Perfect quantum state transfer with randomly coupled quantum chains

Daniel Burgarth; Sougato Bose

doi:10.1088/1367-2630/7/1/135

Abstract

We suggest a scheme that allows arbitrarily perfect state transfer even in the presence of random fluctuations in the couplings of a quantum chain. The scheme performs well for both spatially correlated and uncorrelated fluctuations if they are relatively weak (say 5%). Furthermore, we show that given a quite arbitrary pair of quantum chains, one can check whether it is capable of perfect transfer by only local operations at the ends of the chains, and the system in the middle being a 'black box'. We argue that unless some specific symmetries are present in the system, it will be capable of perfect transfer when used with dual-rail encoding. Therefore our scheme puts minimal demand not only on the control of the chains when using them, but also on the design when building them.

Export citation and abstract BibTeX RIS

1. Introduction

Recently, much interest has been devoted to quantum communication with quantum chains [1]–[24]. The main spirit of these papers is that permanently coupled systems can be used for the transfer of quantum information with 'minimal control', that is, only the sending and the receiving parties can apply gates to the system, but the part of the chain interconnecting them cannot be controlled during the communication process. A scheme with less control is obviously impossible. The first proposals [1, 2] considered a regular spin chain with Heisenberg interactions. A physical implementation of this scheme was discussed in [3], and its channel capacity was derived in [4]. Already in [1, 2] it was realized that such a transfer will, in general, not be perfect. The reason for the imperfect transfer is the dispersion of the information along the chain. This becomes worse as the chains get longer.

Since then, many very interesting methods have been developed to improve the fidelity of the transfer. One method is to use Hamiltonians with engineered couplings [5]–[12] such that the dispersed information will 'refocus' at the receiving end of the chain. Another approach is to encode and decode the information using multiple spins [13, 14] to form Gaussian wave packets (which have a lower dispersion). This has been generalized [14] in an elegant way by using 'phantom' spins such that a multiple-spin encoding can be achieved by only controlling two sending and receiving qubits. By using gapped systems [15]–[18], the intermediate spins are only virtually excited, and the transfer has a very high fidelity. Finally, in [19] we have suggested a 'dual-rail' encoding using two parallel quantum channels. It was shown in [20] that such a protocol achieves arbitrarily perfect transfer for nearly any type of quantum chain, transforming a heavily dispersive dynamic into one that can be used for state transfer. In this scheme, not only the control needed during the transfer is minimized (no local access along the chains is needed), but also the control needed to design the system in the first place.

The main requirement for perfect transfer with dual-rail encoding as presented in the literature till date [19, 20] is that two identical quantum chains have to be designed. While this is not so much a theoretical problem, for possible experimental realizations of the scheme [3], the question arises naturally how to cope with slight asymmetries of the channels. The purpose of this paper is to demonstrate that in many cases, perfect state transfer with dual-rail encoding is possible for quantum chains with differing Hamiltonians.

By doing so, we also offer a solution to another and perhaps more general problem: if one implements any of the above schemes, the Hamiltonians will always be different from the theoretical ones by some random perturbation. This will lead to a decrease of fidelity in particular where specific energy levels were assumed (see [25] for an analysis of fluctuations affecting the scheme [5]). Also in general, random systems can lead to an Anderson localization [26] of the eigenstates (and therefore to low-fidelity transport of quantum information). This problem can be avoided using the scheme described below. We will show numerically that the dual-rail scheme can still achieve arbitrarily perfect transfer for a uniformly coupled Heisenberg Hamiltonian with random noise on the coupling strengths (both for the case of spatially correlated and uncorrelated fluctuations). Moreover, for any two quantum chains, we show that Bob and Alice can check whether their system is capable of dual-rail transfer without directly measuring their Hamiltonians or local properties of the system along the chains but by only measuring their part of the system.

2. Conclusive transfer

By 'conclusive transfer', we mean that the receiver has a certain probability for obtaining the perfectly transferred state, and some way of checking whether this has happened. Conclusive transfer is more valuable than simple state transfer with the same fidelity, because errors are detected, and memory effects [27] are unimportant if the transfer was successful. A single spin-1/2 quantum chain could however not be used for conclusive transfer, because any measurement would destroy the unknown quantum state that is being transferred. The simplest quantum chain for conclusive transfer is a system consisting of two uncoupled quantum chains (1) and (2), as shown in figure 1. The chains are described by the two Hamiltonians H⁽¹⁾ and H⁽²⁾ acting on the corresponding Hilbert spaces $\cal{H}_{1}$ and $\cal{H}_{2}$ . The total Hamiltonian is thus

$\begin{equation} H=H^{(1)}\,{\otimes}\, I^{(2)}+I^{(1)}\,{\otimes}\, H^{(2)} \end{equation}\tag{1}$

and the time evolution operator factorizes as

$\begin{equation} U(t) = \exp(-{\rm i}Ht), \end{equation}\tag{2}$

$\begin{equation} \hphantom{U(t)} = \exp(-{\rm i}H^{(1)}t)\,{\otimes}\,\exp(-{\rm i}H^{(2)}t). \end{equation}\tag{3}$

For the moment, we assume that both chains have equal length N, but it will become clear in section 4 that this is not a requirement of our scheme. Like in [19, 20], we assume that the quantum chains consist of qubits and that both Hamiltonians commute with the z-component of the total spin of the chains,

$\begin{equation} \left\lbrack H^{(1)},\sum_{n=1}^{N}\sigma_{z,n}^{(1)}\right\rbrack =\left\lbrack H^{(2)},\sum_{n=1}^{N}\sigma_{z,n}^{(2)}\right\rbrack =0. \end{equation}\tag{4}$

