When a Coarse-Grained Detector Miscounts Dissipation
How hidden quantum information can leak back into visible dynamics—and how detector-level tests can keep the thermodynamic accounting honest
Keywords: quantum thermodynamics, coarse graining, entropy production, open quantum systems, quantum information, Lindblad dynamics, statistical physics, preprint
Preprint note — 7 September 2026: This essay is a nontechnical account of my research manuscript, Robust resource-loss bounds under approximate quantum coarse-graining covariance. The manuscript has been submitted to Physical Review Research as a Regular Article and has not yet completed peer review. It should therefore be read as a preprint account, not as an accepted or published result. Statements, terminology, and interpretations may change in response to editorial and referee feedback.
Two states that look the same
Imagine a detector that reports only whether a quantum system is in a low-energy sector or a high-energy sector. Each sector contains several microscopic energy levels, but the detector does not distinguish them. It also misses some quantum coherences. Many genuinely different states therefore produce the same visible record.
At the initial time, two such states might give exactly the same histogram: 70 percent low sector, 30 percent high sector. From the detector’s point of view, they are the same macrostate.
Now let both states evolve and measure them again.
Must the later histograms still agree?
If they do, the result is consistent with closure for that tested pair and time. Establishing global dynamical closure requires the later records to agree for every pair of coarse-equivalent inputs across the relevant evolution times. Under that stronger condition, the present detector record contains everything needed to predict the future detector record. If the histograms separate even once, information that was initially hidden has returned to influence what the detector can see.
That possibility creates a subtle problem in quantum thermodynamics. We often treat the information discarded by coarse graining as part of an entropy-production ledger. But losing access to microscopic detail is not automatically the same thing as physically dissipating it. Hidden information may persist, affect later populations, or be converted into visible structure. A detector can forget something that the dynamics has not destroyed.
The important question is therefore not simply what the detector cannot see. It is whether what the detector cannot see can later change what it does see.
My preprint develops a quantitative way to ask that question. Under a precise class of coarse-graining maps, it separates total relaxation into three contributions, identifies the error produced by failure of dynamical closure, and turns that error into something that can be measured from two detector histograms. It also gives worst-case and finite-sample bounds when closure is only approximate.
What coarse graining actually does
Coarse graining is often described as throwing information away. That is correct, but incomplete. A usable coarse model must also say what state we assign after observing a coarse outcome.
In the construction used in the paper, the detector divides the system into mutually exclusive sectors. It records the probability of each sector and then represents the unresolved state inside each sector by a fixed reference state. In a thermal setting, that reference can be the equilibrium state conditioned on the observed sector.
This is a measure-and-reprepare description. First ask which sector occurred; then replace everything the detector did not resolve with an agreed representative state. The resulting reconstructed macrostate preserves the detector histogram while discarding inter-sector coherence and microscopic variation inside each sector.
The gap between the original microscopic state and this reconstructed state can be quantified using relative entropy, an information-theoretic measure of distinguishability. I call this the hidden information of the state. It includes both coherence between sectors and mismatch between the actual within-sector state and the representative state assigned by the coarse model.
A second relative-entropy quantity measures how far the reconstructed macrostate remains from a stationary reference, such as a Gibbs state. That part is visible at detector level because it depends only on the sector probabilities.
The information geometry behind this hidden-plus-visible decomposition is established theory, connected to conditional expectations, observational entropy, and Petz recovery. The novelty claimed in the preprint is not that decomposition itself. The new issue is dynamical: what happens to the thermodynamic accounting when the evolution and the coarse-graining operation are not exactly compatible?
One compatibility assumption hides two different questions
It is tempting to demand that coarse graining and evolution commute: evolve first and coarse grain later, or coarse grain first and evolve later, and obtain the same result. But prior work on resource-destroying maps shows that this apparently single demand contains two independent conditions.
The first is macroscopic closure. It asks whether future detector statistics depend only on the current detector description. If two microscopic states have the same coarse record now, closure says that their later coarse records must also agree. This is the quantum-detector analogue of strong lumpability in classical Markov chains.
