One of the persistent difficulties in discussing the emergence of macroscopic irreversibility from microscopic quantum dynamics is that we often talk about “macroscopic states”, “macroscopic operations”, or “macroscopic correlations” without a fully satisfactory definition of what these terms actually mean.

This paper, written together with Teruaki Nagasawa, Eyuri Wakakuwa, and Kohtaro Kato, grew out of a very concrete problem related to the second law of thermodynamics. In earlier work, we showed that for an isolated quantum system the macroscopic entropy does not increase if and only if the system remains in a macroscopic state throughout its evolution. The problem was that, at that stage, “macroscopic state” was defined only implicitly. This made it hard to push the discussion further, because it was unclear what structural, algebraic, or operational features such states really had.

The goal of this work was therefore not to introduce yet another entropy, but to understand macroscopicity itself in a precise way. We wanted a definition that was algebraic, constructive, and operational, and that could serve as a solid foundation for discussing emergence.

Macroscopicity as an inferential boundary

One of the main messages of the paper is that the divide between “macro” and “micro” is not absolute. It depends on the observer. This statement should be taken in a very literal and modest sense.

By an observer we do not mean anything related to the measurement problem or consciousness. We simply mean a specification of which physical quantities can be simultaneously measured. These are what von Neumann already called macroscopic observables. Before discussing emergence, one must decide which variables one is actually focusing on.

Once this choice is made, the distinction between macroscopic and microscopic degrees of freedom coincides exactly with a boundary between what can be inferred and what cannot be inferred. Given the measured variables and some prior information, certain microscopic details can be retrodicted from macroscopic data, while others are irretrievably lost. What counts as macroscopic is determined by this inferential boundary. It is not that macroscopic variables “emerge” from microscopic ones that are “hidden” underneath. Rather, the choice of macroscopic variables determines what is macro and what is micro.

Interestingly, this inferential and retrodictive perspective was not imposed from the outside. It emerged naturally from the mathematics. In hindsight, this was quite telling, especially in light of recent work we did on the role of prediction and retrodiction in fluctuation relations.

From observational entropy to observational deficit

Observational entropy already captures part of this story. It explains how entropy can increase under unitary dynamics when one restricts attention to macroscopic measurements. However, in its standard formulation it relies on a uniform prior.

This is a serious limitation. In thermodynamic settings, a thermal prior is often the natural choice. In infinite-dimensional systems, the uniform state may not even exist. For this reason, we introduced the notion of observational deficit, defined relative to an arbitrary prior. Conceptually, the observational deficit measures how much information about a state is lost when one moves from a microscopic description to macroscopic data, taking into account the observer’s prior knowledge.

Macroscopic states are then precisely those for which this deficit vanishes. They are the states that can be perfectly retrodicted from macroscopic data alone.

Inferential reference frames and the MPPP

A central technical result of the paper is the existence and uniqueness of what we call the maximal projective post-processing (MPPP) of a measurement, relative to a given prior. This object plays a key conceptual role.

The result shows that any observer, defined by a measurement and a prior, can be represented by a suitable projective measurement. This projective measurement captures exactly the information that is inferentially accessible. For this reason, we interpret it as an inferential reference frame.

Just as a symmetry reference frame distinguishes between speakable and unspeakable information, an inferential reference frame distinguishes between what can be retrodicted from macroscopic data and what cannot. It is this structure then that decides what counts as macroscopic and what counts as microscopic for a given observer.

A resource theory of microscopicity

Once macroscopic states are clearly identified, it becomes natural to ask what makes one state more microscopic than another. Answering this question requires more than a classification. It requires an operational framework.

This is why we developed a resource theory of microscopicity. In this theory, macroscopic states are free states, and microscopicity is the resource. The framework forces one to think in terms of operations that do not generate microscopicity and to clarify what is really at stake when microscopic details matter.

An important outcome is that several well-known resource theories appear as special cases. Coherence, athermality, and asymmetry all fit naturally into this framework once appropriate choices of measurements and priors are made. Seeing these theories as instances of microscopicity provides a unified operational interpretation, and sheds light, for example, on the distinction between speakable and unspeakable coherence.

Observer-dependent correlations

The same perspective can be applied to correlations. Entanglement, discord, and related notions are often treated as absolute properties of states. Our results support a different view. The visibility and usefulness of correlations depend on the observer’s inferential reference frame.

Correlations are, after all, correlations among observers, so it is entirely natural to ask what it means for correlations themselves to be macroscopic. What matters is not an abstract, observer-independent property of a state, but which correlations are actually accessible given a specific inferential reference frame. From this perspective, it is natural to start from what correlations look like from the viewpoint of a single observer with limited access, since this already determines which inter-observer correlations can meaningfully be discussed. Our framework provides a starting point for such an analysis, and may eventually lead to something like a relativity theory for inferential reference frames.

What this enables

At a broad level, this work provides a rigorous solution to a basic mathematical problem: what does it mean for a state, an operation, or a correlation to be macroscopic?

Without a precise answer to this question, discussions of the emergence of macroscopic behavior risk remaining vague. With it, we can finally say what we are talking about. This does not solve the problem of emergence by itself, but it clarifies the language and the structure needed to address it in a meaningful way.

The paper is titled Macroscopicity and observational deficit in states, operations, and correlations and appeared in Reports on Progress in Physics earlier this year. It is available for free on the arXiv.