It then follows that the state

$\begin{equation} |\mathbf{0}\rangle ^{(i)}\equiv |00 \ldots\,0\rangle ^{(i)} \end{equation}\tag{5}$

is an eigenstate of H⁽ⁱ⁾ and that the dynamics of an initial state of the form

$\begin{equation} |\mathbf{1}\rangle^{(i)}\equiv |10 \ldots\,0\rangle ^{(i)} \end{equation}\tag{6}$

is restricted to the subspace of single excitations,

$\begin{equation} |\mathbf{m}\rangle^{(i)}\equiv \left| \begin{array}{@{}c@{}c@{}c@{}c@{}c@{}} 0 \ldots &0 &1 &0 \ldots &0\\ &&\,m\textrm{th}\, \end{array} \right\rangle^{(i)}\quad (1\leqslant m\leqslant N). \end{equation}\tag{7}$

In the following, we will omit all indices and write the states of the full Hilbert space $\cal{H}_{1}\,{\otimes}\,\cal{H}_{2}$ as

$\begin{equation} |\mathbf{m,n}\rangle \equiv|\mathbf{m}\rangle ^{(1)}\,{\otimes}\,|\mathbf{n}\rangle^{(2)}. \end{equation}\tag{8}$

**Figure 1.** Two quantum chains interconnecting A and B. Control of the systems is only possible at the two qubits of either end.
Download figure:
Standard Export PowerPoint slide

We assume that the sender, Alice, has full access to the first qubit of each chain, and that the receiver, Bob, has full access to the last qubit of each chain. With 'full access' we mean that they can perform a two-qubit gate (say, a CNOT), and arbitrary single-qubit operations. Bob also needs the ability to perform single-qubit measurements.

Initially, Alice encodes the state as

$\begin{equation} \alpha|\mathbf{0,1}\rangle +\beta|\mathbf{1,0}\rangle. \end{equation}\tag{9}$

This is a superposition of an excitation in the first qubit of the first chain and an excitation in the first qubit of the second chain. The state will evolve into

$\begin{equation} \sum_{n=1}^{N}\{\alpha g_{n,1}(t)|\mathbf{0,n}\rangle +\beta f_{n,1}(t)|\mathbf{n,0}\rangle\} . \end{equation}\tag{10}$

Note that there is only one excitation in the system. The probability amplitudes are given by

$\begin{equation} f_{n,1}(t) \equiv \langle \mathbf{n,0}|U(t)|\mathbf{1,0}\rangle, \end{equation}\tag{11}$

$\begin{equation} g_{n,1}(t) \equiv\langle \mathbf{0,n}|U(t)|\mathbf{0,1}\rangle. \end{equation}\tag{12}$

In [19], these functions were identical. For differing chains this is no longer the case. We may, however, find a time t₁ such that the modulus of their amplitudes at the last spins are the same (see figure 2),

$\begin{equation} g_{N,1}(t_{1})={\rm e}^{{\rm i}\phi_{1}}f_{N,1}(t_{1}). \end{equation}\tag{13}$

At this time, the state (10) can be written as

$\begin{equation} \sum_{n=1}^{N-1}\{\alpha g_{n,1}(t_{1})|\mathbf{0,n}\rangle +\beta f_{n,1}(t_{1})|\mathbf{n,0}\rangle\} +f_{N,1}(t_{1})\{{\rm e}^{{\rm i}\phi_{1}}\alpha|\mathbf{0,N}\rangle +\beta|\mathbf{N,0}\rangle\}. \end{equation}\tag{14}$

Bob decodes the state by applying a CNOT gate on his two qubits, with the first qubit as the control bit. The state thereafter is

$\begin{equation} \sum_{n=1}^{N-1}\{\alpha g_{n,1}(t_{1})|\mathbf{0,n}\rangle +\beta f_{n,1}(t_{1})|\mathbf{n,0}\rangle\} +f_{N,1}(t_{1})\{{\rm e}^{{\rm i}\phi_{1}}\alpha|\mathbf{0}\rangle ^{(1)}+\beta|\mathbf{N}\rangle^{(1)}\} \,{\otimes}\,|\mathbf{N}\rangle^{(2)}. \end{equation}\tag{15}$

Bob then measures his second qubit. Depending on the outcome of this measurement, the systems will either be in the state

$\begin{equation} \frac{1}{\sqrt{p_{1}}}\sum_{n=1}^{N-1}\{\alpha g_{n,1}(t_{1})|\mathbf{0,n}\rangle +\beta f_{n,1}(t_{1})|\mathbf{n,0}\rangle\} \end{equation}\tag{16}$

or in

$\begin{equation} \{{\rm e}^{{\rm i}\phi_{1}}\alpha|\mathbf{0}\rangle^{(1)}+\beta|\mathbf{N}\rangle^{(1)}\} \,{\otimes}\,|\mathbf{N}\rangle^{(2)}, \end{equation}\tag{17}$

where p₁ = 1 − |f_N,1(t₁)|² = 1 − |g_N,1(t₁)|² is the probability that Bob has not received the state. The state (17) corresponds to the correctly transferred state with a known phase error (which can be corrected by Bob using a simple phase gate). If Bob finds the system in the state (16), the transfer has been unsuccessful, but the information is still in the chain. We thus see that conclusive transfer is still possible with randomly coupled chains as long as the requirement (13) is met. This requirement will be further discussed and generalized in the next section.

**Figure 2.** The absolute values of the transition amplitudes f_N,1(t) and g_N,1(t) for two Heisenberg chains of length N = 10. The couplings strengths of both chains were chosen randomly from the interval [0.8J, 1.2J]. The circles show times where Bob can perform measurements without gaining information on α and β.
Download figure:
Standard Export PowerPoint slide

3. Arbitrarily perfect transfer

If the transfer was unsuccessful, the state (16) will evolve further, offering Bob further opportunities to receive Alice's message. For identical quantum chains, this must ultimately lead to a success for any reasonable Hamiltonian [20]. For differing chains, this is not necessarily the case, because measurements are only allowed at times where the probability amplitude at the end of the chains is equal, and there may be systems where this is never the case. In this section, we will develop a criterion that generalizes equation (13) and allows us to check numerically whether a given system is capable of arbitrarily perfect state transfer.

