Abstract
In this paper, we propose a novel protection scheme for N-qubit Greenberger–Horne–Zeilinger (GHZ) states against amplitude damping noise, using unilateral operations. The key innovation lies in the implementation of local operations on a single qubit, irrespective of the total number of qubits, thereby significantly reducing the operational complexity and resource requirements compared to existing methods. The scheme involves a unilateral rotation to transform the GHZ state into a less vulnerable configuration before exposing it to noise, followed by a weak measurement and another unilateral rotation to recover the initial state. We provide a comprehensive mathematical analysis of the proposed unilateral protection scheme, deriving expressions for both fidelity and success probability. The practical feasibility of our scheme is demonstrated through simulations using the Qiskit SDK, in which we provide the corresponding quantum circuit for performing the partial weak measurement used in our scheme. Comparative analysis with pioneer protection schemes reveals a substantial enhancement in success probability and fidelity for both GHZ and generalized GHZ states, emphasizing the superiority of the proposed approach.







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23 May 2024
A Correction to this paper has been published: https://doi.org/10.1007/s11227-024-06179-6
Notes
MATLAB and Qiskit codes for regenerating the results of this article are available from https://github.com/Sjd-Hz/Unilateral-protection-scheme.
References
Raussendorf R, Briegel HJ (2001) A one-way quantum computer. Phys Rev Lett 86(22):5188
Nezami S, Walter M (2020) Multipartite entanglement in stabilizer tensor networks. Phys Rev Lett 125(24):241602
Hu X-M, Zhang C, Liu B-H, Cai Y, Ye X-J, Guo Y, Xing W-B, Huang C-X, Huang Y-F, Li C-F et al (2020) Experimental high-dimensional quantum teleportation. Phys Rev Lett 125(23):230501
Paulson K, Panigrahi PK (2019) Tripartite non-maximally-entangled mixed states as resource for optimally controlled quantum teleportation fidelity. Phys Rev A 100(5):052325
Harraz S, Zhang J-Y, Cong S (2022) High-fidelity quantum teleportation with w states through noisy channels. arXiv preprint arXiv:2206.14463
Song D, He C, Cao Z, Chai G (2018) Quantum teleportation of multiple qubits based on quantum Fourier transform. IEEE Commun Lett 22(12):2427–2430
Bennett CH, Wiesner SJ (1992) Communication via one-and two-particle operators on Einstein–Podolsky–Rosen states. Phys Rev Lett 69(20):2881
Guo Y, Liu B-H, Li C-F, Guo G-C (2019) Advances in quantum dense coding. Adv Quantum Technol 2(5–6):1900011
Yeo Y, Chua WK (2006) Teleportation and dense coding with genuine multipartite entanglement. Phys Rev Lett 96(6):060502
Grasselli F, Kampermann H, Bruß D (2018) Finite-key effects in multipartite quantum key distribution protocols. New J Phys 20(11):113014
Epping M, Kampermann H, Bruß D et al (2017) Multi-partite entanglement can speed up quantum key distribution in networks. New J Phys 19(9):093012
Pivoluska M, Huber M, Malik M (2018) Layered quantum key distribution. Phys Rev A 97(3):032312
Han J, Liu Y, Sun X, Chen A (2020) A self-adjusting quantum key renewal management scheme in classical network symmetric cryptography. J Supercomput 76:4212–4230
Haist T, Osten W (2012) White-light interferometric method for secure key distribution. J Supercomput 62:656–662
Greenberger DM, Horne MA, Zeilinger A (1989) Going beyond bell’s theorem. In: Bell’s Theorem, Quantum Theory and Conceptions of the Universe. Springer, Berlin, pp 69–72
Tavakoli A, Herbauts I, Żukowski M, Bourennane M (2015) Secret sharing with single d-level quantum system. Phys Rev A 92(3):030302