The second is image invariance. It asks whether a reconstructed macrostate remains inside the chosen family of reconstructed states as it evolves. The dynamics may take a perfectly valid coarse state and create new microscopic structure that the reconstruction does not contain. Image invariance rules that out.
These conditions are easy to conflate, but they answer different physical questions. Closure concerns prediction: can the coarse variables predict themselves? Image invariance concerns model preservation: does evolution keep us inside the reconstructed-state family? Full compatibility requires both.
For thermodynamic inference, closure is the crucial condition controlling whether a reduction in hidden information can safely be credited toward total dissipation. Image invariance controls the opposite side of the ledger: whether evolution can create new hidden structure from a reconstructed state.
Neither condition can silently stand in for the other. The paper gives explicit four-state channels showing both logical separations. Closure can hold even while hidden information increases. Image invariance can hold even while hidden information changes later detector probabilities. Treating commutation as a single yes-or-no assumption conceals which inference has failed.
A three-part ledger
The central result is an exact accounting identity. Under the paper’s assumptions, total contraction of relative entropy toward a stationary reference separates into three terms:
Total contraction = change in hidden information + resolved relaxation + closure correction.
The first term records how much information hidden by the detector decreases or increases during evolution.
The second is the relaxation visible when the reconstructed macrostate is itself evolved. This term is always nonnegative. It is ordinary contraction toward the stationary state along the coarse-prepared route.
The third is a signed correction caused by failure of macroscopic closure. It records hidden-to-visible leakage: the extent to which microscopic details omitted by the coarse state alter later detector statistics.
“Signed” matters. The correction may be positive or negative. If it is negative enough, the decrease in hidden information can appear larger than the total contraction. A naive inequality then overcounts dissipation—not because relative entropy ceased to contract, but because some of the change was assigned to the wrong column of the ledger.
This is the main conceptual point. Coarse graining does not merely omit part of the thermodynamic account. When closure fails, it can move information between the hidden and visible columns.
For a general channel with a fixed reference state, the total quantity here should be called relative-entropy contraction. It becomes physical integrated entropy production only under additional thermodynamic assumptions. In the paper’s fixed-Hamiltonian, undriven thermal-relaxation model, it equals the system entropy change minus inverse temperature times the heat absorbed by the system. I do not claim that interpretation for every fixed-point-preserving quantum channel.
A two-route detector test
The signed correction becomes useful because it can be obtained from an operational comparison.
In the first route, prepare the microscopic input, let it evolve for a chosen time, and then measure which detector sector it occupies.
In the second route, begin with the reconstructed coarse state corresponding to the same input, apply the same evolution, and measure the same sectors.
If the two histograms agree, hidden microscopic details have not affected the visible detector statistics for that input and time. If they differ, their total-variation distance gives a direct state-dependent measure of closure failure. The signed correction itself can be calculated from the two histograms together with the stationary sector weights.
This is much cheaper than reconstructing both complete output states. It turns a structural assumption about two quantum operations into a falsifiable comparison of counts.
There is an important limitation. The test does not make the whole dissipation detector-only. The change in hidden information still requires microscopic knowledge, tomography, or an independent resource witness. What the two-route test does is calibrate whether that hidden-information change may be interpreted as part of the total contraction. It prevents an untested closure assumption from being smuggled into the conclusion.
The route correction gives the tightest certificate developed in the paper when both preparations are available. Because the resolved-relaxation term is nonnegative, removing it leaves a guaranteed lower bound on the total contraction. Instead of assuming exact compatibility, one measures the correction produced by its failure.
From exact closure to finite-error guarantees
Directly evaluating the signed correction requires the stationary weights as well as the two histograms. Sometimes one wants a simpler worst-case guarantee based only on how far apart those histograms are.
The paper derives such a bound. Its penalty depends on three things: the histogram discrepancy, the number of detector sectors, and the stationary probability of those sectors. It does not depend directly on the full microscopic Hilbert-space dimension or on the smallest eigenvalue hidden inside each unresolved sector.
