),


where 𝐌̃_𝐤 is a time independent matrix given by Eq. (<ref>)
such that 𝐌̃_𝐤 = 𝐌̃_𝐤-1 + δ𝐌̃_𝐤.
Here 𝐌̃_0 is associated to the optimal ordered Hamiltonian, 
Eq. (<ref>), and δ𝐌̃_𝐤 is the modification to 𝐌̃_𝐤-1
coming from the presence of the dynamical disorder given by Eqs. (<ref>)-(<ref>) 
and (<ref>).





 §.§ Fluctuating disorder
 

This is the case where both aspects of static and dynamic disorders are combined, namely,
where we have time and position dependent noise or time and position 
dependent fluctuations in the coupling constants. The coupling constants between qubits as well as 
the strength of the noise are given by Eqs. (<ref>)-(<ref>), with
δ J_i,j(k), δ h_j(k), and δΔ_i,j(k) depending on the position of the qubits
and on time in the same way already described, respectively, for the cases of static and dynamic disorders.
The time evolution of the system is given by Eq. (<ref>) 
and 𝐌̃_𝐤 is a time independent matrix with  
position dependent coefficients given by Eq. (<ref>). 





§ RESULTS




We first study static disorder and dynamic and fluctuating disorders for
τ = 10% T_max=10%Jt/ħ 
(Figs. <ref> and <ref>).






This value of τ means that the Hamiltonian changes 10 times during
the time evolution of the system from t=0 to t=T_max according to the prescription 
given above. 



Looking at the upper panels of 
Figs. <ref> and <ref>,
we note the following trends for the entanglement 
transmitted to Bob. For small values of J_m/J the presence of noise barely affects the efficiency of the protocol, 
i.e., the presence of random  external magnetic fields and σ_i^zσ_j^z interactions do not affect 
considerably the system's performance up to a deviation of 10% about the optimal settings for the ordered case. 
On the other hand, the presence of disorder in the coupling constants affects the system performance much more strongly than the 
presence of noise. However, for deviations of 1% about the optimal settings, the efficiency is barely reduced.

Moreover, among the three types of disorder in the coupling constants, static disorder is the least severe. 
In this scenario, even for a deviation of 
10% about the optimal settings, we still beat the entanglement transmitted using the optimal ordered standard model (dotted lines).
Note that dynamic and fluctuating disorders in the coupling constants affect the system nearly in the same way. 
When noise and disorder in the coupling constants act simultaneously, the reduction in efficiency is almost the same as if the
noise was absent, another confirmation of the fact that the types of noise here studied barely affect the system's efficiency
for small values of J_m/J.

































































Moving to the lower panels of 
Figs. <ref> and <ref>,
we note different trends for the entanglement 
transmitted to Bob. 
Indeed, for high values of J_m/J the presence of noise becomes relevant in almost all cases, reducing the efficiency of the protocol,
and the presence of disorder in the coupling constants no longer dominates the reduction of the system's performance. 
Of all three types of disorder, dynamic disorder is now the least severe. In this case, even for a deviation of 
10% about the optimal settings, the entanglement transmitted to Bob is very high, close to the optimal value for the 
clean system and way above the entanglement reaching Bob if we employ the standard model (dotted lines).
Dynamic and fluctuating disorders, on the other hand, no longer affect the system in the same way,
with the latter being much more severe.
When noise and disorder in the coupling constants act simultaneously, we get almost always the worst scenario, 
where the reduction in efficiency is most severe (square-blue curves). 
However, even for high values of J_m/J, deviations of 1% about the optimal settings do
not affect much the amount of entanglement reaching Bob.


In Fig. <ref> we study how the rate of change of the Hamiltonian during the time evolution affects the system's 
performance. We work with the worst of the six scenarios shown in 
Figs. <ref> and <ref>,
namely, fluctuating
disorder with noise and disorder in the coupling constants simultaneously affecting the system. 
Looking at Fig. <ref> it is not difficult to convince ourselves that the higher the frequency of changes in the
Hamiltonian or, equivalently, the smaller the period τ of changes,  
the lower the entanglement transmitted to Bob. 

