Observational Entropy in Coarse-Grained Systems

• Observational entropy is a coarse-grained measure that quantifies uncertainty relative to selected measurements, unifying Boltzmann, Gibbs/Shannon, and quantum entropy frameworks.
• It decomposes macroscopic uncertainty into a Shannon term for outcome probabilities and a Boltzmann-type term reflecting macrostate volumes in both single and sequential coarse-grainings.
• Its applications span thermodynamics, chaos diagnostics, and quantum correlations, offering actionable insights into equilibrium and out-of-equilibrium dynamics.

Searching arXiv for primary and papers on observational entropy to ground the article in the cited literature. Observational entropy is a coarse-grained entropy for classical and quantum systems that quantifies uncertainty relative to a chosen measurement or sequence of measurements. In the quantum setting, for a POVM or projective coarse-graining, it combines a Shannon term over observed macrostates with a Boltzmann-type term determined by macrostate volumes, and it can be written as the deficit from maximal entropy measured by a quantum-to-classical channel (Nagasawa et al., 2024). In this sense it unifies Boltzmann entropy, Gibbs/Shannon entropy, von Neumann’s macroscopic entropy, and diagonal entropy through different choices of coarse-graining (Nagasawa et al., 2024). Across the literature, it is used as a framework for equilibrium and non-equilibrium thermodynamics, information extraction under generalized measurements, continuity analysis, random-unitary typicality, localization and chaos diagnostics, and extensions based on non-uniform priors and maximum-entropy principles (Šafránek et al., 2020).

1. Formal definition and basic structure

For a finite-dimensional Hilbert space $H$ of dimension $d$, a state is a density operator $\rho$, and a measurement is a POVM $P=\{P_x\}_x$ with $P_x\ge 0$ and $\sum_x P_x=I$. The outcome probabilities and volumes are

$$p_x=\mathrm{Tr}[P_x\rho], \qquad V_x=\mathrm{Tr}[P_x].$$

The observational entropy with respect to $P$ is

$$S_P(\rho)=-\sum_x p_x \ln\!\left(\frac{p_x}{V_x}\right) =\ln d-D\!\big(P(\rho)\,\big\|\,P(u)\big),$$

where $u=I/d$ is the maximally mixed state and $d$0 is the quantum-to-classical channel associated with the measurement (Nagasawa et al., 2024).

This formula exhibits the standard decomposition into outcome uncertainty and unresolved intra-macrostate uncertainty: $d$1 The first term is the Shannon entropy of the measurement outcomes, while the second is the average Boltzmann contribution arising from coarse macrostate volumes (Schindler, 2020). A directly equivalent interpretation is that $d$2 quantifies the distinguishability of $d$3 from the maximally mixed state under the chosen observation (Nagasawa et al., 2024).

Several universal bounds follow. Since relative entropy is nonnegative,

$d$4

with equality if and only if $d$5, equivalently $d$6 for all outcomes (Nagasawa et al., 2024). Observational entropy also obeys the lower bound

$d$7

where $d$8 is the von Neumann entropy, by data processing under the measurement channel (Nagasawa et al., 2024). In the broader projective formalism, the minimum over all sufficiently fine coarse-grainings equals the von Neumann entropy, while the coarsest partition yields the maximal value $d$9 (Schindler, 2020).

The same structure carries over to classical systems. For a partition $\rho$0 of phase space with phase-space density $\rho$1, one defines

$\rho$2

and

$\rho$3

so the classical and quantum expressions are formally parallel, with the main quantum novelty being the role of noncommuting sequential coarse-grainings (Šafránek et al., 2019).

2. Sequential coarse-grainings and structural relations to standard entropies

Although many recent rigorous results focus on a single coarse-graining, the standard extension to sequential coarse-grainings is central to the subject. For projective coarse-grainings $\rho$4, the joint probabilities and joint volumes are

$\rho$5

$\rho$6

and the corresponding entropy is

$\rho$7

For commuting projectors this reduces to a classical partition entropy, whereas for noncommuting projectors the order matters because the products encode a noncommutative refinement (Nagasawa et al., 2024).

This order dependence is one of the defining differences from the classical theory. In classical phase space, multiple coarse-grainings simply intersect, and there is always a joint partition. In the quantum setting, ordered products of projectors can yield different probabilities and different effective volumes, and a single joint partition need not exist (Šafránek et al., 2019). The operational reading is that observational entropy is entropy relative to a specific measurement protocol, not merely to a static partition (Šafránek et al., 2020).

