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Query complexities of quantum channel discrimination and estimation: A unified approach

Zixin Huang, Johannes Jakob Meyer, Theshani Nuradha, Mark M. Wilde

TL;DR

This work provides a unified framework for quantum channel discrimination and estimation by formulating all results in terms of isometric extensions of channels. It derives lower bounds on query complexities in both parallel and adaptive access models using upper bounds on the squared Bures distance and the SLD Fisher information, and connects discrimination bounds to estimation limits. The authors also present upper bounds for channel distance and information through SDP-representable expressions, enabling practical computation, and they develop binary-search and SDP-based methods to compute these bounds. The framework yields a consistent set of results that subsume and clarify prior approaches, with potential extensions to energy-constrained settings and multi-channel discrimination, thereby advancing the understanding of quantum sensing and information-processing limits.

Abstract

The goal of quantum channel discrimination and estimation is to determine the identity of an unknown channel from a discrete or continuous set, respectively. The query complexity of these tasks is equal to the minimum number of times one must call an unknown channel to identify it within a desired threshold on the error probability. In this paper, we establish lower bounds on the query complexities of channel discrimination and estimation, in both the parallel and adaptive access models. We do so by establishing new or applying known upper bounds on the squared Bures distance and symmetric logarithmic derivative Fisher information of channels. Phrasing our statements and proofs in terms of isometric extensions of quantum channels allows us to give conceptually simple proofs for both novel and known bounds. We also provide alternative proofs for several established results in an effort to present a consistent and unified framework for quantum channel discrimination and estimation, which we believe will be helpful in addressing future questions in the field.

Query complexities of quantum channel discrimination and estimation: A unified approach

TL;DR

This work provides a unified framework for quantum channel discrimination and estimation by formulating all results in terms of isometric extensions of channels. It derives lower bounds on query complexities in both parallel and adaptive access models using upper bounds on the squared Bures distance and the SLD Fisher information, and connects discrimination bounds to estimation limits. The authors also present upper bounds for channel distance and information through SDP-representable expressions, enabling practical computation, and they develop binary-search and SDP-based methods to compute these bounds. The framework yields a consistent set of results that subsume and clarify prior approaches, with potential extensions to energy-constrained settings and multi-channel discrimination, thereby advancing the understanding of quantum sensing and information-processing limits.

Abstract

The goal of quantum channel discrimination and estimation is to determine the identity of an unknown channel from a discrete or continuous set, respectively. The query complexity of these tasks is equal to the minimum number of times one must call an unknown channel to identify it within a desired threshold on the error probability. In this paper, we establish lower bounds on the query complexities of channel discrimination and estimation, in both the parallel and adaptive access models. We do so by establishing new or applying known upper bounds on the squared Bures distance and symmetric logarithmic derivative Fisher information of channels. Phrasing our statements and proofs in terms of isometric extensions of quantum channels allows us to give conceptually simple proofs for both novel and known bounds. We also provide alternative proofs for several established results in an effort to present a consistent and unified framework for quantum channel discrimination and estimation, which we believe will be helpful in addressing future questions in the field.

Paper Structure

This paper contains 36 sections, 23 theorems, 217 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Summary of results
  4. Paper organization
  5. Preliminaries
  6. Quantum channel discrimination and estimation
  7. Quantum channel discrimination
  8. Parallel and adaptive settings of channel discrimination
  9. Query complexities of channel discrimination
  10. Quantum channel estimation
  11. Parallel setting
  12. Adaptive setting
  13. Connecting channel estimation and discrimination
  14. Fidelity and Bures distance of states and channels
  15. Fidelity and Bures distance of states
  16. ...and 21 more sections

Key Result

Proposition 7

For $d$-dimensional states $\rho$ and $\sigma$ with canonical purifications in eq:canonical-pure-rho and eq:canonical-pure-sigma, respectively, the following equalities hold: In eq:halmos-dilation-fidelity, $|0\rangle|\psi^{\sigma}\rangle,|0\rangle|\psi^{\rho}\rangle\in\mathbb{C}^{2}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ and $U_{W}$ acts nontrivially on the first two systems of $\mathbb{C}^{

Figures (1)

  • Figure 1: Depiction of (a) parallel and (b) adaptive strategies for quantum channel discrimination and estimation. In both cases, the unknown channel is queried $n$ times, where $n=3$ in the figure. In discrimination, $\theta \in \{ 1,2\}$ is from a discrete set, and the goal is for the guess $\hat{\theta}$ to equal $\theta$. In estimation, $\theta \in \Theta \subseteq \mathbb{R}$ is from a continuous set, and the goal is for the guess $\hat{\theta}$ to approximate $\theta$ within some tolerance. In each protocol, $\mathcal{Q}$ is a measurement performed in order to determine the value of $\theta$. In the case of channel discrimination, $\mathcal{Q}$ is a measurement with two outcomes, while in the case of channel estimation, $\mathcal{Q}$ is a measurement with outcomes in $\Theta$.

Theorems & Definitions (64)

  • Definition 1
  • Remark 2: Trivial cases
  • Definition 3: Minimax error probability for parallel channel estimation
  • Definition 4: Query complexity of parallel channel estimation
  • Definition 5: Minimax error probability for adaptive channel estimation
  • Definition 6: Query complexity of adaptive channel estimation
  • Proposition 7
  • proof
  • Proposition 8
  • proof
  • ...and 54 more