That scaling distinction is important. A macroscopic detector may compress an enormous microscopic space into a handful of outcomes. A continuity bound formulated on the full quantum state space pays for every microscopic dimension. The detector-level bound instead pays for the alphabet the experiment actually distinguishes.
This should not be oversold as dimension independence in every possible sense. Sector degeneracy can affect the stationary sector weights, which in turn affect the penalty. The result also relies on the detector-simplex construction: orthogonal sectors and an idempotent measure-and-reprepare assignment. Generalized measurements and noncommuting-prior recovery maps require different geometry.
For uniform stationary sector weights, the underlying probability-distribution penalty is mathematically sharp over the mismatch interval stated in the paper. No uniformly smaller function of the same limited information works throughout that interval and setting. The manuscript also supplies a unitary construction that attains equality at the endpoint. This does not mean ordinary physical systems saturate the bound; it means the proved worst-case dependence cannot simply be wished away.
Real histograms also contain sampling noise. The finite-shot version uses standard concentration inequalities to enlarge the observed mismatch by a confidence-qualified allowance. In the two-sector example, 10,000 repetitions of each route and a 95 percent confidence level add approximately 0.032 to the allowed histogram discrepancy.
That confidence statement applies to sampling of the two route histograms. It does not automatically include uncertainty in the hidden-information estimate, stationary weights, state preparation, detector calibration, or dynamical model. Those errors must be propagated separately in an actual experiment.
A thermally consistent counterexample
A correction is only interesting if the uncorrected statement can genuinely fail. The paper therefore constructs a four-level open quantum system divided into two detector sectors. Thermal transition rates obey detailed balance, and the Gibbs state remains fixed and unique. The model is thermodynamically consistent throughout the parameter range studied.
When two related cross-sector transition rates are symmetric, coarse graining and dynamics are exactly compatible. Introducing a controlled asymmetry lets a population imbalance hidden inside a sector influence the later sector probabilities. Closure and image invariance then fail even though detailed balance remains intact.
For a deliberately selected stress-test state, the naive hidden-information bound exceeds the total entropy production near the beginning of the trajectory. The largest overcount is about 0.000467 natural-log information units at dimensionless time 0.35. This is small, but it is genuine: it is neither rounding error nor the output of an unphysical channel.
At that same point, the correction based on the two preparation routes retains 99.95 percent of the true total contraction. The simpler histogram-distance certificate retains 91.31 percent. For an illustrative finite-shot calculation, I treat the exact model discrepancy as the observed discrepancy and add the 95 percent sampling allowance for 10,000 repetitions per route. The resulting lower bound retains 69.35 percent. Actual sampled histograms would fluctuate and therefore produce a random retained fraction.
These percentages describe one intentionally sensitive model trajectory. They are not claims about the size or frequency of violations in nature.
Figure 1. A deliberately chosen four-level thermal stress test. In panel (a), the corrected certificates remain below the total contraction while the raw hidden-information curve briefly overcounts it. Panel (b) identifies the cause: an early negative closure correction outweighs the positive resolved contribution. Panel (c) shows the detector-level mismatch and the corresponding worst-case continuity penalty. Time is expressed in units of the model’s internal relaxation rate.
Beyond one carefully chosen trajectory
A single example can establish logical failure, but it cannot show whether the correction remains useful away from that example. I therefore tested 2,000 independently sampled four-level quantum states. Each state was evaluated at six times and four rate asymmetries, producing 48,000 state-time-asymmetry evaluations.
The distinction matters: these are 48,000 evaluations, not 48,000 independent states.
At the largest tested asymmetry, the uncorrected inequality failed in 1.30 percent of the specified evaluations. The route certificate retained a median 86.72 percent of total contraction, with a fifth percentile of 57.63 percent. The norm-only certificate retained a median 76.87 percent, with a fifth percentile of 48.54 percent. The norm certificate remained positive in every sampled case.
These are conditional statistics for one Hilbert–Schmidt state ensemble, one four-level model, and one fixed time-and-parameter grid. They are not universal probabilities for quantum systems or experiments. Their purpose is narrower: to test the implementation across diverse inputs and show how the certificates behave within the declared numerical design.