The reason why a small period in the fluctuating disorder leads to a poor transmission of entanglement 
is the fact that the smaller the period of changes, the higher the frequency of changes in the Hamiltonian.
This higher number of changes in the Hamiltonian is cumulative and ultimately leads to a greater deviation from 
the optimal values of the coupling constants. 
For longer periods, nevertheless, the Hamiltonian changes only once or twice
during the whole time evolution, leading to almost no change in the optimal settings when we work with small disorder. 
This fact reflects in a better performance for systems with longer periods of changes in the Hamiltonian when
compared with the shorter ones.







In order to make it clearer to analyze how different types of disorder affect the efficiency of the protocol, 
we show in Fig. <ref> and in the same graphic the entanglement transmitted for the 
three different types of disorder studied here. We fix 
the value of τ such that it is 10% of Jt/ħ, when dealing with dynamic and fluctuating disorders,
and work with the scenario where noise and disorder in the coupling constants are simultaneously affecting the system.

For J_m/J = 2.86, upper panel of Fig. <ref>, we note that for small fluctuations about the optimal settings
(<5%), dynamic disorder is the most severe type of disorder. For greater values of deviations about the optimal
settings, fluctuating disorder becomes the worst case. Also, for small values of J_m/J, static disorder is the 
least severe type of disorder.
Looking at the lower panel of Fig. <ref>, when J_m/J = 49.98, we note that dynamic disorder becomes very
benign, little affecting the system's performance. We also note that fluctuating disorder is the most severe 
type of disorder for all range of fluctuations about the optimal settings of the clean system.  





§ CONCLUSION



Summing up, we have extensively studied how the protocol presented in Ref. <cit.>,
specially devised to transmit maximally entangled Bell states, responds to disorder and 
noise for systems of size of the order of a hundred qubits. 
We have studied three types of disorder, namely, static, dynamic, and fluctuating disorders 
<cit.> and two types of noise that may affect the system. In one type of noise we have 
random external magnetic fields acting on the qubits of the system and in the 
other one we investigated how spurious σ^z_iσ^z_j interactions reduce the system's
capacity to transmit entanglement. 

For all the several cases of noise and disorder 
presented here, we showed that for deviations of up to 1% about the optimal settings of the clean
system, we still have excellent entanglement transmission, with little or no reduction in the efficiency
of the protocol. Moreover, up to deviations about the
optimal settings of the order of  3%, the proposed model (upper panel of Fig. <ref>)
still beats the optimal entanglement transmission efficiency of the 
standard model (lower panel of Fig. <ref>). For a system's size of the order of
a thousand qubits, we obtain excellent transmission of entanglement for deviations in the optimal
settings of up to 0.1% and acceptable ones for deviations of up to 0.5% (see the Appendix).   

All those results show that the proposed model is very robust to
noise and disorder and we believe it might prove fruitful to test in future works its robustness when
the number of excitations is no longer conserved and when we have residual Dzyaloshinskii-Moriya interactions
affecting the system <cit.>.










§ ACKNOWLEDGMENTS

RV thanks CNPq (Brazilian National Council for Scientific and Technological Development)
for funding and GR thanks CNPq and CNPq/FAPERJ (State of Rio de Janeiro Research Foundation) for financial support through the National Institute of
Science and Technology for Quantum Information.






§ THE 1000 + 2 QUBITS CASE



Whenever we have dynamic or fluctuating disorders, specially for small values of τ, the computational resources 
needed to handle systems of the order of thousands of qubits are very demanding. 
For that reason, we report here only two calculations for the N = 1000 + 2 qubits system by implementing  
one tenth of the realizations made for the N = 100 + 2 qubits case, namely, instead of 1000 realizations for
each disorder percentage, we work with 100 realizations. We have also worked with the 
worst possible scenario, in which the system is most severely affected, i.e., fluctuating disorder affecting the coupling constants as well
as fluctuating random external fields and spurious σ^z_iσ^z_j interactions. Note that for less stringent
disorder and noise scenarios, the N = 1000 + 2 qubits case gives much better results.