The quantity of interest for conclusive state transfer is the joint probability $\cal{P}(l)$ that after having checked l times, Bob still has not received the proper state at his ends of the chain. Optimally, this should approach zero if l tends to infinity. In order to derive an expression for $\cal{P}(l),$ let us assume that the transfer has been unsuccessful for l − 1 times with time intervals t_i between the ith and the (i − 1)th measurement, and calculate the probability of failure at the lth measurement. In a similar manner, we assume that all the l − 1 measurements have met the requirement of conclusive transfer (that is, Bob's measurements are unbiased with respect to α and β) and derive the requirement for the lth measurement.

To calculate the probability of failure for the lth measurement, we need to take into account that Bob's measurements disturb the unitary dynamics of the chain. If the state before a measurement with the outcome 'failure' is |ψ rang , the state after the measurement will be

$\begin{equation} \frac{1}{\sqrt{p_{l}}}Q|\psi\rangle, \end{equation}\tag{18}$

where Q is the projector

$\begin{equation} Q=I-|\mathbf{0,N}\rangle\langle \mathbf{0,N}|-|\mathbf{N,0}\rangle \langle \mathbf{N,0}| \end{equation}\tag{19}$

and p_l is the probability of failure at the lth measurement. The dynamics of the chain is alternating between unitary and projective, such that the state before the lth measurement is given by

$\begin{equation} \frac{1}{\sqrt{\cal{P}(l-1)}}\prod_{j=1}^{l}\{U(t_{j})Q\} \{\alpha|\mathbf{1,0}\rangle +\beta|\mathbf{0,1}\rangle\}, \end{equation}\tag{20}$

where we have used that

$\begin{equation} Q\{\alpha|\mathbf{1,0}\rangle +\beta|\mathbf{0,1}\rangle\} =\alpha|\mathbf{1,0}\rangle +\beta|\mathbf{0,1}\rangle \end{equation}\tag{21}$

for the first factor j = 1, and

$\begin{equation} \cal{P}(l-1)=\prod_{j=1}^{l-1}p_{j}. \end{equation}\tag{22}$

Note that the operators in (20) do not commute and that the time ordering of the product (the index j increases from right to left) is important. The probability that there is an excitation at the Nth site of either chain is given by

$\begin{equation} \frac{1}{\cal{P}(l-1)}\{|\alpha|^{2}|F(l)|^{2}+|\beta|^{2}|G(l)|^{2}\}, \end{equation}\tag{23}$

with

$\begin{equation} F(l)\equiv\langle \mathbf{N,0}|\prod_{j=1}^{l}\{U(t_{j})Q\} |\mathbf{1,0}\rangle \end{equation}\tag{24}$

and

$\begin{equation} G(l)\equiv\langle \mathbf{0,N}|\prod_{j=1}^{l}\{U(t_{j})Q\} |\mathbf{0,1}\rangle. \end{equation}\tag{25}$

Bob's measurements are therefore unbiased with respect to α and β if and only if

$\begin{equation} |F(l)|=|G(l)|\quad \forall l. \end{equation}\tag{26}$

In this case, the state can still be transferred conclusively (up to a known phase). The probability of failure at the lth measurement is given by

$\begin{equation} p_{l}=1-\frac{|F(l)|^{2}}{\cal{P}(l-1)}. \end{equation}\tag{27}$

We will show in the appendix that the condition (26) is equivalent to

$\begin{equation} \left\Vert \prod_{j=1}^{l}\{ U(t_{j})Q\} |\mathbf{1,0}\rangle \right\Vert =\left\Vert \prod_{j=1}^{l}\{U(t_{j})Q\} |\mathbf{0,1}\rangle \right\Vert \quad \forall l \end{equation}\tag{28}$

and that the joint probability of failure is simply

$\begin{equation} \cal{P}(l)=\left\Vert \prod_{j=1}^{l+1}\{U(t_{j})Q\} |\mathbf{1,0}\rangle \right\Vert ^{2}. \end{equation}\tag{29}$

It may look as if equation (28) was a complicated multi-time condition for the measuring times t_i, that becomes increasingly difficult to fulfil with a growing number of measurements. This is not the case. If proper measuring times have been found for the first l − 1 measurements, a trivial time t_l that fulfils equation (28) is t_l = 0. In this case, Bob measures immediately after the (l − 1)th measurement and the probability amplitudes on his ends of the chains will be equal—and zero (a useless measurement). But since the left- and right-hand sides of equation (28) when seen as functions of t_l are both quasi-periodic functions with initial value zero, it is likely that they intersect many times, unless the system has some specific symmetry or the systems are completely different. Furthermore, for the lth measurement, equation (28) is equivalent to

$\begin{equation} \left\Vert QU(t_{l})|\xi\rangle \right\Vert \equiv\left\Vert QU(t_{l})|\zeta\rangle \right\Vert \end{equation}\tag{30}$

with

$\begin{equation} |\xi\rangle =Q\prod_{j=1}^{l-1}U(t_{j})Q|\mathbf{1,0}\rangle \end{equation}\tag{31}$

and

$\begin{equation} |\zeta\rangle =Q\prod_{j=1}^{l-1}U(t_{j})Q|\mathbf{0,1}\rangle. \end{equation}\tag{32}$

From this we can see that if the system is ergodic, the condition for conclusive transfer is fulfilled at many different times.

Note that we do not claim at this point that any pair of chains will be capable of arbitrary perfect transfer. We will discuss in the next system how one can check this for a given system by performing some simple experimental tests.