The framework subsumes several standard entropy notions as special cases. If the outcome distribution is concentrated on one macrostate $\rho$8, then

$\rho$9

which is Boltzmann entropy (Nagasawa et al., 2024). If all macrostates have unit volume, then

$P=\{P_x\}_x$0

the Gibbs/Shannon entropy of the outcome distribution (Nagasawa et al., 2024). If $P=\{P_x\}_x$1 is the energy-eigenprojector measurement with $P=\{P_x\}_x$2, observational entropy reduces to diagonal entropy (Nagasawa et al., 2024). For general projective coarse-grainings, it coincides with von Neumann’s macroscopic entropy (Nagasawa et al., 2024).

A related structural identity appears for local product coarse-grainings. For $P=\{P_x\}_x$3 and $P=\{P_x\}_x$4,

$P=\{P_x\}_x$5

where $P=\{P_x\}_x$6 is the Shannon mutual information of the induced joint local measurement statistics (Schindler, 2020). This implies additivity on product states and subadditivity on correlated states (Schindler, 2020).

3. Macrostates, coarse-grained states, and Petz recovery

A major information-theoretic development is the relation between observational entropy, coarse-grained states, and Petz recovery. For a POVM $P=\{P_x\}_x$7, the Petz recovery map relative to the maximally mixed state is

$P=\{P_x\}_x$8

and the corresponding coarse-graining operator is

$P=\{P_x\}_x$9

A state $P_x\ge 0$0 is macroscopic for $P_x\ge 0$1 if and only if

$P_x\ge 0$2

Equivalently, $P_x\ge 0$3 is a fixed point of the coarse-graining map (Nagasawa et al., 2024).

The explicit structure theorem states that $P_x\ge 0$4 is macroscopic for $P_x\ge 0$5 if and only if there exists a PVM $P_x\ge 0$6 with $P_x\ge 0$7, meaning $P_x\ge 0$8 for a partition of outcomes, and coefficients $P_x\ge 0$9 such that

$\sum_x P_x=I$0

As a consequence, any macroscopic state commutes with all POVM elements, and the maximally mixed state is always macroscopic (Nagasawa et al., 2024).

A closely related construct is the coarse-grained or Bayesian-retrodicted state

$\sum_x P_x=I$1

for a coarse-graining with POVM elements $\sum_x P_x=I$2 and probabilities $\sum_x P_x=I$3. This state depends only on the measurement and its outcome statistics and can be obtained as the Petz-recovered state of the measurement channel relative to the uniform prior (Buscemi et al., 2022). The gap between observational entropy and von Neumann entropy is controlled by the distinguishability between the true state and this recovered state: $\sum_x P_x=I$4 There is also an upper continuity-type bound

$\sum_x P_x=I$5

with $\sum_x P_x=I$6 (Buscemi et al., 2022).

This perspective sharpens the statement $\sum_x P_x=I$7: the excess entropy is not merely a generic data-processing remainder, but is quantitatively linked to recoverability under the chosen coarse-graining (Buscemi et al., 2022). A plausible implication is that observational entropy is best viewed not only as a coarse-grained thermodynamic entropy, but also as a recoverability-sensitive measure of how much information remains inaccessible under a specified observation.

4. Continuity, generalized measurements, and general quantum priors

For general POVMs $\sum_x P_x=I$8, observational entropy is

$\sum_x P_x=I$9

The measurement channel

$$p_x=\mathrm{Tr}[P_x\rho], \qquad V_x=\mathrm{Tr}[P_x].$$0

makes explicit the identity

$$p_x=\mathrm{Tr}[P_x\rho], \qquad V_x=\mathrm{Tr}[P_x].$$1

so observational entropy is the observed entropy deficit relative to the maximally mixed state (Schindler et al., 2023).

A central finite-dimensional stability result is the measurement-independent continuity bound

$$p_x=\mathrm{Tr}[P_x\rho], \qquad V_x=\mathrm{Tr}[P_x].$$2

where

$$p_x=\mathrm{Tr}[P_x\rho], \qquad V_x=\mathrm{Tr}[P_x].$$3

This bound does not depend on the number of outcomes or on the volumes $$p_x=\mathrm{Tr}[P_x\rho], \qquad V_x=\mathrm{Tr}[P_x].$$4; it follows from a bounded-concavity property of observational entropy and an Alicki–Fannes–Winter–type argument (Schindler et al., 2023). The same work also shows that $$p_x=\mathrm{Tr}[P_x\rho], \qquad V_x=\mathrm{Tr}[P_x].$$5 is uniformly continuous as a function of the measurement in finite dimension, but that no universal Fannes-type asymptotic bound of the form $$p_x=\mathrm{Tr}[P_x\rho], \qquad V_x=\mathrm{Tr}[P_x].$$6 can hold for measurement continuity (Schindler et al., 2023).

Observational entropy also extends naturally to generalized measurements and measurement sequences formulated at the level of instruments. For an instrument with Kraus operators $$p_x=\mathrm{Tr}[P_x\rho], \qquad V_x=\mathrm{Tr}[P_x].$$7, one has

$$p_x=\mathrm{Tr}[P_x\rho], \qquad V_x=\mathrm{Tr}[P_x].$$8

and the same entropy formula applies (Schindler, 2020). In this framework, observational entropy quantifies how influential a given series of generalized measurements is in information extraction, and many of the familiar properties from projective measurements persist for POVM sequences (Šafránek et al., 2020).

A more recent line of work replaces the implicit uniform prior with a general prior state $$p_x=\mathrm{Tr}[P_x\rho], \qquad V_x=\mathrm{Tr}[P_x].$$9. In the commuting case, the generalized quantity

$P$0

admits both a statistical-deficiency and a Bayesian-retrodiction interpretation (Bai et al., 2023). For noncommuting $P$1 and $P$2, three candidates are proposed, including

$P$3

and a Belavkin–Staszewski-based version

$P$4

with the latter giving a unified fully quantum expression that preserves both major interpretations (Bai et al., 2023). This development is especially relevant in infinite-dimensional or energy-constrained settings, where the uniform prior is not physically meaningful (Bai et al., 2023).

A broader reformulation replaces the standard macrostate volume $P$5 by a prior-dependent volume

$P$6

and defines generalized observational entropy

$P$7

This unifies measurement-based observational entropy with Jaynes’ maximum-entropy framework and recovers the traditional definition under the uniform prior $P$8 (Schindler et al., 19 Mar 2025). The same framework is presented as resolving pathologies of traditional observational entropy in infinite dimensions by replacing divergent standard volumes with physically meaningful prior-induced effective volumes (Schindler et al., 19 Mar 2025).

5. Thermodynamics, equilibration, and entropy increase

Observational entropy is used as a candidate thermodynamic entropy because it can increase under unitary dynamics even though the von Neumann entropy is constant. In isolated systems, this is not automatic for arbitrary coarse-grainings, but it holds generically for thermodynamically motivated ones and, in a rigorous probabilistic sense, for sufficiently coarse observations under random unitary dynamics (Nagasawa et al., 2024).

A basic deterministic result concerns macroscopic initial states. If $P$9 is macroscopic for $$S_P(\rho)=-\sum_x p_x \ln\!\left(\frac{p_x}{V_x}\right) =\ln d-D\!\big(P(\rho)\,\big\|\,P(u)\big),$$0, $$S_P(\rho)=-\sum_x p_x \ln\!\left(\frac{p_x}{V_x}\right) =\ln d-D\!\big(P(\rho)\,\big\|\,P(u)\big),$$1 is a unitary evolution, and $$S_P(\rho)=-\sum_x p_x \ln\!\left(\frac{p_x}{V_x}\right) =\ln d-D\!\big(P(\rho)\,\big\|\,P(u)\big),$$2, then

$$S_P(\rho)=-\sum_x p_x \ln\!\left(\frac{p_x}{V_x}\right) =\ln d-D\!\big(P(\rho)\,\big\|\,P(u)\big),$$3

Thus, observational entropy never decreases for such initial states, and it increases strictly except on a zero-measure set of unitaries preserving macroscopicity (Nagasawa et al., 2024).

