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Macroscopicity and observational deficit in states, operations, and correlations

Teruaki Nagasawa, Eyuri Wakakuwa, Kohtaro Kato, Francesco Buscemi

TL;DR

This work introduces a principled, observer-centered framework connecting macroscopic descriptions of quantum systems to coarse-graining via POVMs and priors. By defining macroscopic states as zero observational deficit fixed points of coarse-graining maps and establishing multiple equivalent characterizations through Petz recovery and maximal projective post-processing, it unifies resource theories of coherence, athermality, and asymmetry under a single formalism. The paper also develops a resource theory of microscopicity with free macroscopic states and a hierarchy of free operations, and introduces observational discord to quantify observer-dependent quantum correlations under limited measurements. Together, these results provide a comprehensive, inferentially grounded approach to macroscopic irreversibility and context-dependent quantum correlations, with potential implications for thermodynamics, quantum information processing, and foundational studies.

Abstract

To understand the emergence of macroscopic irreversibility from microscopic reversible dynamics, the idea of coarse-graining plays a fundamental role. In this work, we develop a unified inferential framework for macroscopic states, that is, coarse descriptions of microscopic quantum systems that can be inferred from macroscopic measurements. Building on quantum statistical sufficiency and Bayesian retrodiction, we characterize macroscopic states through equivalent abstract (algebraic) and explicit (constructive) formulations. Central to our approach is the notion of observational deficit, which quantifies the degree of irretrodictability of a state relative to a prior and a measurement. This leads to a general definition of macroscopic entropy as an inferentially grounded measure of asymmetry under Bayesian inversion. We formalize this structure in terms of inferential reference frames, defined by the pair consisting of a prior and a measurement, which encapsulate the observer's informational perspective. We then formulate a resource theory of microscopicity, treating macroscopic states as free states and introducing a hierarchy of microscopicity-non-generating operations. This theory unifies and extends existing resource theories of coherence, athermality, and asymmetry. Finally, we apply the framework to study quantum correlations under observational constraints, introducing the notion of observational discord and deriving necessary and sufficient conditions for their vanishing in terms of information recoverability. This work is dedicated to Professor Ryszard Horodecki on the occasion of his 80th birthday, in deep admiration and gratitude for his pioneering contributions to quantum information theory.

Macroscopicity and observational deficit in states, operations, and correlations

TL;DR

This work introduces a principled, observer-centered framework connecting macroscopic descriptions of quantum systems to coarse-graining via POVMs and priors. By defining macroscopic states as zero observational deficit fixed points of coarse-graining maps and establishing multiple equivalent characterizations through Petz recovery and maximal projective post-processing, it unifies resource theories of coherence, athermality, and asymmetry under a single formalism. The paper also develops a resource theory of microscopicity with free macroscopic states and a hierarchy of free operations, and introduces observational discord to quantify observer-dependent quantum correlations under limited measurements. Together, these results provide a comprehensive, inferentially grounded approach to macroscopic irreversibility and context-dependent quantum correlations, with potential implications for thermodynamics, quantum information processing, and foundational studies.

Abstract

To understand the emergence of macroscopic irreversibility from microscopic reversible dynamics, the idea of coarse-graining plays a fundamental role. In this work, we develop a unified inferential framework for macroscopic states, that is, coarse descriptions of microscopic quantum systems that can be inferred from macroscopic measurements. Building on quantum statistical sufficiency and Bayesian retrodiction, we characterize macroscopic states through equivalent abstract (algebraic) and explicit (constructive) formulations. Central to our approach is the notion of observational deficit, which quantifies the degree of irretrodictability of a state relative to a prior and a measurement. This leads to a general definition of macroscopic entropy as an inferentially grounded measure of asymmetry under Bayesian inversion. We formalize this structure in terms of inferential reference frames, defined by the pair consisting of a prior and a measurement, which encapsulate the observer's informational perspective. We then formulate a resource theory of microscopicity, treating macroscopic states as free states and introducing a hierarchy of microscopicity-non-generating operations. This theory unifies and extends existing resource theories of coherence, athermality, and asymmetry. Finally, we apply the framework to study quantum correlations under observational constraints, introducing the notion of observational discord and deriving necessary and sufficient conditions for their vanishing in terms of information recoverability. This work is dedicated to Professor Ryszard Horodecki on the occasion of his 80th birthday, in deep admiration and gratitude for his pioneering contributions to quantum information theory.

Paper Structure

This paper contains 15 sections, 15 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations
  3. Macroscopic states
  4. Observational entropy
  5. Equivalent characterizations of macroscopic states and inferential reference frames
  6. Maximal projective post-processing
  7. Fixed points of the coarse-graining map
  8. Resource theory of microscopicity
  9. Free states and operations
  10. Relative Entropy of Microscopicity
  11. Reduction to other resource theories
  12. Observer-dependent measure of quantum correlations
  13. Locally macroscopic states and operations
  14. Observational discord
  15. Conclusion

Key Result

Lemma 2.1

Suppose that $\boldsymbol{Q}=\{Q_y\}_y$ is such that, for each $y\in\mathcal{Y}$, there exists a unit vector $\ket{\phi_y}$ with $Q_y\ket{\phi_y}=\ket{\phi_y}$. Suppose also that there exists another POVM $\boldsymbol{P}=\{P_x\}_x$ such that $\boldsymbol{Q}\preceq \boldsymbol{P}$. Then, the post-pro for all $x$ and $y$.

Figures (1)

  • Figure 1: A schematic diagram illustrating the relationship between the resource theory of microscopicity and other prominent resource theories. The large vertical ellipse represents the theory of microscopicity, which encompasses several other theories. Within this framework, the inner circle signifies the resource theory of coherence, while the sun symbol represents the resource theory of athermality, a specific case of microscopicity (see Example \ref{['example:athermality']}). The dashed, partially overlapping horizontal ellipse depicts the resource theory of asymmetry, where the cloud denotes a special case in which the multiplicity space of group representations is trivial (see Example \ref{['example:asymmetry']}). Finally, the star symbol stands for the resource theory of block-coherence, which cannot be derived from our framework due to the more constrained structure of macroscopic states (see Remark \ref{['remark:block']}).

Theorems & Definitions (46)

  • Lemma 2.1: Buscemi2024-sharpnessnagasawa2024generic
  • Definition 3.1: Observational deficit
  • Definition 3.2: Coarse-graining maps
  • Definition 3.3: Coarse-grained state and macroscopic state
  • Remark 3.4: Coarse-graining and POVM post-processing
  • Definition 3.5: Observational entropy
  • Definition 4.1: Maximal projective post-processing (MPPP)
  • Theorem 4.2: Macroscopic states with general quantum prior
  • Remark 4.3
  • Remark 4.4
  • ...and 36 more