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Controlled-joint remote implementation of operators and its possible generalization

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Abstract

The existing notion of the shared entangled state-assisted remote preparation of unitary operator (equivalently the existing notion of quantum remote control) using local operation and classical communication is generalized to a scenario where under the control of a supervisor two users can jointly implement arbitrary unitaries (one unknown unitary operation by each or equivalently a single unitary decomposed into two unitaries of the same dimension and given to two users) on an unknown quantum state available with a geographically separated user. It is explicitly shown that the task can be performed using a four-qubit hyperentangled state, which is entangled simultaneously in both spatial and polarization degrees of freedom of photons. The proposed protocol which can be viewed as primitive for distributed photonic quantum computing is further generalized to the case that drops the restrictions on the number of controllers and the number of parties performing unitaries and allows both the numbers to be arbitrary. It is also shown that all the existing variants of quantum remote control schemes can be obtained as special cases of the present scheme.

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Data Availability

This manuscript has no associated data. [Authors’ comment: This is a theoretical work that presents a protocol for a specific task. The scheme is neither experimentally realized nor simulated and consequently no data is produced.]

Notes

  1. Note that if we use particle order permutation technique as described and utilized in [5] and the scheme of [7], a scheme of CRIO of an arbitrary operator would require 2 Bell states and 4 bits of classical communication, whereas a trivial scheme for CRIO obtained by modifying an efficient scheme for controlled bidirectional quantum teleportation [5] would require one more classical bit.

  2. The cross-Kerr nonlinear interaction between an auxiliary coherent state \(|z\rangle \) (\(|z\rangle =\text {exp}(-|z|^2/2)\sum _{n=0}^{\infty }(z^n/\sqrt{n!}|n\rangle )\) where \(|n\rangle \) is a Fock state containing n photons) and a photon path lets say \(|b\rangle \) with interaction parameters \(\theta \) and \(-\theta \) is mathematically represented as \(K_b(\pm \theta )|z\rangle |b\rangle =|ze^{\pm i\theta }\rangle |b\rangle \). The X-quadrature homodyne detection technique is used to measure whether the coherent state is in \(|z\rangle \) or \(|ze^{\pm i\theta }\rangle \). It is to be noted that \(|ze^{i\theta }\rangle \) and \(|ze^{-i\theta }\rangle \) are indistinguishable with this kind of measurement.

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Acknowledgements

Authors acknowledge the support from the QUEST scheme of Interdisciplinary Cyber Physical Systems (ICPS) program of the Department of Science and Technology (DST), India (Grant No.: DST/ICPS/QuST/Theme-1/2019/14 (Q80)).

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Authors and Affiliations

  1. Jaypee Institute of Information Technology, A 10, Sector 62, Noida, UP, 201309, India

    Satish Kumar & Anirban Pathak

  2. Thang Long Institute of Mathematics and Applied Sciences, Thang Long University, Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam

    Nguyen Ba An

  3. Center for Theoretical Physics, Institute of Physics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

    Nguyen Ba An

Authors
  1. Satish Kumar
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  2. Nguyen Ba An
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Contributions

AP and NBA conceptualized the problem and edited the final manuscript. AP supervised the work and verified the calculations. SK performed most of the computation and initial analysis. He has also prepared the first draft of the manuscript.

Corresponding author

Correspondence to Anirban Pathak.

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Kumar, S., An, N.B. & Pathak, A. Controlled-joint remote implementation of operators and its possible generalization. Eur. Phys. J. D 78, 90 (2024). https://doi.org/10.1140/epjd/s10053-024-00883-x

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