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Instrument-based quantum resources: quantification, hierarchies and towards constructing resource theories

Jatin Ghai, Arindam Mitra

TL;DR

The work addresses quantifying instrument-based quantum resources by extending quantum resource theories to quantum instruments themselves. It introduces distance-based resource measures via the diamond norm, provides SDP formulations for computing these measures, and develops concrete instrument-based resource theories for information preservability, entanglement preservability, incompatibility preservability, traditional incompatibility, and parallel incompatibility. The paper establishes hierarchies among free objects and measures, presenting a unified framework to study resource conversion, catalysis, and more within sequential and multi-party quantum settings. Its significance lies in enabling rigorous quantitative analysis of how instruments preserve information, entanglement, or incompatibility across complex quantum networks, with practical implications for programmable devices and distributed quantum protocols.

Abstract

Quantum resources are certain features of the quantum world that provide advantages in certain information-theoretic, thermodynamic, or any other useful operational tasks that are outside the realm of what classical theories can achieve. Quantum resource theories provide us with an elegant framework for studying these resources quantitatively and rigorously. While numerous state-based quantum resource theories have already been investigated, and to some extent, measurement-based resource theories have also been explored, instrument-based resource theories remain largely unexplored, with only a few notable exceptions. As quantum instruments are devices that provide both the classical outcomes of induced measurements and the post-measurement quantum states, they are quite important, especially for scenarios where multiple parties sequentially act on a quantum system. In this work, we study several instrument-based resource theories, namely (1) the resource theory of information preservability, (2) the resource theory of (strong) entanglement preservability, (3) the resource theory of (strong) incompatibility preservability, (4) the resource theory of traditional incompatibility, and (5) the resource theory of parallel incompatibility. Furthermore, we outline the hierarchies of these instrument-based resources and provide measures to quantify them. In short, we provide a detailed framework for several instrument-based quantum resource theories.

Instrument-based quantum resources: quantification, hierarchies and towards constructing resource theories

TL;DR

The work addresses quantifying instrument-based quantum resources by extending quantum resource theories to quantum instruments themselves. It introduces distance-based resource measures via the diamond norm, provides SDP formulations for computing these measures, and develops concrete instrument-based resource theories for information preservability, entanglement preservability, incompatibility preservability, traditional incompatibility, and parallel incompatibility. The paper establishes hierarchies among free objects and measures, presenting a unified framework to study resource conversion, catalysis, and more within sequential and multi-party quantum settings. Its significance lies in enabling rigorous quantitative analysis of how instruments preserve information, entanglement, or incompatibility across complex quantum networks, with practical implications for programmable devices and distributed quantum protocols.

Abstract

Quantum resources are certain features of the quantum world that provide advantages in certain information-theoretic, thermodynamic, or any other useful operational tasks that are outside the realm of what classical theories can achieve. Quantum resource theories provide us with an elegant framework for studying these resources quantitatively and rigorously. While numerous state-based quantum resource theories have already been investigated, and to some extent, measurement-based resource theories have also been explored, instrument-based resource theories remain largely unexplored, with only a few notable exceptions. As quantum instruments are devices that provide both the classical outcomes of induced measurements and the post-measurement quantum states, they are quite important, especially for scenarios where multiple parties sequentially act on a quantum system. In this work, we study several instrument-based resource theories, namely (1) the resource theory of information preservability, (2) the resource theory of (strong) entanglement preservability, (3) the resource theory of (strong) incompatibility preservability, (4) the resource theory of traditional incompatibility, and (5) the resource theory of parallel incompatibility. Furthermore, we outline the hierarchies of these instrument-based resources and provide measures to quantify them. In short, we provide a detailed framework for several instrument-based quantum resource theories.

Paper Structure

This paper contains 16 sections, 26 theorems, 180 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum Measurements, Quantum Channels and Quantum Instruments
  4. A distance measure for measurements and channels via Diamond norm
  5. Formal aspects of a generic instrument-based quantum resource theory
  6. Main Results
  7. Quantification of a generic instrument-based quantum resource
  8. SDP formulation for computation of the resource measures $\mathbbm{R}~ \text{and}~\overline{\mathbbm{R}}$
  9. Some specific instrument-based quantum resources: characterization, quantification, hierarchies, and towards constructing resource theories
  10. Trash-and-prepare instruments and the resource theory of information preservability
  11. (Weak) Entanglement-breaking instruments and the resource theory of (strong) entanglement preservability
  12. (Weak) Incompatibility-breaking instruments and the resource theory of (strong) incompatibility preservability
  13. Traditional compatible instruments and the resource theory of traditional incompatibility
  14. Parallel compatible instruments and the resource theory of parallel incompatibility
  15. Hierarchies among resource measures
  16. ...and 1 more sections

Key Result

Lemma 1

If the instrument $\mathbf{I}_i=\{\Phi^i_a\}\in \mathscr{I}(\mathcal{H},\mathcal{K})$ implements $M_i$ and $\Phi_i$ for all $i\in\{1,2\}$ then

Figures (2)

  • Figure 1: This represents a schematic diagram for the fairly general transformation of a set of quantum channels described in Eq.\ref{['gensupermap']}. In this figure, we have shown the input and output Hilbert spaces of this transformation, as it is more relevant for our purpose of illustration. This general transformation will be used comprehensively throughout the paper.
  • Figure 2: This Venn diagram qualitatively shows the hierarchies (subset relations) among different classes of instruments. More specifically, from the discussion till now, we have $\mathscr{I}_{TP}(\mathcal{H}, \mathcal{K})\subseteq\mathscr{I}_{EB}(\mathcal{H},\mathcal{K})\subseteq \mathscr{I}_{WEB}(\mathcal{H},\mathcal{K})\subseteq\mathscr{I}_{WIB}(\mathcal{H},\mathcal{K})$ and $\mathscr{I}_{TP}(\mathcal{H}, \mathcal{K})\subseteq\mathscr{I}_{EB}(\mathcal{H},\mathcal{K})\subseteq \mathscr{I}_{IB}(\mathcal{H},\mathcal{K})\subseteq\mathscr{I}_{WIB}(\mathcal{H},\mathcal{K})$ for arbitrary $\mathcal{H}$ and $\mathcal{K}$.

Theorems & Definitions (63)

  • Definition 1: Traditional Compatibility
  • Definition 2: Parallel Compatibility
  • Remark 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • ...and 53 more