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Quantum stochastic thermodynamics of macroscopic systems: an algebraic approach

Published 10 Jul 2026 in quant-ph | (2607.09242v1)

Abstract: We build a framework for the thermodynamics of macroscopic quantum systems. In contrast with approaches requiring access to the full density matrix, our framework relies on a coarse-grained description, based on measurement statistics of a few observables. When these observables commute, the outcomes define classical macrostates whose entropy is quantified by observational entropy, accounting for uncertainty about both the macrostate and the microstate within it. We extend this notion to non-commuting observables forming a subalgebra of the operator space, and use Jaynes' principle to define an algebra-dependent entropy interpolating between von Neumann and observational entropies. Given initial and final measurement sets, connected by internal and/or environment-induced dynamics, we derive a second law for the coarse-grained dynamics. Unlike formulations based on von Neumann entropy, our inequality captures irreversibility from both non-unitary environment-induced dynamics and internal equilibration. It takes the usual form of a positive entropy production when the system is initially at internal equilibrium, while correction terms capture nonequilibrium resources ignored by the coarse-graining. We also derive fluctuation theorems for coarse-grained thermodynamic quantities. Along a quasi-static path of measurement schemes, we identify quantum macroscopic notions of work and heat fulfilling the first and second laws, including an additional work contribution from manipulating the algebra to which the system is confined, through external constraints or quantum measurement backaction. Finally, we apply our framework to examples illustrating the impact of varying the coarse-graining scheme. Our approach unifies macroscopic and stochastic thermodynamics in a genuinely quantum framework, laying the basis for a versatile, experimentally friendly toolbox to analyze complex quantum dynamics.

Summary

  • The paper introduces an algebraic framework that defines thermodynamic quantities using coarse-grained observables in macroscopic quantum systems.
  • It rigorously derives second laws and fluctuation theorems that connect microscopic dynamics with measurement-based coarse-graining.
  • The study highlights observer-dependent definitions of heat and work via quasi-static protocols, bridging quantum and classical thermodynamics.

Algebraic Quantum Stochastic Thermodynamics of Macroscopic Systems

Introduction and Motivation

This work proposes a unifying framework for quantum stochastic thermodynamics of macroscopic systems based on an algebraic perspective. Traditional quantum thermodynamic methods typically require full information of the system density matrix. This is largely inaccessible for macroscopic or many-body quantum systems, both theoretically (due to exponential Hilbert space growth) and experimentally (due to limitations in measurement resolution and control). The authors close this gap by developing a coarse-grained, measurement-based framework: thermodynamic quantities are defined with respect to the statistics of accessible, macroscopic observables. This approach covers both stochastic irreversibility, as in open or small quantum systems, and traditional macroscopic irreversibility, originating from internal equilibration and coarse-graining.

Algebraic Approach and Coarse-Graining

Central to the formalism is the concept of coarse-graining via subalgebras of accessible observables. An observer’s capabilities are encoded not by their knowledge of the full Hilbert space, but by which subalgebra A\mathcal{A} of the total operator algebra B(H)\mathcal{B}(\mathcal{H}) they can probe. Given a measurement protocol, the outcome statistics define a "macrostate" (equivalently, a reduced description of the physical system). For commuting observables, the statistics induce a classical probability distribution over macrostates; for more general (noncommuting) accessible algebras, the reduced description is a density operator on a reduced Hilbert space.

This structure is succinctly illustrated in the scheme below.

Figure 1

Figure 1: Schematic of measurement-induced coarse-graining: the accessible observables define subspaces (macrostates) of the full Hilbert space, linked by micro-macro and macro-micro Petz maps. Both open and autonomous (closed bipartite) perspectives are accommodated.

The authors formalize this using the Jaynes maximum entropy principle: for a given set of accessible statistics, the "coarse-grained state" is the density matrix of maximal von Neumann entropy compatible with those statistics. This recovers known results such as the observational entropy in the commutative case and interpolates between Boltzmann and von Neumann entropy for general subalgebras.