4. Tomography

Suppose someone gives you two different experimentally designed spin chains. It may seem from the above that knowledge of the full Hamiltonian of both chains is necessary to check how well the system can be used for state transfer. This would be a very difficult task, because we would need access to all the spins along the channel to measure all the parameters of the Hamiltonian. In fact by expanding the projectors in equation (28) one can easily see that the only matrix elements of the evolution operator which are relevant for conclusive transfer are

$\begin{equation} f_{N,1}(t) = \langle \mathbf{N,0}|U(t)|\mathbf{1,0}\rangle, \end{equation}\tag{33}$

$\begin{equation} f_{N,N}(t) = \langle \mathbf{N,0}|U(t)|\mathbf{N,0}\rangle, \end{equation}\tag{34}$

$\begin{equation} g_{N,1}(t) =\langle \mathbf{0,N}|U(t)|\mathbf{0,1}\rangle, \end{equation}\tag{35}$

$\begin{equation} g_{N,N}(t) = \langle \mathbf{0,N}|U(t)|\mathbf{0,N}\rangle. \end{equation}\tag{36}$

Physically, this means that the only relevant properties of the system are the transition amplitudes to 'arrive' at Bob's ends and to 'stay' there. The modulus of f_N,1(t) and f_N,N(t) can be measured by initializing the system in the states |1,0 rang and |N,0 rang and then performing a reduced density matrix tomography at Bob's site at different times t, and the complex phase of these functions is obtained by initializing the system in $(|\mathbf{0,0}\rangle +|\mathbf{1,0}\rangle)/\sqrt{2}$ and $(|\mathbf{0,0}\rangle +|\mathbf{N,0}\rangle)/\sqrt{2}$ instead. In the same way, g_N,1(t) and g_N,N(t) are obtained. All this can be done in the spirit of 'minimal control' at the sending and receiving ends of the chain only, and needs to be done only once. It is interesting to note that the dynamics in the middle part of the chain is not relevant at all. It is a 'black box' that may involve even completely different interactions, number of spins, etc (see figure 3). Once the transition amplitudes [equations (33)–(36)] are known, one can search numerically for optimized measurement times t_i using equation (29) and the condition from equation (28).

**Figure 3.** The relevant properties for conclusive transfer can be determined by measuring the response of the two systems at their ends only.
Download figure:
Standard Export PowerPoint slide

One weakness of the scheme described here is that the times at which Bob measures have to be very precise, because otherwise the measurements will not be unbiased with respect to α and β. This demand can be relaxed by measuring at times where not only the probability amplitudes are similar, but also their slope (see figure 2). The computation of these optimal timings for a given system may be complicated, but they only need to be done once.

5. Numerical examples

In this section, we show some numerical examples for two chains with Heisenberg couplings J which are fluctuating. The Hamiltonians of the chains are given by

$\begin{equation} H^{(1)} = \sum_{n=1}^{N-1}J(1+\delta_{n}^{(1)})\vec{\sigma}_{n}^{(1)}\cdot\vec{\sigma}_{n+1}^{(1)}, \end{equation}\tag{37}$

$\begin{equation} H^{(2)} = \sum_{n=1}^{N-1}J(1+\delta_{n}^{(2)})\vec{\sigma}_{n}^{(2)}\cdot\vec{\sigma}_{n+1}^{(2)}, \end{equation}\tag{38}$

where δ_n⁽ⁱ⁾ are uniformly distributed random numbers from the interval [−Δ, Δ]. We have considered two different cases: in the first case, the δ_n⁽ⁱ⁾ are completely uncorrelated (i.e. independent for both chains and all sites along the chain); and in the second case, we have taken into account a spatial correlation of the signs of δ_n⁽ⁱ⁾ along each of the chains, while still keeping the two chains uncorrelated. For both cases, we find that arbitrarily perfect transfer remains possible except for some very rare realizations of the δ_n⁽ⁱ⁾.

Because measurements must only be taken at times which fulfil the condition (28), and these times usually do not coincide with the optimal probability of finding an excitation at the ends of the chains, it is clear that the probability of failure at each measurement will on average be higher than for chains without fluctuations. Therefore, more measurements have to be performed in order to achieve the same probability of success. The price for noisy couplings is thus a longer transmission time and a higher number of gating operations at the receiving end of the chains. Some averaged values are given in table 1 for the Heisenberg chain with uncorrelated coupling fluctuations.

Table 1. The total time t and the number of measurements M needed to achieve a probability of success of 99% for different fluctuation strengths Δ (uncorrelated case). Given is the statistical mean ± SD. The length of the chain is N = 20 and the number of random samples is 10. For strong fluctuations Δ = 0.1, we also found particular samples where the success probability could not be achieved within the time range searched by the algorithm.

	Δ = 0	Δ = 0.01	Δ = 0.03	Δ = 0.05	Δ = 0.1
$t\lbrack\frac{\hbar}{J}\rbrack$	377	524 ± 27	694 ± 32	775 ± 40	1106 ± 248
M	28	43 ± 3	58 ± 3	65 ± 4	110 ± 25

For the case where the signs of δ_n⁽ⁱ⁾ are correlated, we have used the same model as in [25], introducing the parameter c such that

$\begin{equation} \delta_{n}^{(i)}\delta_{n-1}^{(i)}>0\quad {\rm with\, probability}\, c \end{equation}\tag{39}$

and

$\begin{equation} \delta_{n}^{(i)}\delta_{n-1}^{(i)}\lt 0\quad {\rm with\, probability} 1-c. \end{equation}\tag{40}$

For c = 1 (c = 0) this corresponds to the case where the signs of the couplings are completely correlated (anti-correlated). For c = 0.5 one recovers the case of uncorrelated couplings. We can see from the numerical results in table 2 that arbitrarily perfect transfer is possible for the whole range of c.

Table 2. The total time t and the number of measurements M needed to achieve a probability of success of 99% for different correlations c between the couplings (see equations (39) and (40)). Given is the statistical mean±SD for a fluctuation strength of Δ = 0.05. The length of the chain is N = 20 and the number of random samples is 20.

	c = 0	c = 0.1	c = 0.3	c = 0.7	c = 0.9	c = 1
$t[\frac{\hbar}{J}]$	666 ± 20	725 ± 32	755 ± 41	797 ± 35	882 ± 83	714 ± 41
M	256 ± 2	62 ± 3	65 ± 4	67 ± 4	77 ± 7	60 ± 4

For Δ = 0, we know from [19] that the time to transfer a state with probability of failure P scales as

$\begin{equation} t(P)=\frac{0.33\hbar N^{1.6}}{J}|\ln\,P|. \end{equation}\tag{41}$

If we want to obtain a similar formula in the presence of noise, we can perform a fit to the exact numerical data. For uncorrelated fluctuations of Δ = 0.05, this is shown in figure 4. The best fit is given by

$\begin{equation} t(P)=\frac{0.2\hbar N^{1.9}}{J}|\ln P|. \end{equation}\tag{42}$

**Figure 4.** Time t needed to transfer a state with a given joint probability of failure P across a chain of length N with uncorrelated fluctuations of Δ = 0.05 and without fluctuations (Δ = 0 ). The points denote exact numerical data, and the fits are given by equations (41) and (42).
Download figure:
Standard Export PowerPoint slide

We conclude that weak fluctuations (say up to 5%) in the coupling strengths do not deteriorate the performance of our scheme much for the chain lengths considered. Both the transmission time and the number of measurements raise, but still in a reasonable way (cf table 1 and equation (4)). For larger fluctuations, the scheme is still applicable in principle, but the amount of junk (i.e. chains not capable of arbitrary perfect transfer) may get too large.