Liao C-H, Yang C-W, Hwang T (2014) Dynamic quantum secret sharing protocol based on GHZ state. Quantum Inf Process 13:1907–1916
Wang X-J, An L-X, Yu X-T, Zhang Z-C (2017) Multilayer quantum secret sharing based on GHZ state and generalized bell basis measurement in multiparty agents. Phys Lett A 381(38):3282–3288
Cherbal S, Zier A, Hebal S, Louail L, Annane B (2023) Security in internet of things: review on approaches based on blockchain, machine learning, cryptography, and quantum computing. J. Supercomput. 80:3738–3816
Christandl M, Wehner S (2005) Quantum anonymous transmissions. In: International Conference on the Theory and Application of Cryptology and Information Security. Springer, Berlin, pp 217–235
Komar P et al (2014) A quantum network of clocks. Nat Phys 10(8):582–587
White AG, James DFV, Munro WJ, Kwiat PG (2001) Exploring Hilbert space: accurate characterization of quantum information. Phys Rev A 65(1):012301
Sharma KK, Gerdt VP (2020) Entanglement sudden death and birth effects in two qubits maximally entangled mixed states under quantum channels. Int J Theor Phys 59:403–414
Nielsen MA, Chuang IL (2001) Quantum computation and quantum information. Phys Today 54(2):60
Sharma KK, Ganguly S (2018) Positive impact of decoherence on spin squeezing in GHZ and W states. J Phys Commun 2(1):015012
Zhang Y-J, Han W, Xia Y-J, Tian J-X, Fan H (2016) Speedup of quantum evolution of multiqubit entanglement states. Sci Rep 6(1):27349
Zhang Y-J, Zou X-B, Xia Y-J, Guo G-C (2010) Different entanglement dynamical behaviors due to initial system-environment correlations. Phys Rev A 82(2):022108
Bennett CH, DiVincenzo DP, Smolin JA, Wootters WK (1996) Mixed-state entanglement and quantum error correction. Phys Rev A 54(5):3824
Krastanov S, Albert VV, Jiang L (2019) Optimized entanglement purification. Quantum 3:123
Fujii K, Yamamoto K (2009) Entanglement purification with double selection. Phys Rev A 80(4):042308
Huang Y-S, Xing H-B, Yang M, Yang Q, Song W, Cao Z-L (2014) Distillation of multipartite entanglement by local filtering operations. Phys Rev A 89(6):062320
Leditzky F, Datta N, Smith G (2017) Useful states and entanglement distillation. IEEE Trans Inf Theory 64(7):4689–4708
Seshadreesan KP, Takeoka M, Wilde MM (2016) Bounds on entanglement distillation and secret key agreement for quantum broadcast channels. IEEE Trans Inf Theory 62(5):2849–2866
Kim Y-S, Lee J-C, Kwon O, Kim Y-H (2012) Protecting entanglement from decoherence using weak measurement and quantum measurement reversal. Nat Phys 8(2):117–120
Harraz S, Cong S, Nieto JJ (2022) Enhancing quantum teleportation fidelity under decoherence via weak measurement with flips. EPJ Quantum Technol 9(1):1–12
Zong X-L, Du C-Q, Yang M, Song W, Yang Q, Cao Z-L (2015) Protecting multipartite entanglement against weak-measurement-induced amplitude damping by local unitary operations. Quantum Inf Process 14:3423–3440
Azuma K, Tamaki K, Lo H-K (2015) All-photonic quantum repeaters. Nat Commun 6(1):6787
Zhou L, Zhang S-S, Zhong W, Sheng Y-B (2020) Multi-copy nested entanglement purification for quantum repeaters. Ann Phys 412:168042
Dias J, Ralph TC (2017) Quantum repeaters using continuous-variable teleportation. Phys Rev A 95(2):022312
Singh MK, Jiang L, Awschalom DD, Guha S (2021) Key device and materials specifications for repeater enabled quantum internet. IEEE Trans Quantum Eng 2:1–9
Munro WJ, Azuma K, Tamaki K, Nemoto K (2015) Inside quantum repeaters. IEEE J Sel Top Quantum Electron 21(3):78–90
Aharonov Y, Albert DZ, Vaidman L (1988) How the result of measurement of component of the spin of spin-1/2 particle can turn out to be 100. Phys Rev Lett 60(14):1351