Derivation of Thermodynamic Laws: Second Law and Fluctuation Theorems

The framework rigorously establishes second laws and fluctuation theorems governing the evolution of the coarse-grained entropy associated with the chosen algebra.

The Two-Point Measurement Protocol

Thermodynamic processes are analyzed through initial and final (possibly distinct, even incompatible) measurement schemes. The observer need not know or control the full quantum dynamics, but must be able to repeat the protocol and obtain consistent macro-observables. By expressing the dynamics in terms of quantum channels connecting initial and final states (with possible environment-induced non-unitarity), the authors derive entropy production inequalities of the form:

Ξ”Scg(ρ)βˆ’Ξ”βŸ¨K⟩ρβ‰₯βˆ’Iextβˆ’Iint\Delta S_\mathrm{cg}(\rho) - \Delta \langle K \rangle_\rho \geq -I_{\mathrm{ext}} - I_{\mathrm{int}}

where ScgS_\mathrm{cg} is the coarse-grained entropy (generalizing observational entropy), KK is the modular Hamiltonian of the equilibrium state for the accessible algebra, IextI_{\mathrm{ext}} accounts for initial system-environment correlations, and IintI_{\mathrm{int}} captures deviation from internal equilibrium at the level of unobserved (ignored) degrees of freedom. When the dynamics is "scale divisible" (i.e., internal degrees of freedom are equilibrated or dynamically decoupled), IintI_\mathrm{int} can be set to zero.

The logical structureβ€”how the initial/final measurement schemes, system dynamics, and coarse-graining compose into a thermodynamic processβ€”is mapped in the following diagram.

Figure 2

Figure 2: Coarse-grained quantum thermodynamic framework structure: vertical arrows represent kinematical (coarse-graining) operations, horizontal arrows denote quantum (possibly open) dynamics, and connections between micro/macro levels are constructed via quantum channels and Petz recovery maps.

Autonomous and Open System Viewpoints

Both open-systems (only partial access to environment) and autonomous (access to both system and environment) settings are analyzed. For the autonomous case, variation in mutual information is shown to bound the entropy changes in system and environment, generalizing classical results to arbitrary quantum coarse-graining.

Stochastic Trajectories and Fluctuation Theorems

At the single trajectory (stochastic) level, the paper generalizes integral fluctuation theorems for entropy production to the quantum, coarse-grained setting. The ratio of forward and backward process probabilities is connected to a stochastic entropy production term, with additional corrections when the initial state is out of internal equilibrium or the process is absolutely irreversible. All such relations are expressed in terms of macroscopic observables, without recourse to the inaccessible microscopic state.

Quasi-Static Transformations and Algebra-Dependent Heat/Work Identification

A crucial theoretical development is the definition of heat and work increments along quasi-static, continuously parametrized paths of accessible algebras, i.e., smoothly time-dependent measurement schemes tuned by the observer. This generalizes classical thermodynamics’ quasi-static protocols to the algebraic quantum setting.

If at each time an accessible algebra A(t)\mathcal{A}(t) is chosen and the system remains at internal equilibrium at each step, the framework identifies

  • Heat: energy changes associated with entropy-variation in the macrostate, proportional to the change in the modular Hamiltonian expectation,
  • Work: energy changes that do not accompany macro-entropy change, including contributions from mechanical-like transformations (time-dependent Hamiltonians) and from the algebra's dynamics (e.g., the act of changing measurement basis),
  • Observer-Dependent Thermodynamic Laws: All these quantities depend on the observer’s operational capabilities as codified by their choice of accessible algebra.

Explicitly, the variation of internal energy is split into heat and work as:

dE=d⟨Ht⟩ρ=Q+Wd E = d \langle H_t \rangle_\rho = \mathcal{Q} + \mathcal{W}

where B(H)\mathcal{B}(\mathcal{H})0 is algebra-dependent heat and B(H)\mathcal{B}(\mathcal{H})1 includes both "classical" and "algebraic" work. This is technically exemplified using models such as a dissipative qubit under time-dependent measurement (see the next section).

Concrete Examples: Qubit, Field Emission, and Many-Body Systems

Several carefully analyzed paradigms demonstrate the breadth of the formalism, emphasizing that all entropy production, heat, and work assignments are observer- and algebra-dependent.

Dissipative Qubit: The dynamical flow and heat/work split for a qubit relaxing towards thermal equilibrium under continuous, basis-rotating measurement is illustrated in Bloch sphere representation.

Figure 3

Figure 3: The evolution of a dissipative qubitβ€”energy flow along radial direction (heat), unitary basis rotation (work), and the structural decomposition of the thermodynamic process within the Bloch sphere.

Emitter–1D Field Coupling: The coarse-graining framework is applied to spontaneous emission, showing how the accessible algebra (e.g., local field modes vs. collective wave trains) determines whether energy released by the qubit is counted as heat or (coherent) work. The field’s observed entropy production and energetic flows crucially depend on the level of access to field observables.

Figure 4

Figure 4

Figure 4: Application of the framework to emission into a 1D field, contrasting single-mode and collective-mode algebra choices for accessible observables, and highlighting their thermodynamic implications.

Noncommutative Algebras and Generalized Entropies

The most general aspectβ€”beyond coarse-graining by commuting observablesβ€”is coarse-graining by arbitrary (possibly noncommutative) subalgebras. By exploiting the structure theorem for type I von Neumann algebras, the authors show that entropy and thermodynamic functionals can be consistently defined for fully quantum reduced descriptions, and that all fluctuation and second-law results generalize to this case. This is essential for describing realistic scenarios where experimentally available observables cannot be simultaneously diagonalized or are distributed over different physical subsystems.

Implications and Outlook

This algebraic coarse-grained formalism systematically unifies the measurement-based "observational" viewpoint with information-theoretic and resource-theoretic quantum thermodynamics. Notably, thermodynamic irreversibility is shown to emerge either from the stochasticity of open quantum dynamics or from internal equilibration with respect to the inaccessible (ignored) part of the system, in a way controllable by the observer's algebraic choices.

By making the dependence of all thermodynamic statements on measurement/control capabilities explicit, the framework provides a robust theoretical foundation for:

  • Extending stochastic quantum thermodynamics to many-body and field systems where fine-grained state tomography is infeasible,
  • Describing experimental setups with limited measurement bandwidth, resolution, or time/frequency selectivity,
  • Investigating fundamental issues, such as the emergence of classicality from coarse-graining, thermodynamics in QFT (including types II and III von Neumann algebras), and observer-dependent resources in quantum engines or batteries.

Conclusion

This paper rigorously establishes a versatile, experimentally accessible algebraic approach to quantum stochastic thermodynamics that interpolates between macroscopic equilibrium, classical stochastic, and full quantum descriptions, depending on the operationally accessible algebra of observables. The theoretical results provide strong guarantees for observer-dependent second laws, clarify the roles of heat, work and entropy production under arbitrary coarse-graining, and lay the groundwork for systematic investigations of thermodynamics in large-scale and strongly quantum systems.

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Overview

This paper builds a simple-to-use toolbox for understanding how big quantum systems behave, without needing to know absolutely everything about them. Instead of trying to keep track of every tiny quantum detail (which is impossible for large systems), the authors focus on a few β€œbig-picture” measurementsβ€”the things you can actually observe in a lab with limited tools. From those few measurements, they define a kind of entropy (a measure of uncertainty or β€œmissing information”) and show that familiar thermodynamic lawsβ€”like the second lawβ€”still hold in this coarse, practical view. They also explain how to talk about heat and work in this setting and how randomness in small systems fits into the picture.