As expected, the results depicted in Figs. <ref> and <ref> show 
that the longer the chain the more the system is affected by disorder and noise. 
However, we still have very good entanglement transmission for disorder of less than 0.1% about the optimal settings
of the clean system and an acceptable one for less than 0.5%. Moreover, for disorder of less than 1%, we still
beat the optimal transmission of the clean standard model (strictly linear chain). 





200

ben00 Bennett, C. H., DiVincenzo, D. P.: Quantum information and computation. 
Nature (London) 404, 247 (2000)

ben93 Bennett, C. H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W. K.: Teleporting an unknown quantum state via dual classical and 
Einstein-Podolsky-Rosen channels. 
Phys. Rev. Lett. 70, 1895 (1993) 

bos03 Bose, S.: Quantum Communication through an Unmodulated Spin Chain. 
Phys. Rev. Lett. 91, 207901 (2003)

chr04 Christandl, M., Datta, N., Ekert, A., Landahl, A. J.: Perfect State Transfer in Quantum Spin Networks. 
Phys. Rev. Lett. 92, 187902 (2004)

nik04 Nikolopoulos, G. M., Petrosyan, D., Lambropoulos, P. L.: Electron wavepacket propagation in a chain of coupled quantum dots. 
J. Phys.: Condens. Matter 16, 4991 (2004)

sub04 Subrahmanyam, V.: Entanglement dynamics and quantum-state transport in spin chains. 
Phys. Rev. A 69, 034304 (2004)

osb04 Osborne, T. J., Linden, N.: Propagation of quantum information through a spin system. 
Phys. Rev. A 69, 052315 (2004)

cir97 Cirac, J. I., Zoller, P., Kimble, H., Mabuchi, H.: Quantum State Transfer and Entanglement Distribution 
among Distant Nodes in a Quantum Network. 
Phys. Rev. Lett. 78, 3221 (1997)

ple04 Plenio, M. B., Hartley, J., Eisert, J.: Dynamics and manipulation of entanglement in coupled harmonic systems with many degrees of freedom. 
New J. Phys. 6, 36 (2004)

sem05 Plenio, M. B., Semião, F. L.: High efficiency transfer of quantum information and multiparticle entanglement generation 
in translation-invariant quantum chains. New J. Phys. 7, 73 (2005)

chr05 Christandl, M., Datta, N., Dorlas, T. C., Ekert, A., Kay, A., Landahl, A. J.: Perfect transfer of arbitrary states in quantum spin networks. 
Phys. Rev. A 71, 032312 (2005)

shi05 Shi, T., Li, Y., Song, Z., Sun, Ch.-P.: Quantum-state transfer via the ferromagnetic chain in a spatially modulated field. 
Phys. Rev. A 71, 032309 (2005)

woj05 Wójcik, A., Łuczak, T., Kurzyński, P., Grudka, A., Gdala, T., Bednarska, M.: Unmodulated spin chains as universal quantum wires. 
Phys. Rev. A 72, 034303 (2005)

li05 Li, Y., Shi, T., Chen, B., Song, Z., Sun, C.-P.: Quantum-state transmission via a spin ladder as a robust data bus. 
Phys. Rev. A 71, 022301 (2005)

kar05 Karbach, P., Stolze, J.: Spin chains as perfect quantum state mirrors. 
Phys. Rev. A 72, 030301 (2005)

kay05 Kay, A., Ericsson, M.: Geometric effects and computation in spin networks. 
New J. Phys. 7, 143 (2005)

har06 Hartmann, M. J., Reuter, M. E., Plenio, M. B.: Excitation and entanglement transfer versus spectral gap. 
New J. Phys. 8, 94 (2006)

huo08 Huo, M. X., Li, Y., Song, Z., Sun, C. P.: The Peierls distorted chain as a quantum data bus for quantum state transfer. 
Europhys. Lett. 84, 30004 (2008)