Xu Y, Shi L, Guan T, Guo C, Li D, Yang Y, Wang X, Xie L, He Y, Xie W (2018) Optimization of quantum weak measurement system with its working areas. Opt Express 26(16):21119–21131
Harraz S, Cong S, Nieto JJ (2022) Comparison of quantum state protection against decoherence via weak measurement, survey. Int J Quantum Inf 20(04):2250007
Gillett G, Dalton R, Lanyon B, Almeida M, Barbieri M, Pryde GJ, O’brien J, Resch K, Bartlett S, White A (2010) Experimental feedback control of quantum systems using weak measurements. Phys Rev Lett 104(8):080503
Pegg D (1989) Some new techniques in nuclear magnetic resonance. Contemp Phys 30(2):101–112
Ladd TD, Jelezko F, Laflamme R, Nakamura Y, Monroe C, O’Brien JL (2010) Quantum computers. Nature 464(7285):45–53
Ballance C, Harty T, Linke N, Sepiol M, Lucas D (2016) High-fidelity quantum logic gates using trapped-ion hyperfine qubits. Phys Rev Lett 117(6):060504
Barends R, Kelly J, Megrant A, Veitia A, Sank D, Jeffrey E, White TC, Mutus J, Fowler AG, Campbell B et al (2014) Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature 508(7497):500–503
Dür W, Briegel H-J, Cirac JI, Zoller P (1999) Quantum repeaters based on entanglement purification. Phys Rev A 59(1):169
Briegel H-J, Dür W, Cirac JI, Zoller P (1998) Quantum repeaters: the role of imperfect local operations in quantum communication. Phys Rev Lett 81(26):5932
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant Number 61973290.
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“The original online version of this article was revised:” In this article the author’s name Harraz Sajede was incorrectly written as Harraz Sajde. The typesetter apologizes for the mistake.
Appendices
Appendix A
The initial N-qubit general GHZ state, as described in Eq. (1), can be represented by its corresponding density matrix, given by
In the first step, a unilateral rotation is applied to the first qubit of the GHZ state. The N-qubit representation of the unilateral rotation is expressed as:
Consequently, the state of the N-qubit GHZ state, following the application of the unilateral rotation, is calculated as:
The notation \(A^T\) denotes the transpose of matrix A, this notation is used here, as all elements within the operation matrices are real.
After the unilateral rotation, the GHZ state transfers through ADC. Considering that each qubit of the N-qubit GHZ state undergoes ADC, the corresponding N-qubit Kraus operators of ADC are expressed as:
Hence, the state of N-qubit GHZ state after passing through ADC is given by
where \(A_{d}=\left| \alpha \right| ^2(1-r),B_{d}=|\beta |^2 (1-r)^{N-1},C_{d}=D_{d}^{\dagger }=\alpha \beta \left( 1-r\right) ^ \frac{N}{2} \text { and } E_d=|\alpha |^2r{\left| 0 \right\rangle } {\left\langle 0 \right| } ^{\otimes N-1} +|\beta |^2(r{\left| 0 \right\rangle } {\left\langle 0 \right| } + (1-r){\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N-1}-\beta ^2 (1-r)^{N-1}({\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N-1}\) After the ADC, the weak measurement operator in Eq. (3) applies to the first qubit. Therefore, the weak measurement operator for the N-qubit GHZ state is expressed as
Hence, the state of N-qubit GHZ state after employing the weak measurement is derived as
where \(A_{m}=|\alpha |^2(1-r),B_{m}=|\beta |^2 (1-q)(1-r)^{N-1},C_{m}=D_{m}^{\dagger }=\alpha \beta (1-q)^\frac{1}{2}\left( 1-r\right) ^ \frac{N}{2} \text { and } E_m=(1-q )(|\alpha |^2r{\left| 0 \right\rangle } {\left\langle 0 \right| } ^{\otimes N-1} +|\beta |^2(r{\left| 0 \right\rangle } {\left\langle 0 \right| } + (1-r){\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N-1}-\beta ^2 (1-r)^{N-1}({\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N-1})\).
Finally, in the last step, we apply the same unilateral operation as the one before the ADC to restore the state to its initial structure. Hence, the final state after the whole process of protection is represented as
where \(A_{f}=|\alpha |^2(1-r),B_{f}=|\beta |^2 (1-q)(1-r)^{N-1},C_{f}=D_{f}^{\dagger }=\alpha \beta (1-q)^\frac{1}{2}\left( 1-r\right) ^ \frac{N}{2} \text { and } E_f=(1-q )(|\alpha |^2r{\left| 0 \right\rangle } {\left\langle 0 \right| } ^{\otimes N-1} +|\beta |^2(r{\left| 0 \right\rangle } {\left\langle 0 \right| } + (1-r){\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N-1}-\beta ^2 (1-r)^{N-1}({\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N-1})\).
Appendix B
In WMRPS, before sending qubits through the ADC, weak measurements are employed on all qubits, followed by a subsequent weak measurement reversal after passing through the ADC. Assuming identical parameters (such as measurement strengths and the decay rate of the ADC) for all qubits, the operators for weak measurement, denoted as \(M_{\text {WM}}\), and weak measurement reversal, denoted as \(M_{\text {WMR}}\), are defined as
where \(p_1\) and \(p_2\) represent the weak measurement and weak measurement reversal strength, respectively. Considering the initial N-qubit GHZ state in Eq. (A1), the density matrix resulting from the application of weak measurement \(M_{\text {WM}}\) on all qubits is represented as
where \(F_{\text {WM}}=|\alpha |^2, G_{\text {WM}}=H^{\dagger }=\alpha \beta (1-p_1)^\frac{N}{2}\ \text {and} \ K_{\text {WM}}=|\beta |^2 (1-p_1)^{N}\), all other elements of the density matrix at this step are zero.
Subsequently, all qubits will be sent through ADCs, resulting in the transformation of the elements of the density matrix to:
where \(F_{\text {ADC}}=|\alpha |^2+|\beta |^2 (1-p_1)^{N}r^{N},G_{\text {ADC}}=H^{\dagger }_{\text {ADC}}=\alpha \beta (1-p_1)^\frac{N}{2}r^\frac{N}{2}, K_{\text {ADC}}=|\beta |^2 (1-p_1)^{N}(1-r )^{N}\text {and}\) \(L_{\text {ADC}}=|\beta |^2(1-p_1)^{N}[(r{\left| 0 \right\rangle } {\left\langle 0 \right| } + (1-r){\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N}-r^{N}{\left| 0 \right\rangle } {\left\langle 0 \right| }^{\otimes N}-(1-r)^{ N}{\left| 1 \right\rangle } {\left\langle 1 \right| } ^{\otimes N}]\).
Finally, by employing the weak measurement reversal \(M_{\text {WMR}}\) on all qubits, the final density matrix after the whole WMRPS is calculated as
where \(F_f=|\alpha |^2(1-p_2)^{N}+|\beta |^2 (1-p_1)^{N}r^{N}(1-p_2)^{N},G_f=H_f^\dagger =\alpha \beta (1-p_1)^\frac{N}{2}(1-p_2)^\frac{N}{2}r^\frac{N}{2}, \text { and }K_f=|\beta |^2 (1-p_1)^{N}(1-r )^{N}\text { and } L_f=|\beta |^2(1-p_1)^{N}((r(1-p_2){\left| 0 \right\rangle } {\left\langle 0 \right| } + (1-r){\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N}-(r(1-p_2))^{\otimes N}{\left| 0 \right\rangle } {\left\langle 0 \right| }^{\otimes N}-(1-r)^{N}{\left| 1 \right\rangle } {\left\langle 1 \right| } ^{N})\). Here, we note that the matrix \(\rho _f^{\text {WMRPS}}\) is a \(2^N \times 2^N\) matrix where the elements \(F_f\), \(G_f\), \(H_f\), and \(K_f\) are scalar values located at the corners of the matrix. The central part of the matrix, denoted by \(L_f\), is a \(2^N \times 2^N\) matrix as well whose corner elements are all 0.
Hence, by considering the final state following the entire WMRPS process in Eq. (B12), the success probability of WMRPS is obtained as:
Also, the fidelity of WMRPS is calculated as
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Sajede, H., Yueyan, W. & Cong, S. Unilateral protection scheme for N-qubit GHZ states against decoherence: a resource-efficient approach. J Supercomput 80, 17004–17020 (2024). https://doi.org/10.1007/s11227-024-06101-0
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DOI: https://doi.org/10.1007/s11227-024-06101-0