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Exact quantum sensing limits for bosonic dephasing channels

Zixin Huang, Ludovico Lami, Mark M. Wilde

TL;DR

This work determines exact limits for discrimination and estimation with bosonic dephasing channels by showing that adaptive quantum strategies reduce to optimal classical tests on the underlying phase densities $p(\phi)$. It establishes equivalences for symmetric and asymmetric hypothesis testing and for channel estimation, providing explicit optimal strategies and finite-energy considerations. Key results include the Chernoff-divergence characterization of symmetric discrimination, a 2nd-order expansion for asymmetric discrimination, and a Cramér–Rao-type bound linked to the classical Fisher information in metrology. The authors construct two attainability schemes (photon-number-superposition and coherent-state) and prove their robustness to loss, making the results practically relevant for quantum sensing under dephasing. This is the first exact solution for a non-Gaussian bosonic channel and lays a foundation for energy-constrained limits and extensions to broader Hamiltonian-driven channels.

Abstract

Dephasing is a prominent noise mechanism that afflicts quantum information carriers, and it is one of the main challenges towards realizing useful quantum computation, communication, and sensing. Here we consider discrimination and estimation of bosonic dephasing channels, when using the most general adaptive strategies allowed by quantum mechanics. We reduce these difficult quantum problems to simple classical ones based on the probability densities defining the bosonic dephasing channels. By doing so, we rigorously establish the optimal performance of various distinguishability and estimation tasks and construct explicit strategies to achieve this performance. To the best of our knowledge, this is the first example of a non-Gaussian bosonic channel for which there are exact solutions for these tasks.

Exact quantum sensing limits for bosonic dephasing channels

TL;DR

This work determines exact limits for discrimination and estimation with bosonic dephasing channels by showing that adaptive quantum strategies reduce to optimal classical tests on the underlying phase densities . It establishes equivalences for symmetric and asymmetric hypothesis testing and for channel estimation, providing explicit optimal strategies and finite-energy considerations. Key results include the Chernoff-divergence characterization of symmetric discrimination, a 2nd-order expansion for asymmetric discrimination, and a Cramér–Rao-type bound linked to the classical Fisher information in metrology. The authors construct two attainability schemes (photon-number-superposition and coherent-state) and prove their robustness to loss, making the results practically relevant for quantum sensing under dephasing. This is the first exact solution for a non-Gaussian bosonic channel and lays a foundation for energy-constrained limits and extensions to broader Hamiltonian-driven channels.

Abstract

Dephasing is a prominent noise mechanism that afflicts quantum information carriers, and it is one of the main challenges towards realizing useful quantum computation, communication, and sensing. Here we consider discrimination and estimation of bosonic dephasing channels, when using the most general adaptive strategies allowed by quantum mechanics. We reduce these difficult quantum problems to simple classical ones based on the probability densities defining the bosonic dephasing channels. By doing so, we rigorously establish the optimal performance of various distinguishability and estimation tasks and construct explicit strategies to achieve this performance. To the best of our knowledge, this is the first example of a non-Gaussian bosonic channel for which there are exact solutions for these tasks.
Paper Structure (28 sections, 2 theorems, 128 equations, 7 figures)

This paper contains 28 sections, 2 theorems, 128 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Quantum channel discrimination and estimation
  3. Quantum channel discrimination
  4. Quantum channel estimation
  5. Main results
  6. Symmetric hypothesis testing
  7. Asymmetric hypothesis testing
  8. Quantum metrology
  9. Examples of bosonic dephasing channels
  10. Optimality
  11. Attainability
  12. Photon-number-superposition method
  13. Coherent-state method
  14. Comparison of methods for finite energy
  15. Bosonic dephasing and loss
  16. ...and 13 more sections

Key Result

Lemma 1

Let $p:[-\pi,\pi]\to \mathbb{R}_+$ be a continuous non-negative function with $p(-\pi)=p(\pi)$ and $\int_{-\pi}^\pi d\phi\ p(\phi) = 1$. For every positive integer $d$, all $k\in \{0,1,\ldots, d-1\}$, and all $\hat{\phi}\in [-\pi,\pi]$, set where $\Pi_d$ is defined by eq:Pi_d, and Then the function $p'_d:[-\pi,\pi]\to \mathbb{R}_+$ defined by where $\mathcal{D}_p$ is the bosonic dephasing chann

Figures (7)

  • Figure 1: A general, adaptive protocol for channel discrimination and parameter estimation, when either $\mathcal{N}_0$ or $\mathcal{N}_1$ is called three times. The initial input state is $\tau$, the adaptive operations are $\mathcal{A}_1$ and $\mathcal{A}_2$, and the final measurement is $\mathcal{Q}$. The final states are denoted by $\rho_0^{(n)}$ and $\rho_1^{(n)}$, and $n=3$ in this case.
  • Figure 2: The Chernoff exponent of different BDCs as a function of $\gamma_2 = \lambda_2 = \kappa_2$, for $\gamma_1 = \lambda_1 = \kappa_1 = 1$.
  • Figure 3: The relative entropy of different BDCs as a function of $\gamma_2 = \lambda_2 = \kappa_2$, for $\gamma_1 = \lambda_1 = \kappa_1 = 1$.
  • Figure 4: The Fisher information of different BDCs as a function of $\gamma = \kappa = \lambda$ shown as a log-log plot. The Fisher information quickly drops to zero as $\gamma$ increases.
  • Figure 5: Channel discrimination and parameter estimation for environment-parameterized channels $\mathcal{D}_p$ and $\mathcal{D}_q$, where the underlying environment states are $\sigma_p$ and $\sigma_q$, respectively. The yellow-shaded boxes denote the underlying environmental states to which we do not have access.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof