---
title: Observational Entropy in Coarse-Grained Systems
url: https://www.emergentmind.com/topics/observational-entropy
type: topic
---

# Observational Entropy in Coarse-Grained Systems

Searching arXiv for primary and recent 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 [2404.11985]. In this sense it unifies Boltzmann entropy, Gibbs/Shannon entropy, von Neumann’s macroscopic entropy, and diagonal entropy through different choices of coarse-graining [2404.11985]. 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 [2008.04409].

## 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 \(P(\cdot)=\sum_x \mathrm{Tr}[P_x\cdot]|x\rangle\langle x|\) is the quantum-to-classical channel associated with the measurement [2404.11985].

This formula exhibits the standard decomposition into outcome uncertainty and unresolved intra-macrostate uncertainty:
\[
S_P(\rho)=-\sum_x p_x\ln p_x+\sum_x p_x\ln V_x.
\]
The first term is the Shannon entropy of the measurement outcomes, while the second is the average Boltzmann contribution arising from coarse macrostate volumes [2010.00142]. A directly equivalent interpretation is that \(\ln d-S_P(\rho)=D(P(\rho)\|P(u))\) quantifies the distinguishability of \(\rho\) from the maximally mixed state under the chosen observation [2404.11985].

Several universal bounds follow. Since relative entropy is nonnegative,
\[
S_P(\rho)\le \ln d,
\]
with equality if and only if \(P(\rho)=P(u)\), equivalently \(p_x=V_x/d\) for all outcomes [2404.11985]. Observational entropy also obeys the lower bound
\[
S_P(\rho)\ge S(\rho),
\]
where \(S(\rho)=-\mathrm{Tr}[\rho\ln\rho]\) is the von Neumann entropy, by data processing under the measurement channel [2404.11985]. 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 \(\ln \dim H\) [2010.00142].

The same structure carries over to classical systems. For a partition \(\Gamma=\bigsqcup_i \Gamma_i\) of phase space with phase-space density \(f\), one defines
\[
p_i=\int_{\Gamma_i} f(x)\,dx,\qquad V_i=\int_{\Gamma_i}dx,
\]
and
\[
S_O=-\sum_i p_i\ln(p_i/V_i),
\]
so the classical and quantum expressions are formally parallel, with the main quantum novelty being the role of noncommuting sequential coarse-grainings [1905.03841].

## 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 \(C_k=\{\Pi^{(k)}_{i_k}\}\), the joint probabilities and joint volumes are
\[
p_{i_1\cdots i_n}
=\mathrm{Tr}\!\left(\Pi^{(n)}_{i_n}\cdots \Pi^{(1)}_{i_1}\,\rho\,\Pi^{(1)}_{i_1}\cdots \Pi^{(n)}_{i_n}\right),
\]
\[
V_{i_1\cdots i_n}
=\mathrm{Tr}\!\left(\Pi^{(n)}_{i_n}\cdots \Pi^{(1)}_{i_1}\right),
\]
and the corresponding entropy is
\[
S_{\mathrm{obs}}(\rho;C_1,\dots,C_n)
=-\sum_{i_1,\dots,i_n}p_{i_1\cdots i_n}\,
\ln\!\left(\frac{p_{i_1\cdots i_n}}{V_{i_1\cdots i_n}}\right).
\]
For commuting projectors this reduces to a classical partition entropy, whereas for noncommuting projectors the order matters because the products encode a noncommutative refinement [2404.11985].

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 [1905.03841]. The operational reading is that observational entropy is entropy relative to a specific measurement protocol, not merely to a static partition [2008.04409].

The framework subsumes several standard entropy notions as special cases. If the outcome distribution is concentrated on one macrostate \(\bar x\), then
\[
S_P(\rho)=\ln V_{\bar x},
\]
which is Boltzmann entropy [2404.11985]. If all macrostates have unit volume, then
\[
S_P(\rho)=-\sum_x p_x\ln p_x,
\]
the Gibbs/Shannon entropy of the outcome distribution [2404.11985]. If \(P\) is the energy-eigenprojector measurement with \(V_x=1\), observational entropy reduces to diagonal entropy [2404.11985]. For general projective coarse-grainings, it coincides with von Neumann’s macroscopic entropy [2404.11985].

A related structural identity appears for local product coarse-grainings. For \(H=\bigotimes_X H_X\) and \(C=\bigotimes_X C_X\),
\[
S_O(C_A\otimes\cdots\otimes C_C;\rho)
=\sum_X S_O(C_X;\rho_X)-I_C(\rho),
\]
where \(I_C(\rho)\) is the Shannon mutual information of the induced joint local measurement statistics [2010.00142]. This implies additivity on product states and subadditivity on correlated states [2010.00142].

