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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Robustness of multiqubit entanglement

Leonid Evgenevich Fedichkin, Arkady Fedorov, Vladimir Privman A

Published May 5, 2006• 3 citations
Publisher

Abstract

We survey results on the decay of multiqubit entanglement due to internal interactions between qubits. Dipole-dipole interaction induces decoherence of strongly entangled nuclear spins. The dynamics of spin clusters can be described as quantum decoherence due to an effective composite bath consisting of fully correlated and uncorrelated parts. The rate of decoherence scales up as a square root of the number of entangled spins, resulting in linear scaling of a measure of quantum noise. Our theory is consistent with a recent experiment.

DOI
10.1117/12.663366

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We survey results on the decay of multiqubit entanglement due to internal interactions between qubits. Dipole-dipole interaction induces decoherence of strongly entangled nuclear spins. The dynamics of spin clusters can be described as quantum decoherence due to an effective composite bath consisting of fully correlated and uncorrelated parts. The rate of decoherence scales up as a square root of the number of entangled spins, resulting in linear scaling of a measure of quantum noise. Our theory is consistent with a recent experiment.
The problem of dipole-dipole decoherence of nuclear spins is considered for strongly entangled spin cluster. Our results show that its dynamics can be described as the decoherence due to interaction with a composite bath consisting of fully correlated and uncorrelated parts. The correlated term causes the slower decay of coherence at larger times. The decoherence rate scales up as a square root of the number of spins giving the linear scaling of the resulting error. Our theory is consistent with recent experiment reported in decoherence of correlated spin clusters.
The problem of dipole-dipole decoherence of nuclear spins is considered for strongly entangled spin cluster. Our results show that its dynamics can be described as the decoherence due to interaction with a composite bath consisting of fully correlated and uncorrelated parts. The correlated term causes the slower decay of coherence at larger times. The decoherence rate scales up as a square root of the number of spins giving the linear scaling of the resulting error. Our theory is consistent with recent experiment reported in decoherence of correlated spin clusters.
We consider pure dephasing of Bell states of electron spin qubits interacting with a sparse bath of nuclear spins. Using the newly developed two-qubit generalization of cluster-correlation expansion method, we calculate the spin-echo decay of $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ and $|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}$ states for various interqubit distances. Comparing the results with calculations in which dephasing of each qubit is treated independently, we identify signatures of influence of common part of the bath on the two qubits. At large interqubit distances, this common part consists of many nuclei weakly coupled to both qubits, so that decoherence caused by it can be modeled by considering multiple uncorrelated sources of noise (clusters of nuclei), each of them weakly affecting the qubits. Consequently, the resulting genuinely two-qubit contribution to decoherence can be described as being caused by classical Gaussian noise. On the other hand, for small interqubit distances the common part of the environment contains clusters of spins that are strongly coupled to both qubits, and their contribution to two-qubit dephasing has visibly non-Gaussian character. We show that one can easily obtain information about non-Gaussianity of environmental noise affecting the qubits from the comparison of dephasing of $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ and $|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}$ Bell states. Numerical results are obtained for two nitrogen vacancy centers interacting with a bath of 13C^{13}\mathrm{C}13C nuclei of natural concentration, for which we obtain that Gaussian description of a correlated part of environmental noise starts to hold for centers separated by about 3 nm.
It has recently been shown that interactions among the spins of a bath can result in an effective suppression of the decoherence of a single qubit embedded in a multi-spin environment. However, the effects of such intrabath interaction on the dynamics of entanglement are still not well understood. Here we study the dynamics of a system composed of two entangled qubits coupled to (1) independent self-interacting spin baths, and (2) a common self-interacting spin bath. We find that, as in the case of a single qubit, strong interactions among the spins in the bath may induce an effective decoupling between the bath and the system—this in turn suppresses the loss of entanglement that a non-, or weakly, interacting bath would typically cause. However we find that the intrabath interaction strength needed to achieve such effects is much larger than in the single qubit case, especially when the two qubits share the same environment. Our results illustrate the system's size effects in the entanglement sharing condition, where the entanglement within the bath limits the entanglement between the bath and the system.
We study decoherence of one, two, and nnn non-interacting qubits. Decoherence, measured in terms of purity, is calculated in linear response approximation, making use of the spectator configuration. The environment and its interaction with the qubits are modelled by random matrices. For two qubits, numerical studies reveal a simple one to one correspondence between its decoherence and its internal entanglement decay. Using this relation we are able to give a formula for concurrence decay. For large environments the evolution induces a unital channel in the two qubits, providing a partial explanation for the relation above. Using a kicked Ising spin network, we study the exact evolution of two non-interacting qubits in the presence of a spin bath. We find that the entanglement (as measured by concurrence) of the two qubits has a close relation to the purity of the pair, and closely follows an analytic relation derived for Werner states. As a collateral result we find that an integrable environment causes quadratic decay of concurrence as well as of purity, while a chaotic environment causes linear decay. Both quantities display recurrences in some integrable environments. Good agreement with the results found using random matrix theory is obtained. Finally, we analyze decoherence of a quantum register in the absence of non-local operations. The problem is solved in terms of a sum rule which implies linear scaling in the number of qubits. Each term involves a single qubit and its entanglement with the remaining ones. Two conditions are essential: decoherence must be small and the coupling of different qubits must be uncorrelated in the interaction picture. We apply the result to the random matrix model, and illustrate its reach considering a GHZ state coupled to a spin bath.
Among the most important parameters for the usefulness of quantum computers are the size of the quantum register and the decoherence time for the quantum information. The decoherence time is expected to get shorter with the number of correlated qubits, but experimental data are only available for small numbers of qubits. Solid-state nuclear magnetic resonance allows one to correlate large numbers of qubits (several hundred) and measure their decoherence rates. We use a modified magnetic dipole-dipole interaction to correlate the proton spins in a solid sample and observe the decay of the resulting highly correlated states. By systematically varying the number of correlated spins, we measure the increase of the decoherence rate with the size of the quantum register.