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A comparative study to analyze efficiencies of \((N+2)\)-qubit partially entangled states in real conditions from the perspective of N controllers

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Abstract

We address the efficiencies of \((N+2)\)-qubit partially entangled states for the involvement of N controllers in a noisy environment from the perspective of controller’s authority and average fidelity of a controlled teleportation protocol. For this, we design a generalized circuit using single- and two-qubit gates and study different cases of two sets of partially entangled multiqubit states. The analysis shows that for a particular set of partially entangled \((N+2)\)-qubit states average fidelity is independent of the state parameter and measurements performed by \(N-1\) controllers in ideal conditions and measurements performed by \(N-1\) controllers in noisy conditions with or without applications of weak measurements and its reversal operations. The results thus facilitate the experimental setups to worry about less number of parameters in the protocol. We further compare the efficiencies of these states from a controller’s perspective to increase his/her authority in the protocol. In addition, we also analyze dense coding protocol with and without the involvement of controllers to demonstrate the usefulness of partially entangled states.

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Acknowledgements

Authors would like to acknowledge IIT Jodhpur, MHRD and SERB (S/SERB/AKR/20180034) for providing the required infrastructure and funding.

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  1. Indian Institute of Technology Jodhpur, Jodhpur, India

    Jyoti Faujdar & Atul Kumar

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  1. Jyoti Faujdar
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Appendices

Appendices

A1 \(\mathbf {GGHZ}\) states as quantum channels with three and four controllers

The following circuit diagrams represent controlled teleportation using GGHZ states as quantum channels for N controllers as given in Fig. 2. Figures 16 and 17 show the circuit diagrams in case of three and four controllers, respectively.

Fig. 16
figure 16

A quantum circuit for controlled teleportation using the five-qubit GGHZ state as a quantum channel with three controllers

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Fig. 17
figure 17

A quantum circuit for controlled teleportation using the six-qubit GGHZ state as a quantum channel with four controllers

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A2 \({|{\Phi }\rangle }\) states as quantum channels with two, three and four controllers

The following circuit diagrams in Figs. 18, 19 and 20 represent controlled teleportation using \({|{\Phi }\rangle }\) states as quantum channels in case of two, three and four controllers, respectively.

Fig. 18
figure 18

A quantum circuit for controlled teleportation using the four-qubit \({|{\Phi }\rangle }\) state as a quantum channel with two controllers

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Fig. 19
figure 19

A quantum circuit for controlled teleportation using the five-qubit \({|{\Phi }\rangle }\) state as a quantum channel with three controllers

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Fig. 20
figure 20

A quantum circuit for controlled teleportation using the six-qubit \({|{\Phi }\rangle }\) state as a quantum channel with four controllers

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A3 \(\mathbf {GGHZ}\) states as quantum channels in noisy environment with weak measurement scheme

The average conditional fidelities of controlled teleportation protocol using GGHZ states as quantum channels in noisy environment with weak measurement scheme are given below, such that

(i) The average condition fidelity considering two controllers is

$$\begin{aligned} \mathrm{CF}_\mathrm{avg}=\dfrac{ 2 + 2d \bar{w} (3 + 4 d \bar{w} + 3 d^2 \bar{w}^2 + d^3 \bar{w}^3) \mathrm{Sin}^2\theta + \mathrm{Sin}2\theta \mathrm{Sin}2 \phi _1 \mathrm{Sin}2 \phi _2}{3 (\mathrm{Cos}^2\theta + (1 + d\bar{w})^4 \mathrm{Sin}^2\theta )}\nonumber \\ \end{aligned}$$
(A1)

where \(\bar{w}=1-w\).