gua08 Gualdi, G., Kostak, V., Marzoli, I., Tombesi, P.: Perfect state transfer in long-range interacting spin chains. 
Phys. Rev. A 78, 022325 (2008)

ban10 Banchi, L., Apollaro, T. J. G., Cuccoli, A., Vaia, R., Verrucchi, P.: Optimal dynamics for quantum-state and entanglement 
transfer through homogeneous quantum systems. 
Phys. Rev. A 82, 052321 (2010)

kur11 Kurzyński, P., Wójcik, A.: Discrete-time quantum walk approach to state transfer. 
Phys. Rev. A 83, 062315 (2011)

kay11 Pemberton-Ross, P. J., Kay, A.: Perfect Quantum Routing in Regular Spin Networks. 
Phys. Rev. Lett. 106, 020503 (2011)
  
god12 Godsil, C., Kirkland, S., Severini, S., Smith, J.: Number-Theoretic Nature of Communication in Quantum Spin Systems. 
Phys. Rev. Lett. 109, 050502 (2012)

apo12 Apollaro, T. J. G., Banchi, L., Cuccoli, A., Vaia, R., Verrucchi, P.: 99%-fidelity ballistic quantum-state transfer 
through long uniform channels. 
Phys. Rev. A 85, 052319 (2012)

lor13 Lorenzo, S., Apollaro, T. J. G., Sindona, A., Plastina, F.: Quantum-state transfer via resonant tunneling through local-field-induced barriers.
Phys. Rev. A 87, 042313 (2013)

sou14 Sousa, R., Omar, Y.: Pretty good state transfer of entangled states through quantum spin chains. 
New J. Phys. 16, 123003 (2014)

hor14 Korzekwa, K., Machnikowski, P., Horodecki, P.: Quantum-state transfer in spin chains via isolated resonance of terminal spins. 
Phys. Rev. A 89, 062301 (2014)

shi15 Shi, Z. C., Zhao, X. L., Yi, X. X.: Robust state transfer with high fidelity in spin-1/2 chains by Lyapunov control. 
Phys. Rev. A 91, 032301 (2015)

lor15 Lorenzo, S., Apollaro, T. J. G., Paganelli, S., Palma, G. M., Plastina, F.: Transfer of arbitrary two-qubit states via a spin chain. 
Phys. Rev. A 91, 042321 (2015)

pou15 Pouyandeh, S., Shahbazi, F.: Quantum state transfer in XXZ spin chains: A measurement induced transport method. 
Int. J. Quantum. Inform. 13, 1550030 (2015)

zha16 Zhang, X.-P., Shao, B., Hu, S., Zou, J., Wu, L.-A.: Optimal control of fast and high-fidelity quantum state transfer in spin-1/2 chains. 
Ann. Phys. (NY) 375, 435 (2016)

che16 Chen, X., Mereau, R., Feder, D. L.: Asymptotically perfect efficient quantum state transfer across uniform chains with two impurities. 
Phys. Rev. A 93, 012343 (2016)

nic16 Nicacio, F., Semião, F. L.: Transport of correlations in a harmonic chain. Phys. Rev. A 94, 012327 (2016)

est17  Estarellas, M. P., D'Amico, I., Spiller, T. P.: Topologically protected localised states in spin chains. Sci. Rep. 7, 42904 (2017)
 
est17b Estarellas, M. P., D'Amico, I., Spiller, T. P.: Robust quantum entanglement generation and generation-plus-storage protocols with spin chains.
Phys. Rev. A 95, 042335 (2017)

apo18 Apollaro, T. J. G., Almeida, G. M. A., Lorenzo, S., Ferraro, A., Paganelli, S.: Spin chains for two-qubit teleportation.
Available from: arXiv:1812.11609 [quant-ph] (2018).
 
