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Wang et al. first studied hierarchical quantum information splitting of an arbitrary single-qubit state via the $\chi$ state as the entangled channel. There exists a hierarchy among the three receivers as far as the power to recover the teleported state is concerned. But the scheme is considered in ideal environment. In this paper, we reinvestigate the scheme in amplitude-damping and phase-damping noises. The fidelity and average fidelity are adopted to quantify the effect of noise. It is found that they are both dependent on the coefficients of the teleported state and the noise parameter. Moreover, we put forward a novel deterministic scheme to realize hierarchical controlled remote preparation of an arbitrary single-qubit state. Comparing with the previous scheme via the $\chi$ state, the sender does not need to perform information dividing due to the subtly constructed measurement basis. We also consider the proposed scheme under noisy environment.

Thermal entanglement of a two-qubit Heisenberg XXZ chain in the presence of different Dzyaloshinskii–Moriya (DM) anisotropic antisymmetric interactions and entanglement teleportation using two independent Heisenberg chains as quantum channel are investigated. It is found that the anisotropic coupling coefficient J_z (z-component exchange constant) and DM interactions can excite the entanglement, and x-component DM interaction D_x has a more remarkable influence than the z-axis DM interaction D_z for exchange constant J > 0 (antiferromagnetic) case, which is contrary to coupling coefficient J < 0 (ferromagnetic) case. The output entanglement and fidelity increase with increasing z-axis coupling parameter J_z for both the antiferromagnetic J > 0 case and ferromagnetic J < 0 case. By introducing the different DM interactions, it is superior to teleportate initial state by using the D_z interaction in the model for J > 0 case. But for J < 0 case, the D_x interaction is better, and the output concurrence and fidelity can reach the maximum value 1 with suitable D_x.

In this paper, the technique of weak measurement is employed in order to enhance the fidelity of joint remote state preparation (JRSP) protocol under decoherence. We design a quantum circuit of joint remote preparation of a single-qubit state through a GHZ state. Then, we analytically derive the average fidelity of the JRSP protocol under four types of noise usually encountered in real-world implementations of quantum communication protocols, i.e. the phase-flip, depolarizing, amplitude-damping and bit-flip noise. Our study shows that the application of weak measurement and measurement reversal could enhance the average fidelity of the JRSP process under the influence of phase-flip, depolarizing, amplitude-damping noise for most values of the decoherence parameters. If the JRSP process suffers from the bit-flip noise, the weak measurement technique is not useful for increasing the average fidelity.

The protocol of the teleportation of an arbitrary unknown one-qubit state is investigated when the quantum channel is subject to the decoherence from the non-Markovian environments with the memory effects. The quality of the teleportation measured by the average fidelity and minimal fidelity can be enhanced in the non-Markovian channel. The memory effects give rise to the revival of the fidelity which is better than that obtained by the Markovian channel.

Randomized benchmarking (RB) is a popular procedure used to gauge the performance of a set of gates useful for quantum information processing (QIP). Recently, Proctor et al. [Phys. Rev. Lett. 119 (2017) 130502] demonstrated a practically relevant example where the RB measurements give a number $r$, very different from the actual average gate-set infidelity $ϵ$, despite past theoretical assurances that the two should be equal. Here, we derive formulas for $ϵ$, and for $r$ from the RB protocol, in a manner permitting easy comparison of the two. We show in general that, indeed, $r\ne ϵ$, i.e. RB does not measure average infidelity, and, in fact, neither one bounds the other. We give several examples, all plausible in real experiments, to illustrate the differences in $ϵ$ and $r$. Many recent papers on experimental implementations of QIP have claimed the ability to perform high-fidelity gates because they demonstrated small $r$ values using RB. Our analysis shows that such a statement from RB alone has to be interpreted with caution.

We review the concept of the quasi-inverse of qubit channels and of higher dimensional channels. Quasi-inverse is a channel which when concatenaded to the original channel, increases its average fidelity in an optimal way. For qubit channels, we fully characterize the quasi-inverse, while for higher dimensional channels, we prove general theorems and provide bounds for the increased average fidelity. Nevertheless, explicit examples are given when exact quasi-inverses can be found.