Note that we have considered the case where the fluctuations δ_nⁱ are time-independent. This is a reasonable assumption if the dynamic fluctuations (e.g. those arising from thermal noise) can be neglected with respect to the constant fluctuations (e.g. those arising from manufacturing errors). If the fluctuations were varying with time, the tomography measurements in section 4 would involve a time-average, and Bob would not measure at exactly the correct times. The transferred state (17) would then be affected by both phase and amplitude noise.

6. Conclusions

We have shown that in many cases, it is possible to perfectly transfer an unknown quantum state along a pair of quantum chains even if their coupling is to some amount random. This is achieved by using a dual-rail encoding combined with measurements at the receiving end of the chains. Since any scheme for quantum communication will suffer from some imperfections when implemented, the dual-rail is a powerful tool to overcome the decrease of fidelity.

Acknowledgments

This work was supported by the UK Engineering and Physical Sciences Research Council through the grant GR/S62796/01 and the QIPIRC.

Appendix

We first use equations (22) and (27) to obtain

$\begin{equation} \cal{P}(l)=\cal{P}(l-1)-|F(l)|^{2} \end{equation}\tag{A.1}$

or using $\cal{P}(0)=1$ ,

$\begin{equation} \cal{P}(l) = 1-\sum_{k=1}^{l}|F(k)|^{2}. \end{equation}\tag{A.2}$

If we introduce

$\begin{equation} v(l) \equiv \left\Vert \prod_{j=1}^{l}\{U(t_{j})Q\} |\mathbf{1,0}\rangle \right\Vert ^{2},\end{equation}\tag{A.3}$

$\begin{equation} w(l) \equiv \left\Vert \prod_{j=1}^{l}\{U(t_{j})Q\} |\mathbf{0,1}\rangle \right\Vert^{2}, \end{equation}\tag{A.4}$

then we can write

$\begin{eqnarray} |F(l)|^{2} &=& \langle \mathbf{1,0}|\left(\prod_{j=1}^{l}\{ U(t_{j})Q\} \right)^{\dagger}|\mathbf{N,0}\rangle \langle \mathbf{N,0}|\prod_{j=1}^{l}\{U(t_{j})Q\}|\mathbf{1,0}\rangle \nonumber \\ &=& v(l)-\langle \mathbf{1,0}|\left(\prod_{j=1}^{l}\{ U(t_{j})Q\}\right)^{\dagger}Q\prod_{j=1}^{l}\{U(t_{j})Q\} |\mathbf{1,0}\rangle \\ &=& v(l)-\left\Vert \prod_{j=1}^{l}Q\{U(t_{j})Q\} |\mathbf{1,0}\rangle \right\Vert^{2} =v(l)-v(l+1),\nonumber \end{eqnarray}\tag{A.5}$

where we have used

$\begin{equation} |\mathbf{N,0}\rangle \langle \mathbf{N,0}| = 1-Q-|\mathbf{0,N}\rangle \langle \mathbf{0,N}|, \end{equation}\tag{A.6}$

$\begin{equation} Q^{2} = Q, \end{equation}\tag{A.7}$

and the fact that the unitary matrix U(t_l+1) does not change the norm of a vector. The same calculation can be done for |G(l)|² such that

$\begin{eqnarray} &&|G(l)|^{2} = w(l)-w(l+1). \end{eqnarray}\tag{A.8}$

If we want to have |F| and |G| matching for all l, we need to have

$\begin{equation} w(l)=v(l)\quad \forall l \end{equation}\tag{A.9}$

$\begin{equation} \left\Vert \prod_{j=1}^{l}\{U(t_{j})Q\} |\mathbf{1,0}\rangle \right\Vert =\left\Vert \prod_{j=1}^{l}\{U(t_{j})Q\} |\mathbf{0,1}\rangle \right\Vert \quad \forall l. \end{equation}\tag{A.10}$

The joint probability of failure is given by

$\begin{equation} \cal{P}(l) = 1-\sum_{k=1}^{l}|F(k)|^{2} \end{equation}\tag{A.11}$

$\begin{equation} \hphantom{\cal{P}(l)}= 1-\sum_{k=1}^{l}v(k)-v(k+1) \end{equation}\tag{A.12}$

$\begin{equation} \hphantom{\cal{P}(l)} = v(l+1) \end{equation}\tag{A.13}$

$\begin{equation} \hphantom{\cal{P}(l)}= \left\Vert \prod_{j=1}^{l+1}Q\{U(t_{j})Q\} |\mathbf{1,0}\rangle \right\Vert^{2}. \end{equation}\tag{A.14}$

References

[1]
Bose S 2003 Phys. Rev. Lett. 91 207901
Crossref PubMed
[2]
Subrahmanyam V 2004 Phys. Rev. A 69 034304
Crossref
[3]
Romito A, Fazio R and Bruder C 2005 Phys. Rev. B 71 100501
Crossref
[4]
Giovannetti V and Fazio R 2005 Phys. Rev. A 71 032314
Crossref
[5]
Christandl M, Datta N, Ekert A and Landahl A J 2004 Phys. Rev. Lett. 92 187902
Crossref PubMed
[6]
Albanese C, Christandl M, Datta N and Ekert A 2004 Phys. Rev. Lett. 93 230502
Crossref PubMed
[7]
Christandl M, Datta N, Dorlas T C, Ekert A, Kay A and Landahl A J 2004 Preprint quant-ph/0411020
Preprint
[8]
Yung M H, Leung D W and Bose S 2004 Quant. Inform. Comput. 4 174
[9]
Yung M H and Bose S 2005 Phys. Rev. A 71 032310
Crossref
[10]
Karbach P and Stolze J 2005 Preprint quant-ph/0501007
Preprint
[11]
Nikolopoulos G M, Petrosyan D and Lambropoulos P 2004 J. Phys.: Condens. Matter 16 4991
IOPscience
[12]
Nikolopoulos G M, Petrosyan D and Lambropoulos P 2004 Europhys. Lett. 65 297
IOPscience
[13]
Osborne T J and Linden N 2004 Phys. Rev. A 69 052315
Crossref
[14]
Haselgrove H L 2004 Preprint quant-ph/0404152
Preprint
[15]
Plenio M B and Semiao F L 2005 New. J. Phys. 7 73
IOPscience
[16]
Li Y, Shi T, Chen B, Song Z and Sun C P 2005 Phys. Rev. A 71 022301
Crossref
[17]
Shi T, Li Y, Song Z and Sun C P 2004 Preprint quant-ph/0408152
Preprint
[18]
Song Z and Sun C P 2004 Preprint quant-ph/0412183
Preprint
[19]
Burgarth D and Bose S 2005 Phys. Rev. A 71 052315
Crossref
[20]
Burgarth D, Bose S and Giovannetti V 2004 Preprint quant-ph/0410175
Preprint
[21]
Paternostro M, Palma G M, Kim M S and Falci G 2004 Preprint quant-ph/0407058
Preprint
[22]
Amico L, Osterloh A, Plastina F, Fazio R and Palma G M 2004 Phys. Rev. A 69 022304
Crossref
[23]
Plenio M B, Hartley J and Eisert J 2004 New J. Phys. 6 36
IOPscience
[24]
Eisert J, Plenio M B, Bose S and Hartley J 2004 Phys. Rev. Lett. 93 190402
Crossref PubMed
[25]
De Chiara G, Rossini D, Montangero S and Fazio R 2005 Preprint quant-ph/0502148
Preprint
[26]
Anderson P W 1958 Phys. Rev. 109 1492
Crossref
[27]
Kretschmann D and Werner R F 2005 Preprint quant-ph/0502106
Preprint

