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Anonymous quantum sensing robust against state preparation errors

Hiroto Kasai, Seiichiro Tani, Yasuhiro Tokura, Yuki Takeuchi

TL;DR

A quantum state verification protocol for a superposition of Greenberger-Horne-Zeilinger and Dicke states is devised and combined with the original protocol of anonymous quantum sensing to improve the performance of anonymous quantum sensing in realistic situations.

Abstract

Networked quantum sensors have several applications such as the mapping of magnetic fields. When the magnetic fields are biomagnetic ones, i.e., they contain some private information, the information of from who non-zero magnetic fields occur has to be protected from eavesdroppers. Anonymous quantum sensing keeps it secret by estimating amplitudes of the magnetic fields without disclosing the positions of non-zero magnetic fields. In this paper, we propose an anonymous quantum sensing protocol that is robust against any independent noise in state preparations. To this end, we devise a quantum state verification protocol for a superposition of Greenberger-Horne-Zeilinger and Dicke states and combine it with the original protocol of anonymous quantum sensing. Our verification protocol can decide whether the fidelity between the ideal and actual states is high or low more efficiently than the direct fidelity estimation. Since the original protocol of anonymous quantum sensing cannot correctly estimate the amplitudes of the magnetic fields under state preparation errors, our results would improve the performance of anonymous quantum sensing in realistic situations.

Anonymous quantum sensing robust against state preparation errors

TL;DR

A quantum state verification protocol for a superposition of Greenberger-Horne-Zeilinger and Dicke states is devised and combined with the original protocol of anonymous quantum sensing to improve the performance of anonymous quantum sensing in realistic situations.

Abstract

Networked quantum sensors have several applications such as the mapping of magnetic fields. When the magnetic fields are biomagnetic ones, i.e., they contain some private information, the information of from who non-zero magnetic fields occur has to be protected from eavesdroppers. Anonymous quantum sensing keeps it secret by estimating amplitudes of the magnetic fields without disclosing the positions of non-zero magnetic fields. In this paper, we propose an anonymous quantum sensing protocol that is robust against any independent noise in state preparations. To this end, we devise a quantum state verification protocol for a superposition of Greenberger-Horne-Zeilinger and Dicke states and combine it with the original protocol of anonymous quantum sensing. Our verification protocol can decide whether the fidelity between the ideal and actual states is high or low more efficiently than the direct fidelity estimation. Since the original protocol of anonymous quantum sensing cannot correctly estimate the amplitudes of the magnetic fields under state preparation errors, our results would improve the performance of anonymous quantum sensing in realistic situations.
Paper Structure (19 sections, 2 theorems, 113 equations, 5 figures)

This paper contains 19 sections, 2 theorems, 113 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Anonymous quantum sensing
  3. Quantum state verification (QSV)
  4. General framework
  5. GHZ-like states
  6. Dicke states
  7. Main result 1: verification of superposition of GHZ and Dicke states
  8. Main result 2: anonymous quantum sensing robust against state preparation errors
  9. Protocol
  10. Optimization of initial state
  11. Conclusion & Discussion
  12. GHZ-like component
  13. Dicke component
  14. Our strategy
  15. Comparison between $\Lambda_{1}( \Omega^{(2)} )$ and $\Lambda( \Omega^{(1)},b,c,1)$
  16. ...and 4 more sections

Key Result

Theorem 1

Let $n$ be any even number and $a$, $b$, $c$, and $d$ be any positive numbers. The eigenvalues and eigenvectors $(\lambda,|\lambda\rangle)$ of the $n$-qubit Hermitian operator are as follows:

Figures (5)

  • Figure 1: Schematic of anonymous quantum sensing. In this figure, we assume that there are six participants, and non-zero magnetic fields occur from the first and fourth participants, i.e., $t_1=1$ and $t_2=4$. The distributor prepares the symmetric quantum state in Eq. (\ref{['superposition']}) and sends each qubit to each participant. Then the participants interact their qubits with their own magnetic fields. As a result, the first and fourth qubits are evolved by the Hamiltonian in Eq. (\ref{['Hamiltonianm']}). Finally, the participants send their qubits to the observer, and the observer measures them with the POVM elements in Eq. \ref{['eq:povm-of-AQS']} to estimate $\omega_1$ and $\omega_4$. Since the output probability distribution does not depend on $t_1$ and $t_2$, any eavesdropper cannot obtain any information about $t_1$ and $t_2$ even if all classical information is given.
  • Figure 2: The values of $H_{\rm{min}}$ for $3 \le n \le 50$.
  • Figure 3: The values of $H_{\rm{min}}$ for $50 \le n \le 500$.
  • Figure 4: The values of $q_H$ and $q_G$ for $3\le n\le50$. For $x\in\{A,B,\cdots,L\}$, the labels $(H,x)$ and $(G,x)$ mean that the corresponding plots represent the values of $q_H$ and $q_G$, respectively.
  • Figure 5: The values of $q_H$ and $q_G$ for $50\le n\le500$. For $x\in\{A,B,\cdots,L\}$, the labels $(H,x)$ and $(G,x)$ mean that the corresponding plots represent the values of $q_H$ and $q_G$, respectively.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2