The full reproducibility suite performs 45 registered checks. They cover the validity of the coarse-graining channel, complete positivity and trace preservation of representative propagators, Gibbs stationarity, pairwise detailed balance, the exact three-part identity, both certificates, randomized theorem cases, raw matrix diagnostics, finite-shot arithmetic, and generator estimates. All 45 checks pass. The largest residual in the central accounting identity is approximately 0.000000000000000666, at the scale expected from floating-point arithmetic.
[[FIGURE_2]]
Figure 2. Ensemble and generator audit. Panel (a) reports the median and fifth-percentile fractions certified across 48,000 evaluations of 2,000 independent states. Panel (b) shows that raw-bound failures become more common as the chosen rate asymmetry increases, while the corrected norm certificate remains positive throughout the tested ensemble. Panel (c) compares the directly calculated closure defect with two model-based upper bounds.
When the dynamics is known but the routes are unavailable
The two-route comparison is attractive because it measures closure failure directly. It may not always be experimentally available. If a trusted Markovian model of the dynamics exists, the paper also bounds the error using the mismatch between the coarse-graining operation and the dynamical generator.
The simplest generator bound grows linearly with time. It is uniform and easy to state, but eventually becomes useless: a bound that keeps growing cannot describe a relaxing system whose memory of the initial state disappears.
The mixing-aware alternative incorporates relaxation toward the stationary state. Under the stated primitive-mixing assumptions, it vanishes both at the initial time and at late times. In the numerical example it captures the correct late-time decay and substantially improves on the linear estimate. At dimensionless time 6, the directly calculated defect is about 0.000573, the mixing-aware upper estimate is about 0.224, and the linear estimate is about 4.74.
The improvement is large, but the mixing estimate is still conservative. It should not be described as a tight reconstruction of the actual error. The practical hierarchy is simple: use route histograms when both preparations are available; use the detector-distance penalty when the mismatch can be estimated; use a generator bound when direct route calibration is unavailable but the model is trusted.
What the result does—and does not—say
The preprint establishes an exact ledger and several rigorous certificates within a defined setting. It shows that detector-level closure failure has a signed contribution, that the contribution can be read from sector histograms, and that approximate closure can support finite-error and finite-shot lower bounds.
It does not show that every loss of microscopic information is heat, irreversibility, or entropy production. It does not reconstruct total dissipation from detector counts alone. It does not yet cover arbitrary generalized measurements or every prior-dependent recovery map. It does not claim that the generator estimates are tight. And it does not turn the ensemble’s 1.30 percent failure frequency into a universal probability.
The work is best understood as a calibration framework. If hidden-information loss is going to be counted toward dissipation, the dynamical closure connecting those ideas should be tested rather than assumed.
This distinction extends beyond the particular four-level model. Coarse descriptions appear wherever experiments have finite resolution, reduced models eliminate variables, or macrostates stand in for microscopic states. In every such case there are two separate acts: choosing what to observe and assuming how the unobserved degrees of freedom influence future observations. The first is a statement about resolution. The second is a statement about dynamics.
Exact closure is an idealization. Approximate closure is the experimentally relevant question: how badly is the condition broken, how much can that alter the information ledger, and what can the available data or model certify?
The two-route protocol turns that question into a comparison of histograms. The continuity bound translates their disagreement into a worst-case correction. The finite-shot result adds honest counting uncertainty. The generator analysis supplies a fallback when one must rely on a trusted dynamical description.
The broader lesson is modest but important:
When coarse-grained information loss is counted as dissipation, dynamical closure should be measured, bounded, or explicitly declared—not silently assumed.
About the manuscript
The underlying manuscript contains the formal assumptions, proofs, counterexamples, full numerical methods, and row-level reproducibility data. A permanent public preprint and archival data link will be added when available.
The work builds on established results concerning relative-entropy geometry, observational entropy, Petz recovery, resource-destroying maps, open-system entropy production, and classical lumpability. Relevant starting points include Resource Destroying Maps, Observational entropy with general quantum priors, Observational entropy and the Petz recovery map, and Macroscopicity and observational deficit in states, operations, and correlations.