## 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 \(P=\{P_x\}\), the Petz recovery map relative to the maximally mixed state is
\[
R_{P,u}(\cdot)=\sum_x \langle x|(\cdot)|x\rangle\,\frac{P_x}{\mathrm{Tr}[P_x]},
\]
and the corresponding coarse-graining operator is
\[
[R_{P,u}\circ P](\rho)=\sum_x \mathrm{Tr}[P_x\rho]\frac{P_x}{\mathrm{Tr}[P_x]}.
\]
A state \(m\) is macroscopic for \(P\) if and only if
\[
m=\sum_x \mathrm{Tr}[P_x m]\,\frac{P_x}{V_x}.
\]
Equivalently, \(m\) is a fixed point of the coarse-graining map [2404.11985].

The explicit structure theorem states that \(m\) is macroscopic for \(P\) if and only if there exists a PVM \(\Pi=\{\Pi_y\}\) with \(\Pi\preceq P\), meaning \(\Pi_y=\sum_{x\in X_y}P_x\) for a partition of outcomes, and coefficients \(c_y\ge 0\) such that
\[
m=\sum_y c_y\,\Pi_y.
\]
As a consequence, any macroscopic state commutes with all POVM elements, and the maximally mixed state is always macroscopic [2404.11985].

A closely related construct is the coarse-grained or Bayesian-retrodicted state
\[
\sigma=\sum_i \frac{p_i}{V_i}\,\Pi_i,
\]
for a coarse-graining with POVM elements \(\Pi_i\) and probabilities \(p_i=\mathrm{Tr}[\Pi_i\rho]\). 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 [2209.03803]. The gap between observational entropy and von Neumann entropy is controlled by the distinguishability between the true state and this recovered state:
\[
S_{\mathcal C}(\rho)-S(\rho)\ge D(\rho\|\sigma).
\]
There is also an upper continuity-type bound
\[
S_{\mathcal C}(\rho)-S(\rho)\le T(\rho,\sigma)\ln(d-1)+h[T(\rho,\sigma)],
\]
with \(T(\rho,\sigma)=\frac12\|\rho-\sigma\|_1\) [2209.03803].

This perspective sharpens the statement \(S_{\mathcal C}(\rho)\ge S(\rho)\): the excess entropy is not merely a generic data-processing remainder, but is quantitatively linked to recoverability under the chosen coarse-graining [2209.03803]. 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 \(M=(M_i)\), observational entropy is
\[
S_M(\rho)=-\sum_i p_i\log\!\Big(\frac{p_i}{V_i}\Big),
\qquad
p_i=\operatorname{tr}(M_i\rho),\quad
V_i=\operatorname{tr}(M_i).
\]
The measurement channel
\[
\Phi_M(\sigma)=\sum_i \operatorname{tr}(\sigma M_i)\,|i\rangle\langle i|
\]
makes explicit the identity
\[
D\!\big(\Phi_M(\rho)\big\Vert \Phi_M(\tfrac{1}{d})\big)=\log d-S_M(\rho),
\]
so observational entropy is the observed entropy deficit relative to the maximally mixed state [2302.00400].

A central finite-dimensional stability result is the measurement-independent continuity bound
\[
|S_M(\rho)-S_M(\sigma)|\le g(\delta)+\delta\log d,
\qquad
\delta=\tfrac12\|\rho-\sigma\|_1,
\]
where
\[
g(x)=
\begin{cases}
-x\log x+(1+x)\log(1+x), & x>0,\\
0, & x=0.
\end{cases}
\]
This bound does not depend on the number of outcomes or on the volumes \(V_i\); it follows from a bounded-concavity property of observational entropy and an Alicki–Fannes–Winter–type argument [2302.00400]. The same work also shows that \(S_M(\rho)\) is uniformly continuous as a function of the measurement in finite dimension, but that no universal Fannes-type asymptotic bound of the form \(f(\gamma)\log d\) can hold for measurement continuity [2302.00400].

Observational entropy also extends naturally to generalized measurements and measurement sequences formulated at the level of instruments. For an instrument with Kraus operators \(K_i\), one has
\[
p_i=\mathrm{Tr}(K_i\rho K_i^\dagger),\qquad
V_i=\mathrm{Tr}(K_iK_i^\dagger),
\]
and the same entropy formula applies [2010.00142]. 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 [2007.07246].

A more recent line of work replaces the implicit uniform prior with a general prior state \(\gamma\). In the commuting case, the generalized quantity
\[
S_{\mathcal M,\gamma}^{\mathrm{clax}}(\rho)
= S(\rho)+D(\rho\|\gamma)-D(M(\rho)\|M(\gamma))
\]
admits both a statistical-deficiency and a Bayesian-retrodiction interpretation [2308.08763]. For noncommuting \(\rho\) and \(\gamma\), three candidates are proposed, including
\[
S^{(1)}_{\mathcal M,\gamma}(\rho)
= S(\rho)+D(\rho\|\gamma)-D(M(\rho)\|M(\gamma))
\]
and a Belavkin–Staszewski-based version
\[
S^{(3)}_{\mathcal M,\gamma}(\rho)
= S(\rho)+D_{BS}(\rho\|\gamma)-D(M(\rho)\|M(\gamma)),
\]
with the latter giving a unified fully quantum expression that preserves both major interpretations [2308.08763]. This development is especially relevant in infinite-dimensional or energy-constrained settings, where the uniform prior is not physically meaningful [2308.08763].

A broader reformulation replaces the standard macrostate volume \(W_\alpha=\mathrm{Tr}(M_\alpha)\) by a prior-dependent volume
\[
V_\alpha=\mathrm{Tr}(\tau M_\alpha)e^{S(\tau)},
\]
and defines generalized observational entropy
\[
S_M^\tau(\rho)=S(\tau)-D_M(\rho\|\tau)
=-\sum_\alpha p_\alpha \ln\!\left(\frac{p_\alpha}{V_\alpha}\right).
\]
This unifies measurement-based observational entropy with Jaynes’ maximum-entropy framework and recovers the traditional definition under the uniform prior \(\tau\propto I/d\) [2503.15612]. 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 [2503.15612].

## 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 [2404.11985].

A basic deterministic result concerns macroscopic initial states. If \(\rho_0\) is macroscopic for \(P\), \(U\) is a unitary evolution, and \(\rho_1=U\rho_0 U^\dagger\), then
\[
S_P(\rho_1)=S_{U^\dagger P U}(\rho_0)\ge S_P(\rho_0)=S(\rho_0)=S(\rho_1).
\]
Thus, observational entropy never decreases for such initial states, and it increases strictly except on a zero-measure set of unitaries preserving macroscopicity [2404.11985].

For arbitrary initial states, the 2024 random-unitary theorem shows that observational entropy generically approaches its maximum very quickly for sufficiently coarse observations. Writing
\[
\kappa(P)=\frac{1}{d}\min_x V_x,
\]
one obtains the Haar-random tail bound
\[
\Pr_H\!\left\{S_P(U\rho U^\dagger)\le (1-\delta)\ln d\right\}
\le
\frac{4}{\kappa(P)}
\exp\!\left(-\frac{\delta}{18\pi^3}\kappa(P)^2 d\ln d\right),
\]
and for \(\varepsilon\)-approximate unitary \(2\)-designs,
\[
\Pr_{\mathcal E}\!\left\{S_P(U\rho U^\dagger)\le (1-\delta)\ln d\right\}
\le
\frac{1}{\kappa(P)^3 d\ln d}\frac{4(1+\varepsilon)}{\delta}.
\]
Hence, for asymptotically coarse observations, random evolution makes the state macroscopically indistinguishable from the maximally mixed macrostate distribution with high probability [2404.11985].

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 \(S_{xE}\), built from position followed by energy, were argued to rise and approach the equilibrium thermodynamic entropy in closed non-integrable systems [1803.00665]. This suggests that observational entropy can supply a microscopic, coarse-grained second-law-like quantity even when the microscopic entropy \(S(\rho)\) 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
\[
S_{\mathrm{obs}}(t)
=
-\sum_{s,E}p_{sE}(t)\ln\!\left(\frac{p_{sE}(t)}{V_{E,\delta}}\right)
\]
satisfies
\[
\Delta S_{\mathrm{obs}}(t)\ge 0
\]
under broad assumptions on the initial product state and the bath energy coarse-graining [1906.09933]. In the weak-coupling limit this recovers the standard entropy balance
\[
\Delta S_{\mathrm{obs}}\approx \Delta S_{\mathrm{Sh}}[p_s(t)]-\beta Q,
\]
and with multiple baths,
\[
\Delta S_{\mathrm{obs}}(t)\approx \Delta S_{\mathrm{Sh}}[p_s(t)]-\sum_\nu \beta_\nu Q_\nu
\]
[1906.09933].

A more recent unification with maximum-entropy principles extends these second-law statements. In the generalized prior-based framework,
\[
S_M^\tau(\rho)=S(\tau)-D_M(\rho\|\tau)
\]
obeys
\[
S(\tau)\ge S_M^\tau(\rho)\ge S(\rho)
\]
under the constraint \(S(\rho;\tau)\le S(\tau)\), admits a sequential chain rule, and supports fluctuation and equilibration bounds relative to the time-averaged state or a maximum-entropy prior [2503.15612]. 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 \(S_{xE}\), built from coarse-grained position and total energy, behaves like Boltzmann entropy. Typical long-time states have
\[
S_{xE}(\mathrm{average})\approx S_{xE}(\max)\approx S_{\mathrm{th}}(A+B)
\]
at high temperature, while minima correspond to states that localize as many particles as possible into one box [1908.07083]. This contrasts with bipartite entanglement entropy, whose extremal configurations and scaling behavior differ qualitatively [1908.07083].

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 [2209.10273]. For fixed coarse-graining, it scales logarithmically with system size in the delocalized phase and obeys an area law in the localized phase [2209.10273]. 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 [2209.10273]. The same work also uses momentum-space coarse-graining to probe the self-dual structure of the model [2209.10273].

In the quantum kicked top, observational entropy under \(J_z\)-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 [2212.01585]. 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 [2212.01585]. 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 [2212.01585].

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
\[
S_H(\rho)=-\sum_{k,l} p_{kl}\ln\!\left(\frac{p_{kl}}{V_{kl}}\right)
\]
is used to define an observable Lyapunov exponent through the linear Ehrenfest-regime growth rate
\[
\lambda_{OE}\equiv \frac{dS_O}{dt},
\]
which quantitatively reproduces the classical Lyapunov exponent in the standard and singular kicked rotors when the observational resolution exceeds a finite threshold [2605.23585]. The same paper uses derivatives of observational entropy as transition diagnostics in the kicked rotor and Aubry–André models [2605.23585]. 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
\[
\Delta_{\mathsf M}(\rho)=S_O^{\min,\mathsf M}(\rho)-S(\rho).
\]
For bipartite pure states, the gaps for LO\(^*\), LO, LOCC, and SEP all equal the entanglement entropy [2510.10058]. More generally, the SEP-based gap is lower-bounded by the relative entropy of entanglement, while the LO\(^*\) gap coincides with the relative entropy of quantumness [2510.10058]. These gaps are not entanglement monotones in general, but they provide a measurement-restriction-based way to quantify inaccessible correlations [2510.10058].

## 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 \(\alpha\)-observational entropy,
\[
S_\alpha^O(\rho;x)
=
-\frac{1}{\alpha-1}
\ln\!\left(\sum_i p_i^\alpha V_i^{1-\alpha}\right),
\]
defined via the Petz–Rényi relative entropy of the measured output distributions. It reduces to standard observational entropy as \(\alpha\to 1\), is monotone under coarse-graining, satisfies
\[
S_\alpha^O(\rho;x)\ge S_\alpha^R(\rho),
\]
and is non-increasing in \(\alpha\) [2312.03572]. This provides a tunable family emphasizing either typical or rare macrostates depending on the parameter regime [2312.03572].

Another extension comes from the decomposition
\[
\mathcal O_{\mathcal C}(\rho)
=
S_{\mathcal C}(\rho)-S(\rho)
=
\mathcal C_{\mathrm{rel}}(\rho;\mathcal C)+\mathcal D_{\mathrm{rel}}(\rho),
\]
where
\[
\mathcal C_{\mathrm{rel}}(\rho;\mathcal C)=S(\Delta[\rho])-S(\rho)
\]
is inter-block coherence and
\[
\mathcal D_{\mathrm{rel}}(\rho)=\sum_x p_x D(\rho_x\|\kappa_x)
\]
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 [2511.22350]. A resource purity ratio
\[
\eta(\rho)=\frac{\mathcal C_{\mathrm{rel}}(\rho;\mathcal C)}{\mathcal O_{\mathcal C}(\rho)}
\]
is used there to diagnose quality degradation in variational quantum algorithms [2511.22350].

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 [2404.11985]. Many thermodynamic arguments rely on non-integrability, weak interactions between coarse subsystems, or random-unitary surrogates rather than explicit few-body Hamiltonians [1803.00665]. Continuity in the measurement argument is subtler than continuity in the state argument, since no universal Fannes-type asymptotic bound exists across all POVMs [2302.00400]. And while generalized prior frameworks resolve certain infinite-dimensional divergences, the finite-outcome restriction remains important in existing continuity and concentration results [2503.15612].

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 [2404.11985]. 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,
\[
S_P(\rho)
=
-\sum_x \mathrm{Tr}[P_x\rho]\,
\ln\!\left(\frac{\mathrm{Tr}[P_x\rho]}{\mathrm{Tr}[P_x]}\right)
=
\ln d-D(P(\rho)\|P(u)),
\]
it quantifies macroscopic uncertainty under coarse observation while interpolating among several classical and quantum entropy notions [2404.11985]. In its generalized forms, it incorporates non-uniform priors, maximum-entropy constraints, Rényi deformations, locality restrictions, and resource decompositions [2503.15612]. 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.

Source: https://www.emergentmind.com/topics/observational-entropy