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Existence of universal resource and uselessness of too entangled states for quantum metrology

Rina Miyajima, Yuki Takeuchi, Seiseki Akibue

TL;DR

The paper investigates whether quantum metrology can leverage universal resource states for linear Hamiltonians and analyzes how entanglement impacts metrological advantage. It introduces and analyzes locally diagonalizable Hamiltonians, using epsilon-nets and measure concentration to show that universal resource states exist for a broad class of linear Hamiltonians, while too much entanglement generally harms metrological performance. For random pure states, the QFI remains at most that of an optimal separable state for broad Hamiltonian families, whereas random symmetric states concentrate near Heisenberg-limited scaling for linear Hamiltonians, establishing a nuanced picture where some structured states outperform others depending on the Hamiltonian class. The results illuminate the limits of entanglement as a resource in quantum metrology, with implications for secure delegation of metrology tasks and for understanding metrological power in complex many-body systems.

Abstract

We show (i) the existence of universal resource states for a certain class of linear Hamiltonians and (ii) the uselessness of highly entangled states for quantum metrology of linear Hamiltonians. We also show that random pure states are basically not useful even if we consider more general Hamiltonians. Since random pure states have high entanglement, this result strengthens the uselessness of highly entangled states for quantum metrology.

Existence of universal resource and uselessness of too entangled states for quantum metrology

TL;DR

The paper investigates whether quantum metrology can leverage universal resource states for linear Hamiltonians and analyzes how entanglement impacts metrological advantage. It introduces and analyzes locally diagonalizable Hamiltonians, using epsilon-nets and measure concentration to show that universal resource states exist for a broad class of linear Hamiltonians, while too much entanglement generally harms metrological performance. For random pure states, the QFI remains at most that of an optimal separable state for broad Hamiltonian families, whereas random symmetric states concentrate near Heisenberg-limited scaling for linear Hamiltonians, establishing a nuanced picture where some structured states outperform others depending on the Hamiltonian class. The results illuminate the limits of entanglement as a resource in quantum metrology, with implications for secure delegation of metrology tasks and for understanding metrological power in complex many-body systems.

Abstract

We show (i) the existence of universal resource states for a certain class of linear Hamiltonians and (ii) the uselessness of highly entangled states for quantum metrology of linear Hamiltonians. We also show that random pure states are basically not useful even if we consider more general Hamiltonians. Since random pure states have high entanglement, this result strengthens the uselessness of highly entangled states for quantum metrology.

Paper Structure

This paper contains 49 sections, 15 theorems, 284 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Summary of our contribution
  3. Preliminary
  4. Details of Result 1 and Result 3
  5. The expectation of the quantum Fisher information (QFI) of random pure states and that of random symmetric states random_qfi
  6. The concentration of the QFI of random pure states and that of random symmetric states random_qfi
  7. The class of Hamiltonians for our analysis
  8. Evaluation of the expectation of QFI of random pure states and that of random symmetric states
  9. Definition of a set of local diagonalizable Hamiltonians
  10. Definition of a set $S_{LD}^{\mathrm{Result\ 3}}$ of local diagonalizable Hamiltonians
  11. Definition of a set $S_L$ of linear Hamiltonians in Result 1
  12. Construction of an $\epsilon$-net $\mathcal{N}_S$ for sets of Hamiltonians $S_{LD}^{\mathrm{Result\ 1}},S_{LD}^{\mathrm{Result\ 3}}$
  13. Property of an $\epsilon$-net $\mathcal{N}_S$
  14. Proof of Result 1
  15. Proof of Result 3
  16. ...and 34 more sections

Key Result

Lemma 1

(The expectation of the QFI of random pure states and that of random symmetric states) The expectation of QFI $f(\psi)$ of Haar random states is That of random symmetric states is where $\Pi_{Sym^n(\mathbb{C}^d)}: (\mathbb{C}^d)^{\otimes n} \rightarrow Sym^n(\mathbb{C}^d)$ is a projection: and $|D|=\dim Sym^n(\mathbb{C}^d)=_{n+d-1}C_n$. $\blacksquare$

Figures (2)

  • Figure 1: Schematic of Result 1. Result 1 means that random symmetric states are useful even if the most unsuitable Hamiltonian is chosen from a set of linear Hamiltonians for each sampled probe.
  • Figure 2: Examples of 2-body Hamiltonians with 5 qubits. From left to right, the graphs represent star-shaped, chain-shaped, ring-shaped, and fully connected Hamiltonians.

Theorems & Definitions (17)

  • Claim 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Proposition 9
  • ...and 7 more