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Optimal quantum metrology under energy constraints

Longyun Chen, Yuxiang Yang

Abstract

The traditional framework of quantum metrology commonly assumes unlimited access to resources, overlooking resource constraints in realistic scenarios. As such, the optimal strategies therein can be infeasible in practice. Here, we investigate quantum metrology where the total energy consumption of the probe state preparation, intermediate control operations, and the final measurement is subject to a constraint. We establish a comprehensive theoretical framework for characterizing energy-constrained multi-step quantum processes, based on which we develop a general optimization method for energy-constrained quantum metrology that determines both the optimal precision and the corresponding strategy. Using the method, we determine the ultimate precision limit of energy-constrained phase estimation and identify a novel advantage of quantum superpositions of causal orders in enhancing the energy efficiency of adaptive quantum estimation.

Optimal quantum metrology under energy constraints

Abstract

The traditional framework of quantum metrology commonly assumes unlimited access to resources, overlooking resource constraints in realistic scenarios. As such, the optimal strategies therein can be infeasible in practice. Here, we investigate quantum metrology where the total energy consumption of the probe state preparation, intermediate control operations, and the final measurement is subject to a constraint. We establish a comprehensive theoretical framework for characterizing energy-constrained multi-step quantum processes, based on which we develop a general optimization method for energy-constrained quantum metrology that determines both the optimal precision and the corresponding strategy. Using the method, we determine the ultimate precision limit of energy-constrained phase estimation and identify a novel advantage of quantum superpositions of causal orders in enhancing the energy efficiency of adaptive quantum estimation.

Paper Structure

This paper contains 17 sections, 18 theorems, 196 equations, 12 figures.

Table of Contents

  1. Introduction to Quantum Combs
  2. Quantum Combs with Definite Causal Orders
  3. Quantum Combs Extended to Indefinite Causal Order Processes
  4. Proof of Theorem \ref{['thm:energy_constrained_comb']}
  5. Proof of Theorem \ref{['thm:comb_implementation']}
  6. Proof of Theorem \ref{['thm:qubit_optimal_estimator']}
  7. Energy-constrained Qubit Phase Estimation
  8. Phase Estimation Precision Scaling with Energy Constraints for Large Dimensions
  9. Proof of Lemma \ref{['lem:cost_lower_bound']}
  10. Lower bound for the average cost of the infinite-dimensional phase channel
  11. Attainability of the precision scaling with respect to energy
  12. Different Battery Models for Causal Superposition Processes
  13. Spacetime-sharing Battery
  14. Separation between Battery Models
  15. Global vs. Local
  16. ...and 2 more sections

Key Result

Theorem 1

For an $N$-step quantum comb $C$, $C \in \operatorname{Comb}^{\leq E}_{\text{gl/loc}}$ if and only if $E_{\text{gl/loc}}(C) \leq E$. Here, the energy consumption can be determined by $E_{\text{gl}}(C)=\max_{n\in[N]}\{\lambda_{\max}(O_n)\}$ and $E_{\text{loc}}(C)=\sum_{n=1}^N\max\{\lambda_{\max}(O'_n

Figures (12)

  • Figure 1: An $N$-step energy-constrained process, with a global battery with initial energy $E_{\text{gl}}$ shared across all steps (the purple solid lines) or $N$ isolated local batteries with initial energy $E_n \geq 0$ respectively, where the $n$-th battery exclusively interacts with the $n$-th step (the green dashed lines). Energy can be transferred bidirectionally.
  • Figure 2: Our goal is to estimate $\theta$ from $C_\theta$ with an energy-constrained probe process $T$, consisting of state preparation and multi-step control, and a final measurement $\mathcal{M}$ in the energy eigenbasis $\{|x\rangle\}$.
  • Figure 3: The quantum circuit to implement a causal superposition strategy with $N=3$. A projective measurement in an arbitrary basis on the degenerate system $\mathcal{H}_{\text{oc}}$ is allowed. The black (blue) part represents a definite causal order strategy without (with) an accessible ancilla $\mathcal{H}_{\text{oc}}$. The orange part represents the strategy with quantum control of the causal order, which is a superposition of different causal orders $\tau \in S_{N-1}$.
  • Figure 4: The strict hierarchy of the optimal average cost $\bar{c}(E)$ for the different strategies and battery models.
  • Figure 5: Given two quantum combs $C_\theta$ and $C_{\theta'}$, we can concatenate each output of $C_\theta$ with each input of $C_{\theta'}$ to form a new comb denoted by $C_\theta \circ C_{\theta'}$.
  • ...and 7 more figures

Theorems & Definitions (31)

  • Definition 1: Battery models
  • Theorem 1: Energy-constrained combs
  • Theorem 2: Implementation of energy-constrained combs
  • Theorem 3: A strict hierarchy of energy-constrained strategies
  • Definition 2
  • Theorem 4: Optimal estimator for covariant comb with non-positive energy consumption on qubit
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 21 more