Export references: BibTeX RIS

Citations

Localization properties and high-fidelity state transfer in hopping models with correlated disorder
G.M.A. Almeida et al 2018 Annals of Physics
Crossref
Almost perfect transport of an entangled two-qubit state through a spin chain
Rafael Vieira and Gustavo Rigolin 2018 Physics Letters A 382 2586
Crossref
Photon-assisted quantum state transfer and entanglement generation in spin chains
A. Gratsea et al 2018 Physical Review A 98
Crossref
Well-protected quantum state transfer in a dissipative spin chain
Naghi Behzadi et al 2018 Scientific Reports 8
Crossref
Quantum-state transfer through long-range correlated disordered channels
Guilherme M.A. Almeida et al 2018 Physics Letters A
Crossref
Perfect coding for dephased quantum state transfer
Alastair Kay 2018 Physical Review A 97
Crossref
Disorder-assisted distribution of entanglement in XY spin chains
Guilherme M. A. Almeida et al 2017 Physical Review A 96
Crossref
Tailoring Spin Chain Dynamics for Fractional Revivals
Alastair Kay 2017 Quantum 1 24
Crossref
Generating quantum states through spin chain dynamics
Alastair Kay 2017 New Journal of Physics 19 043019
IOPscience
Non-adiabatic quantum state preparation and quantum state transport in chains of Rydberg atoms
Maike Ostmann et al 2017 New Journal of Physics 19 123015
IOPscience
Designing spin-channel geometries for entanglement distribution
E. K. Levi et al 2016 Physical Review A 94
Crossref
Time-independent quantum circuits with local interactions
Sahand Seifnashri et al 2016 Physical Review A 93
Crossref
Quantum state transfer in optomechanical arrays
G. D. de Moraes Neto et al 2016 Physical Review A 93
Crossref
Permutation-invariant codes encoding more than one qubit
Yingkai Ouyang and Joseph Fitzsimons 2016 Physical Review A 93
Crossref
Quantum error correction for state transfer in noisy spin chains
Alastair Kay 2016 Physical Review A 93
Crossref
High Fidelity Symmetric Telecloning and Entanglement Distribution of Spin Quantum States by Weak Measurement and Reversal
Qiong Wang et al 2016 International Journal of Theoretical Physics
Crossref
Transferring information through a mixed-five-spin chain channel
Hamid Arian Zad and Hossein Movahhedian 2016 Chinese Physics B 25 080307
IOPscience
Robust quantum state transfer between two superconducting qubits via partial measurement
Yan-Ling Li et al 2016 Laser Physics Letters 13 125202
IOPscience
Time independent universal computing with spin chains: quantum plinko machine
K F Thompson et al 2016 New Journal of Physics 18 073044
IOPscience
Quantum state transfer in a disordered one-dimensional lattice
S. Ashhab 2015 Physical Review A 92
Crossref
Quantum efficiencies in finite disordered networks connected by many-body interactions
Adrian Ortega et al 2015 Annalen der Physik n/a
Crossref
Parity-based mirror inversion for efficient quantum state transfer and computation in nearest-neighbor arrays
P. Kumar and S. Daraeizadeh 2015 Physical Review A 91
Crossref
Quantum State Transfer in Atom-Cavity Systems with Uncolored Cayley Interacting Networks
R. Sufiani 2015 International Journal of Theoretical Physics 54 185
Crossref
Measurement-assisted quantum communication in spin channels with dephasing
Abolfazl Bayat and Yasser Omar 2015 New Journal of Physics 17 103041
IOPscience
Atomic and Photonic Entanglement Generation in n Coupled Atom-Cavity Systems
R. Sufiani and A. Darkhosh 2014 International Journal of Theoretical Physics
Crossref
Universal scheme for finite-probability perfect transfer of arbitrary multispin states through spin chains
Zhong-Xiao Man et al 2014 Annals of Physics 351 739
Crossref
Controllable entanglement transfer via two parallel spin chains
Zhong-Xiao Man et al 2014 Physics Letters A
Crossref
Excitation and state transfer through spin chains in the presence of spatially correlated noise
Jan Jeske et al 2013 Physical Review A 88
Crossref
Optimally Designed Quantum Transport across Disordered Networks
Mattia Walschaers et al 2013 Physical Review Letters 111
Crossref
Quantum-state transfer via resonant tunneling through local-field-induced barriers
S. Lorenzo et al 2013 Physical Review A 87
Crossref
Quantum Tasks Using Symmetric Cluster States
Ming-Xing Luo 2013 International Journal of Theoretical Physics
Crossref
Colored channels for high-fidelity information transfer and processing between remote multi-branch quantum circuits
G. D. de Moraes Neto et al 2013 EPL (Europhysics Letters) 103 43001
IOPscience
TRANSPORT OF QUANTUM CORRELATIONS ACROSS A SPIN CHAIN
TONY J. G. APOLLARO et al 2012 International Journal of Modern Physics B 1345035
Crossref
Number-Theoretic Nature of Communication in Quantum Spin Systems
Chris Godsil et al 2012 Physical Review Letters 109
Crossref
High fidelity and flexible quantum state transfer in the atom-coupled cavity hybrid system
B. F. C. Yabu-uti and J. A. Roversi 2012 Quantum Information Processing
Crossref
Dynamical Gaussian state transfer with quantum-error-correcting architecture
Go Tajimi and Naoki Yamamoto 2012 Physical Review A 85 022303
Crossref
Production of high-quality metal-coated pyroelectric SrTiO3 films
A. Yoffe et al 2012 Technical Physics 57 134
Crossref
Concurrence in disordered systems
Jenny Hide 2012 Journal of Physics A: Mathematical and Theoretical 45 115302
IOPscience
Quantum discord and classical correlations in the bond-charge Hubbard model: Quantum phase transitions, off-diagonal long-range order, and violation of the monogamy property for discord
Michele Allegra et al 2011 Physical Review B 84 245133
Crossref
Engineering interactions for quasiperfect transfer of polariton states through nonideal bosonic networks of distinct topologies
G. de Moraes Neto et al 2011 Physical Review A 84 032339
Crossref
Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework
S. Machnes et al 2011 Physical Review A 84 022305
Crossref
Perfect state transfer over interacting boson networks associated with group schemes
Mohamad Ali Jafarizadeh et al 2011 Quantum Information Processing
Crossref
Impurity entanglement in the J–J₂–δ quantum spin chain
Andreas Deschner and Erik S Sørensen 2011 Journal of Statistical Mechanics: Theory and Experiment 2011 P10023
IOPscience ADS
Generating a GHZ state in 2m-qubit spin network
M A Jafarizadeh et al 2011 Journal of Statistical Mechanics: Theory and Experiment 2011 P05014
IOPscience
From decoherence-free channels to decoherence-free and quasi-free subspaces within bosonic dissipative networks
G D M Neto et al 2011 Journal of Physics B: Atomic, Molecular and Optical Physics 44 145502
IOPscience
Quantum Entanglement Channel based on Excited States in a Spin Chain
Zhang Shao-Liang et al 2011 Chinese Physics Letters 28 120301
IOPscience
Communication at the quantum speed limit along a spin chain
Michael Murphy et al 2010 Physical Review A 82 022318
Crossref
Large effects of boundaries on spin amplification in spin chains
Benoit Roubert et al 2010 Physical Review A 82 022302
Crossref
Modelling the electric field applied to a tokamak
R. W. Johnson 2010 The European Physical Journal D
Crossref ADS
State transfer in dissipative and dephasing environments
M. L. Hu 2010 The European Physical Journal D 59 497
Crossref ADS
Quantum entanglement in photosynthetic light-harvesting complexes
Mohan Sarovar et al 2010 Nature Physics
Crossref