chi05 De Chiara, G., Rossini, D., Montangero, S., Fazio, R.: From perfect to fractal transmission in spin chains. 
Phys. Rev. A 72, 012323 (2005)

fit05 Fitzsimons, J., Twamley, J.: Superballistic diffusion of entanglement in disordered spin chains. Phys. Rev. A 72, 050301 (2005)

bur05 Burgarth, D., Bose, S.: Perfect quantum state transfer with randomly coupled quantum chains. New J. Phys. 7, 135 (2005)

pet10 Petrosyan, D., Nikolopoulos, G. M., Lambropoulos, P.: State transfer in static and dynamic spin chains with disorder. 
Phys. Rev. A 81, 042307 (2010)

zwi11 Zwick, A., Álvarez, G. A., Stolze, J., Osenda, O.: Robustness of spin-coupling distributions for perfect quantum state transfer. 
Phys. Rev. A 84, 022311 (2011); Spin chains for robust state transfer: Modified boundary couplings versus completely engineered chains. 
85, 012318 (2012)

bru12 Bruderer, M., Franke, K., Ragg, S., Belzig, W., Obreschkow, D.: 
Exploiting boundary states of imperfect spin chains for high-fidelity state transfer. 
Phys. Rev. A 85, 022312 (2012)

nik13 Nikolopoulos, G. M.: Characterizing quantum gates via randomized benchmarking. Phys. Rev. A 87, 042311 (2013)

kur14 Zwick, A., Álvarez, G. A., Bensky, G., Kurizki, G.: Optimized dynamical control of state transfer through noisy spin chains. 
New. J. Phys. 16, 065021 (2014)

ash15 Ashhab, S.: Quantum state transfer in a disordered one-dimensional lattice. Phys. Rev. A 92, 062305 (2015)

pav16 Pavlis, A. K., Nikolopoulos, G. M., Lambropoulos, P.: Evaluation of the performance of two state-transfer Hamiltonians in the presence of static disorder. 
Quantum Inf Process 15, 2553 (2016)

ron16 Ronke, R., Estarellas, M. P., D'Amico, I., Spiller, T. P., Miyadera, T.: Anderson localisation in spin chains for perfect state transfer. 
Eur. Phys. J. D 70, 189 (2016)

lyr17 Almeida, G. M. A., de Moura, F. A. B. F., Apollaro, T. J. G., Lyra, M. L.: Disorder-assisted distribution of entanglement in XY spin chains. 
Phys. Rev. A 96, 032315 (2017)

lyr17a Almeida, G. M. A., de Moura, F. A. B. F., Lyra, M. L.: Quantum-state transfer through long-range correlated disordered channels. 
Phys. Lett. A 382, 1335 (2018)

lyr17b Almeida, G. M. A., de Moura, F. A. B. F., Lyra, M. L.: Entanglement generation between distant parties via disordered spin chains. 
Available from: arXiv:1711.08553 [quant-ph] (2017).

ben96 Bennett, C. H., DiVincenzo, D. P., Smolin, J. A., Wootters, W. K.: Mixed-state entanglement and quantum error correction. 
Phys. Rev. A 54, 3824 (1996)

woo98 Wootters, W. K.: Entanglement of Formation of an Arbitrary State of Two Qubits. Phys. Rev. Lett. 80, 2245 (1998)

vie18 Vieira, R., Rigolin, G.: Almost perfect transport of an entangled two-qubit state through a spin chain. Phys. Lett. A 382, 2586 (2018)

vie13 Vieira, R., Amorim, E. P. M., Rigolin, G.: Dynamically Disordered Quantum Walk as a Maximal Entanglement Generator. 
Phys. Rev. Lett. 111, 180503 (2013)

vie14 Vieira, R., Amorim, E. P. M., Rigolin, G.: Entangling power of disordered quantum walks.  Phys. Rev. A 89, 042307 (2014)

dzy58 Dzyaloshinsky, I.: A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics.  J. Phys. Chem. Solids 4, 241 (1958)
		
mor60 Moriya, T: Anisotropic Superexchange Interaction and Weak Ferromagnetism. Phys. Rev. 120, 91 (1960)