For arbitrary initial states, the 2024 random-unitary theorem shows that observational entropy generically approaches its maximum very quickly for sufficiently coarse observations. Writing

$$S_P(\rho)=-\sum_x p_x \ln\!\left(\frac{p_x}{V_x}\right) =\ln d-D\!\big(P(\rho)\,\big\|\,P(u)\big),$$4

one obtains the Haar-random tail bound

$$S_P(\rho)=-\sum_x p_x \ln\!\left(\frac{p_x}{V_x}\right) =\ln d-D\!\big(P(\rho)\,\big\|\,P(u)\big),$$5

and for $$S_P(\rho)=-\sum_x p_x \ln\!\left(\frac{p_x}{V_x}\right) =\ln d-D\!\big(P(\rho)\,\big\|\,P(u)\big),$$6-approximate unitary $$S_P(\rho)=-\sum_x p_x \ln\!\left(\frac{p_x}{V_x}\right) =\ln d-D\!\big(P(\rho)\,\big\|\,P(u)\big),$$7-designs,

$$S_P(\rho)=-\sum_x p_x \ln\!\left(\frac{p_x}{V_x}\right) =\ln d-D\!\big(P(\rho)\,\big\|\,P(u)\big),$$8

Hence, for asymptotically coarse observations, random evolution makes the state macroscopically indistinguishable from the maximally mixed macrostate distribution with high probability (Nagasawa et al., 2024).

The thermodynamic program predates these random-matrix-style concentration results. In isolated many-body systems, physically motivated coarse-grainings such as factorized observational entropy (FOE), built from local energy coarse-grainings, and $$S_P(\rho)=-\sum_x p_x \ln\!\left(\frac{p_x}{V_x}\right) =\ln d-D\!\big(P(\rho)\,\big\|\,P(u)\big),$$9, built from position followed by energy, were argued to rise and approach the equilibrium thermodynamic entropy in closed non-integrable systems (Šafránek et al., 2018). This suggests that observational entropy can supply a microscopic, coarse-grained second-law-like quantity even when the microscopic entropy $u=I/d$0 is invariant.

In open systems, observational entropy was used to derive entropy production as a change in the observational entropy of the universe. For a system coupled to thermal baths and coarse-grained by a fine-grained system measurement together with bath energy measurements, the observational entropy

$u=I/d$1

satisfies

$u=I/d$2

under broad assumptions on the initial product state and the bath energy coarse-graining (Strasberg, 2019). In the weak-coupling limit this recovers the standard entropy balance

$u=I/d$3

and with multiple baths,

$u=I/d$4

(Strasberg, 2019).

A more recent unification with maximum-entropy principles extends these second-law statements. In the generalized prior-based framework,

$u=I/d$5

obeys

$u=I/d$6

under the constraint $u=I/d$7, admits a sequential chain rule, and supports fluctuation and equilibration bounds relative to the time-averaged state or a maximum-entropy prior (Schindler et al., 19 Mar 2025). This suggests a broader thermodynamic role in which equilibrium ensembles, coarse observations, and information-theoretic priors are treated within a single formalism.

6. Applications: localization, chaos, and quantum correlations

Observational entropy has been applied to out-of-equilibrium dynamics, localization transitions, and quantum chaos because it is directly defined from coarse-grained measurement outcomes and does not require state tomography.

In a one-dimensional interacting lattice of spinless fermions, the sequential entropy $u=I/d$8, built from coarse-grained position and total energy, behaves like Boltzmann entropy. Typical long-time states have

$u=I/d$9

at high temperature, while minima correspond to states that localize as many particles as possible into one box (Faiez et al., 2019). This contrasts with bipartite entanglement entropy, whose extremal configurations and scaling behavior differ qualitatively (Faiez et al., 2019).

For the Aubry–André model, observational entropy under real-space coarse-graining distinguishes the delocalized and localized phases. In the delocalized phase, it grows rapidly with coarse-grain size and saturates to the maximal value, whereas in the localized phase the growth is logarithmic in the coarse-grain size (Modak et al., 2022). For fixed coarse-graining, it scales logarithmically with system size in the delocalized phase and obeys an area law in the localized phase (Modak et al., 2022). After a quench from a localized initial state, the entropy grows logarithmically in time in the delocalized phase and at the transition point, but oscillates in the localized phase (Modak et al., 2022). The same work also uses momentum-space coarse-graining to probe the self-dual structure of the model (Modak et al., 2022).

In the quantum kicked top, observational entropy under $d$00-basis coarse-graining witnesses the crossover from regular to chaotic dynamics. In the regular phase, it grows logarithmically with the coarse-graining length beyond a critical value, while in the chaotic regime the growth is much faster and the short-time growth rate acts as a measure of chaoticity (PG et al., 2022). The work compares this behavior with OTOCs and argues that observational entropy is more robust in the deep quantum regime, where OTOC-based diagnostics show strong revivals (PG et al., 2022). Long-time fluctuations of observational entropy also distinguish saddle-point scrambling from true chaos: the former exhibits large persistent fluctuations, while the latter saturates with smaller fluctuations (PG et al., 2022).