(ii) The average condition fidelity considering three controllers is

$$\begin{aligned} \mathrm{CF}_\mathrm{avg}=\dfrac{2 + 2d \bar{w} (4 + 7 d \bar{w} + 7 d^2 \bar{w}^2 + 4 d^3 \bar{w}^3 + d^4 \bar{w}^4) \mathrm{Sin}^2\theta + \mathrm{Sin}2\theta \mathrm{Sin}2 \phi _1 \mathrm{Sin}2 \phi _2 \mathrm{Sin}2 \phi _3}{3 (\mathrm{Cos}^2\theta + (1 + d \bar{w})^5 \mathrm{Sin}^2\theta )}\nonumber \\ \end{aligned}$$
(A2)

(iii) The average condition fidelity considering four controllers is

$$\begin{aligned} \begin{aligned} \mathrm{CF}_\mathrm{avg}=\frac{{2 + 2d \bar{w} (5 + 11 d \bar{w} + 14 d^2 \bar{w}^2 + 11 d^3 \bar{w}^3 + 5 d^4 \bar{w}^4 + d^5 \bar{w}^5) \mathrm{Sin}^2\theta }{+\mathrm{Sin}2\theta \mathrm{Sin}2 \phi _1 \mathrm{Sin}2 \phi _2 \mathrm{Sin}2 \phi _3 \mathrm{Sin}2 \phi _4}}{3 (\mathrm{Cos}^2\theta + (1 + d \bar{w})^6 \mathrm{Sin}^2\theta )} \end{aligned}\nonumber \\ \end{aligned}$$
(A3)

A4 \({{|{{\varvec{\Phi }}}\rangle }}\) states as quantum channels in noisy environment with weak measurement scheme

The average conditional fidelities and controller’s powers of controlled teleportation protocol using \({|{\Phi }\rangle }\) states as quantum channels in noisy environment with weak measurement scheme are represented below, such that

(i) The average condition fidelity and controller’s power considering two controllers are

$$\begin{aligned} \mathrm{CF}_\mathrm{avg}=\frac{{3 (4 + 8 d \bar{w} + 7 d^2 \bar{w}^2 + 2 d^3 \bar{w}^3) - 3 d \bar{w} (4 + 5 d \bar{w} + 2 d^2 \bar{w}^2)\mathrm{Cos} 2 \theta }{- (2 - d \bar{w})^2 (3 + 2 d \bar{w})\mathrm{Cos}2 \phi _2 + d^2 \bar{w}^2 (-7 + 2 d \bar{w})\mathrm{Cos} 2 \theta \mathrm{Cos} 2 \phi _2}}{12 ((2 + 2 d \bar{w} + d^2 \bar{w}^2) Cos^2\theta +2 (1 + d\bar{w})^3 \mathrm{Sin}^2\theta )}\nonumber \\ \end{aligned}$$
(A4)

and

$$\begin{aligned} CP=\frac{{d \bar{w} (12 + d \bar{w} - 2 d^2 \bar{w}^2) - (12 + 8 d \bar{w} + 3 d^2 \bar{w}^2 - 2 d^3 \bar{w}^3)\mathrm{Cos} 2 \theta }{- (2 - d \bar{w})^2 (3 + 2 d \bar{w}) \mathrm{Cos} 2 \phi _2 + d^2 \bar{w}^2 (-7 + 2 d \bar{w})\mathrm{Cos} 2 \theta \mathrm{Cos} 2 \phi _2}}{12 ((2 + 2 d \bar{w} + d^2 \bar{w}^2)\mathrm{Cos}^2\theta +2 (1 + d\bar{w})^3 \mathrm{Sin}^2\theta )},\nonumber \\ \end{aligned}$$
(A5)

respectively.

(ii) The average condition fidelity and controller’s power considering three controllers are

and

respectively. (iii) The average condition fidelity and controller’s power considering four controllers are

and

respectively.

A5 Tables to illustrate the influence of each factor involved in the controlled teleportation protocol on average fidelity and controllers power

See Tables 2 and 3.

Table 2 \({|{\Phi }\rangle }\) states in noisy environment with weak measurement
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Table 3 GGHZ states in noisy environment with weak measurement
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Faujdar, J., Kumar, A. A comparative study to analyze efficiencies of \((N+2)\)-qubit partially entangled states in real conditions from the perspective of N controllers. Quantum Inf Process 20, 64 (2021). https://doi.org/10.1007/s11128-021-02993-6

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