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Robust and efficient transport of two-qubit entanglement via disordered spin chains

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Abstract

We investigate how robust is the modified XX spin-1/2 chain of Vieira and 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.

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Acknowledgements

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.

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  1. Departamento de Física, Universidade Federal de São Carlos, São Carlos, SP, 13565-905, Brazil

    Rafael Vieira & Gustavo Rigolin

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  1. Rafael Vieira
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Correspondence to Gustavo Rigolin.

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A The 1000 + 2 qubits case

A The 1000 + 2 qubits case

Whenever we have dynamic or fluctuating disorders, specially for small values of \(\tau \), 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 \(\sigma ^z_i\sigma ^z_j\) interactions. Note that for less stringent disorder and noise scenarios, the \(N = 1000 + 2\) qubits case gives much better results.

Fig. 7
figure 7

For \(N = 100 + 2\) qubits, we have \(J_m/J=49.98\), the optimal setting for the clean system when \(J_m/J\le 50\), giving a transmitted entanglement of EoF \(= 0.98\) at the time \(Jt/\hbar = 20.53\) (square-black curve). For \(N = 1000 + 2\) qubits, we have \(J_m/J=298.27\), the optimal setting for the clean system when \(J_m/J\le 300\), with transmitted entanglement given by EoF \(= 0.99\) at the time \(Jt/\hbar = 118.55\) (circle-red curve). Solid curves are related to the proposed model when affected by fluctuating disorder and noise about those optimal settings, while the dotted curves refer to the optimal entanglement transmitted if we employ the clean standard model (strictly linear chain). The period \(\tau \) of changes in the Hamiltonian is shown in the figure (Color figure online)

Full size image

As expected, the results depicted in Figs. 6 and 7 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).

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Vieira, R., Rigolin, G. Robust and efficient transport of two-qubit entanglement via disordered spin chains. Quantum Inf Process 18, 135 (2019). https://doi.org/10.1007/s11128-019-2254-1

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