A 2026 extension develops a phase-space POVM version based on Pretty Good Measurement corrections to coherent-state Husimi sampling. There, the phase-space observational entropy

$d$01

is used to define an observable Lyapunov exponent through the linear Ehrenfest-regime growth rate

$d$02

which quantitatively reproduces the classical Lyapunov exponent in the standard and singular kicked rotors when the observational resolution exceeds a finite threshold (Kannan et al., 22 May 2026). The same paper uses derivatives of observational entropy as transition diagnostics in the kicked rotor and Aubry–André models (Kannan et al., 22 May 2026). This suggests that observational entropy can function not only as a thermodynamic entropy but also as an experimentally accessible dynamical complexity observable.

Finally, locality-restricted variants relate observational entropy to entanglement and nonclassical correlations. Minimizing observational entropy over local measurement classes defines an entropy gap

$d$03

For bipartite pure states, the gaps for LO$d$04, LO, LOCC, and SEP all equal the entanglement entropy (Rossetti et al., 11 Oct 2025). More generally, the SEP-based gap is lower-bounded by the relative entropy of entanglement, while the LO$d$05 gap coincides with the relative entropy of quantumness (Rossetti et al., 11 Oct 2025). These gaps are not entanglement monotones in general, but they provide a measurement-restriction-based way to quantify inaccessible correlations (Rossetti et al., 11 Oct 2025).

7. Generalizations, limitations, and open directions

Several extensions indicate that observational entropy is better understood as a family of related coarse-grained entropies than as a single fixed formula. One such extension is $d$06-observational entropy,

$d$07

defined via the Petz–Rényi relative entropy of the measured output distributions. It reduces to standard observational entropy as $d$08, is monotone under coarse-graining, satisfies

$d$09

and is non-increasing in $d$10 (Sinha et al., 2023). This provides a tunable family emphasizing either typical or rare macrostates depending on the parameter regime (Sinha et al., 2023).

Another extension comes from the decomposition

$d$11

where

$d$12

is inter-block coherence and

$d$13

is intra-block noise. This decomposition has been proposed as the basis of a resource degradation theory in which coherent resources can degrade into blockwise classical noise while the total inconsistency remains approximately conserved (Zhou, 27 Nov 2025). A resource purity ratio

$d$14

is used there to diagnose quality degradation in variational quantum algorithms (Zhou, 27 Nov 2025).

The literature also identifies clear limitations. Concentration theorems for entropy increase require sufficiently coarse observations; fine-grained partitions need not exhibit concentration near the maximum (Nagasawa et al., 2024). Many thermodynamic arguments rely on non-integrability, weak interactions between coarse subsystems, or random-unitary surrogates rather than explicit few-body Hamiltonians (Šafránek et al., 2018). Continuity in the measurement argument is subtler than continuity in the state argument, since no universal Fannes-type asymptotic bound exists across all POVMs (Schindler et al., 2023). And while generalized prior frameworks resolve certain infinite-dimensional divergences, the finite-outcome restriction remains important in existing continuity and concentration results (Schindler et al., 19 Mar 2025).

Open directions stated in the literature include extending concentration results from random unitaries and approximate designs to concrete local Hamiltonian dynamics, sharpening finite-size constants, clarifying the precise relation to ETH, developing fluctuation theorems for time-dependent generalized observational entropy, and applying prior-based coarse-grained entropies to field theory or gravity (Nagasawa et al., 2024). A plausible implication is that future work will continue to shift the emphasis from a single entropy formula toward a modular framework in which measurements, priors, dynamical constraints, and operational tasks determine the relevant observational entropy.

Observational entropy therefore occupies a distinctive position at the intersection of statistical mechanics, quantum information, and measurement theory. In its standard finite-dimensional form,

$d$15

it quantifies macroscopic uncertainty under coarse observation while interpolating among several classical and quantum entropy notions (Nagasawa et al., 2024). In its generalized forms, it incorporates non-uniform priors, maximum-entropy constraints, Rényi deformations, locality restrictions, and resource decompositions (Schindler et al., 19 Mar 2025). Across these variants, the recurring theme is the same: entropy is defined not only by the state itself, but also by what is observable, at what resolution, and relative to which physically meaningful prior.