Quasi-perfect state transfer in a bosonic dissipative network
A Cacheffo et al 2010 Journal of Physics B: Atomic, Molecular and Optical Physics 43 105503
IOPscience
Quantum system under the actions of two counteracting baths: A model for the attenuation-amplification interplay
F. Lorenzen et al 2009 Physical Review A 80 062103
Crossref
Quantum-state transfer on spin-chain channels with random imperfections
D. X. Kong and A. M. Wang 2009 The European Physical Journal D
Crossref
State transfer in intrinsic decoherence spin channels
M. L. Hu and H. L. Lian 2009 The European Physical Journal D
Crossref
Perfect state transfer without state initialization and remote collaboration
Marcin Markiewicz and Marcin Wieśniak 2009 Physical Review A 79 054304
Crossref
Quantum Communication beyond the Localization Length in Disordered Spin Chains
Jonathan Allcock and Noah Linden 2009 Physical Review Letters 102 110501
Crossref
Branching spin chain dynamics
Irene D’Amico et al 2009 Journal of Magnetism and Magnetic Materials 321 949
Crossref
Entanglement entropy in quantum impurity systems and systems with boundaries
Ian Affleck et al 2009 Journal of Physics A: Mathematical and Theoretical 42 504009
IOPscience
Temperature effects on a network of dissipative quantum harmonic oscillators: collective damping and dispersion processes
M A de Ponte et al 2009 Journal of Physics A: Mathematical and Theoretical 42 365304
IOPscience
Perfect state transfer of a qudit over underlying networks of group association schemes
M A Jafarizadeh et al 2009 Journal of Statistical Mechanics: Theory and Experiment 2009 P04004
IOPscience
Heisenberg chains cannot mirror a state
Marcin Wieśniak 2008 Physical Review A 78 052334
Crossref
Creating and preserving multi-partite entanglement with spin chains
Irene D'Amico et al 2008 physica status solidi (c) 5 2481
Crossref
Perfect state transfer over distance-regular spin networks
M. A. Jafarizadeh and R. Sufiani 2008 Physical Review A 77 022315
Crossref
Adiabatic quantum transport in a spin chain with a moving potential
Vinitha Balachandran and Jiangbin Gong 2008 Physical Review A 77 012303
Crossref
Conclusive and perfect quantum state transfer with a single spin chain
Jiankui He et al 2008 Physics Letters A 372 185
Crossref
Optimal transfer of a d-level quantum state over pseudo-distance-regular networks
M A Jafarizadeh et al 2008 Journal of Physics A: Mathematical and Theoretical 41 475302
IOPscience
Trapped Rydberg ions: from spin chains to fast quantum gates
Markus Müller et al 2008 New Journal of Physics 10 093009
IOPscience
Symmetric Telecloning and Entanglement Distribution of Spin Quantum States
Wang Qiong et al 2008 Chinese Physics Letters 25 2770
IOPscience
Freezing distributed entanglement in spin chains
Brendon W. Lovett et al 2007 Physical Review A 76 030302
Crossref
Bounds on the Speed of Information Propagation in Disordered Quantum Spin Chains
Christian K. Burrell and Tobias J. Osborne 2007 Physical Review Letters 99 167201
Crossref ADS
Information-flux approach to multiple-spin dynamics
G. M. Palma et al 2007 Physical Review A 76 042316
Crossref
Role of interference in quantum state transfer through spin chains
D. Braun et al 2007 Physical Review A 76 022321
Crossref
Quantum communication via a continuously monitored dual spin chain
Daniel Burgarth et al 2007 Physical Review A 75 062328
Crossref
Optimal quantum-chain communication by end gates
Vittorio Giovannetti et al 2007 Physical Review A 75 062327
Crossref
Perfect state transfer in networks of arbitrary topology and coupling configuration
G. M. Nikolopoulos et al 2007 Physical Review A 75 042319
Crossref
Spin chains with electrons in Penning traps
I. Marzoli et al 2007 Physical Review A 75 032348
Crossref
Multiuser quantum communication networks
Tomasz Gdala et al 2007 Physical Review A 75 022330
Crossref
Relation between phase-space coverage and entanglement for spin- 1∕2 systems
Stefan Schenk and Gert-Ludwig Ingold 2007 Physical Review A 75 022328
Crossref ADS APS Article
Fractional revivals of the quantum state in a tight-binding chain
Z. Song et al 2007 Physical Review A 75 012113
Crossref
Unifying Quantum State Transfer and State Amplification
Alastair Kay 2007 Physical Review Letters 98 010501
Crossref
INFORMATION TRANSFER RATES IN SPIN QUANTUM CHANNELS
DAVIDE ROSSINI et al 2007 International Journal of Quantum Information 05 439
Crossref
Quantum communication through spin chain dynamics: an introductory overview
Sougato Bose 2007 Contemporary Physics 48 13
Crossref
Quantum impurity entanglement
Erik S Sørensen et al 2007 Journal of Statistical Mechanics: Theory and Experiment 2007 P08003
IOPscience
The generalized Lyapunov theorem and its application to quantum channels
Daniel Burgarth and Vittorio Giovannetti 2007 New Journal of Physics 9 150
IOPscience
Use of dynamical coupling for improved quantum state transfer
A. O. Lyakhov and C. Bruder 2006 Physical Review B 74 235303
Crossref
Improved Transfer of Quantum Information Using a Local Memory
Vittorio Giovannetti and Daniel Burgarth 2006 Physical Review Letters 96
Crossref ADS APS Article
Quantum communication in spin systems with long-range interactions
A. J. Fisher et al 2006 Physical Review A 74 012321
Crossref ADS APS Article
Perfect state transfer: Beyond nearest-neighbor couplings
Alastair Kay 2006 Physical Review A 73 032306
Crossref ADS APS Article
EFFICIENT AND PERFECT STATE TRANSFER IN QUANTUM CHAINS
DANIEL BURGARTH et al 2006 International Journal of Quantum Information 04 405
Crossref
Universal destabilization and slowing of spin-transfer functions by a bath of spins
Daniel Burgarth and Sougato Bose 2006 Physical Review A 73 062321
Crossref ADS APS Article
Globally Controlled Quantum Wires for Perfect Qubit Transport, Mirroring, and Computing
Fitzsimons, Joseph and Twamley, Jason 2006 Physical Review Letters 97
ADS APS Article
Excitation and entanglement transfer versus spectral gap
M J Hartmann et al 2006 New Journal of Physics 8 94
IOPscience
Controlling Quantum State Transfer in Spin Chain with Confined Field
Chen Bing and Song Zhi 2006 Communications in Theoretical Physics 46 749
IOPscience
Superballistic diffusion of entanglement in disordered spin chains
J. Fitzsimons and J. Twamley 2005 Physical Review A 72 050301
Crossref ADS APS Article
Unmodulated spin chains as universal quantum wires
Tomasz Gdala et al 2005 Physical Review A 72 034303
Crossref ADS APS Article
Entanglement creation and distribution on a graph of exchange-coupled qutrits
Alessio Serafini et al 2005 Physical Review A 72 052333
Crossref ADS APS Article
Spin chains as perfect quantum state mirrors
Peter Karbach and Joachim Stolze 2005 Physical Review A 72 030301
Crossref ADS APS Article
Arbitrarily perfect quantum communication using unmodulated spin chains, a collaborative approach
B Vaucher et al 2005 Journal of Optics B: Quantum and Semiclassical Optics 7 S356
IOPscience
Quantum state transfer in arrays of flux qubits
A Lyakhov and C Bruder 2005 New Journal of Physics 7 181
IOPscience
Geometric effects and computation in spin networks
Alastair Kay and Marie Ericsson 2005 New Journal of Physics 7 143
IOPscience
Efficient and perfect state transfer in quantum chains
Daniel Burgarth et al 2005 Journal of Physics A: Mathematical and General 38 6793
IOPscience

Export citations: BibTeX RIS