Key objectives and questions

The paper aims to answer these questions in everyday terms:

  • How can we describe the thermodynamics (heat, work, entropy) of a big quantum system if we can only measure a few β€œcoarse” features of it?
  • Can we define an entropy that uses only what we can actually observe and still behaves like thermodynamic entropy should?
  • Will a version of the second law of thermodynamics (that entropy tends to increase) still hold when we only look at coarse information?
  • Can we say something about the size and likelihood of fluctuations (random ups and downs) in energy and entropy at this coarse level?
  • How should we define heat and work along slow, controlled changes when what we can measure or control is itself changing?

Approach and methods (explained simply)

The authors use a few core ideas. Here’s the picture:

  • Coarse-graining: Think of a giant library you can’t fully catalog. Instead of listing every single book (the β€œmicrostates”), you just count how many books are on each shelf (the β€œmacrostates”). You lose detail, but you keep the big picture. In physics, this means grouping many tiny quantum details into a few measured quantities.
  • Accessible vs. inaccessible: The β€œaccessible” part is what your measurement setup can see and control. The β€œinaccessible” part is everything it can’t. Mathematically, the accessible set is described by an β€œalgebra” of observablesβ€”a structured set of things you can measure. Sometimes these observables get along (β€œcommute”) and sometimes they don’t (β€œnon-commute”), like trying to measure both position and momentum precisely at onceβ€”you can’t.
  • Macrostates and coarse-grained states: A macrostate is the summary of what you measured. The authors then build a β€œcoarse-grained state”: among all the detailed quantum states that match your measurements, they pick the most unbiased oneβ€”the one with the largest entropy. This follows Jaynes’ maximum-entropy principle: don’t assume anything you didn’t measure. For the hidden parts you can’t see, they fill them with a uniform β€œdon’t-know” distribution.
  • An entropy that fits the situation: Their entropy splits into three intuitive pieces: 1) a Shannon/Gibbs term that measures your uncertainty about which big category (β€œshelf”) the system is in, 2) a von Neumann term that measures quantum uncertainty within the accessible part of each category, 3) a Boltzmann-like term that counts how many hidden microstates sit behind each category (how much detail you’re ignoring). This entropy smoothly connects two familiar extremes: if you can see everything, it becomes the usual von Neumann entropy; if you only count categories, it becomes β€œobservational entropy.”
  • Two viewpoints for dynamics:
    • Open system: Your system may be nudged by an environment you don’t control, making its behavior noisy or irreversible.
    • Autonomous pair: Two systems push and pull on each other in isolation, and you measure each one coarsely.
  • Second law and fluctuations: They look at what you measure at the start and the end of a process, and show that coarse-grained entropy obeys a second-law-like inequality. They also establish fluctuation theoremsβ€”rules about how likely it is to see rare β€œentropy-decreasing” events versus ordinary β€œentropy-increasing” ones.
  • Heat and work along slow changes: When you change things gradually (quasi-statically), you can split energy changes into β€œheat” and β€œwork” at the coarse level. Surprisingly, they find an extra kind of β€œwork” that comes from changing what you can measure or constrainβ€”like moving walls to confine the system or changing the measurement that defines your coarse view.

Main findings and why they matter

Here are the key takeaways in plain language:

  • A practical entropy that matches what you can actually measure: The paper defines a coarse-grained entropy that treats both kinds of uncertainty fairlyβ€”the uncertainty about which big category the system is in and the uncertainty inside each category. This extends β€œobservational entropy” to cases where your different measurements don’t all align.
  • A second law that captures two sources of irreversibility at once:
    • External irreversibility: randomness injected by the environment (noise, dissipation).
    • Internal irreversibility: the natural β€œsmoothing out” that happens when a big system’s hidden details settle into an even spread (internal equilibration).
    • Their second-law inequality covers both effects, which standard quantum approaches (based only on the full density matrix) can miss at the β€œinternal” level.
  • Corrections when the hidden parts aren’t equilibrated: If the invisible details haven’t settled, the inequality includes extra correction terms. This tells you exactly how much your coarse picture can deviate from the neat textbook version of thermodynamicsβ€”and why.
  • Fluctuation theorems at the coarse level: Even when you only see a few big features, the paper shows that powerful relations still constrain the ups and downs of heat, work, and entropy. This generalizes well-known results to a more realistic, measurement-limited setting.
  • Work and heat emerge naturallyβ€”and there’s a new β€œalgebra-change” work: Along slow, controlled changes, energy splits into heat and work as expected. But there’s an additional work-like cost if you change the very way you measure or constrain the system (for example, adding/removing restrictions or performing measurements that reshape what’s accessible). This is a distinctly quantum-and-operational insight.
  • A clear hierarchy of entropies: The more you coarse-grain (the less you can see), the larger your entropy. Symbolically: von Neumann entropy ≀ algebra-based entropy ≀ observational entropy. This matches the idea that less information means more uncertainty.

Implications and potential impact

  • Fits real experiments: In big or many-body quantum systems, you usually can’t reconstruct the full quantum state. This framework works with just a few coarse measurementsβ€”the kinds that are actually feasibleβ€”yet still gives you honest thermodynamics.
  • Bridges two worlds: It connects macroscopic thermodynamics (where lots of details are averaged out) and quantum stochastic thermodynamics (which tracks everything), letting you slide between them by changing how coarse your view is.
  • Explains irreversibility without magic: It clarifies that entropy can grow because of outside noise and also because you choose to ignore internal detailsβ€”both contribute to the β€œarrow of time” you observe.
  • Guides design and interpretation: It offers an β€œexperimental-friendly” way to define heat, work, and entropy in complex quantum devices, helping researchers analyze and compare setups even when only partial information is available.
  • Foundations and the quantum-to-classical transition: By showing how classical-like thermodynamic behavior emerges from quantum rules when you look coarsely, it strengthens our understanding of how the familiar macroscopic world arises from the strange quantum one.

In short, the paper shows how to do honest, useful thermodynamics with the information you can actually get from large quantum systemsβ€”and it tells you exactly how your choices of what to measure affect what you see as heat, work, and entropy.

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Knowledge Gaps

Knowledge gaps, limitations, and open questions

Below is a single, consolidated list of what remains missing, uncertain, or unexplored in the paper, expressed as concrete, actionable items for future research.

  • Extension beyond type-I, finite-dimensional von Neumann algebras: how to generalize the framework (definitions of macrostates, Petz-based coarse-graining, entropy hierarchy, second law and fluctuation theorems) to infinite-dimensional systems and to type-II/III algebras (relevant for QFT and thermodynamic limits).
  • Realistic measurements (POVMs): the paper defers a general treatment to future work; a complete theory is needed for POVMs that do not generate subalgebras, including (i) systematic construction of coarse-grained states and entropies under non-projective, non-idempotent measurements, (ii) stability under repeated measurements, and (iii) implications for second-law inequalities and fluctuation theorems.
  • Operational identification of the accessible algebra: concrete procedures to infer, certify, or learn the accessible subalgebra from finite, noisy data (including partial tomographic sets within sectors), and to reconstruct macrostates without requiring full knowledge of all operators in the algebra.
  • Robustness to experimental imperfections: quantitative bounds on how errors in estimating sector probabilities, within-sector states, or sector dimensions propagate to the coarse-grained entropy, entropy production, and the derived inequalities.
  • Choice and determination of sector β€œvolumes” VJ and Vβ€²J: practical methods to determine these dimensions in macroscopic or many-body systems (where symmetries, constraints, or selection rules may limit accessible microstates), and how misestimation affects the Boltzmann terms and thermodynamic conclusions.
  • Internal equilibrium criterion: precise dynamical conditions (mixing/ergodicity, timescale separation, locality, unitality) that guarantee relaxation to the algebra-dependent coarse-grained state ρcgA, including convergence rates and finite-size corrections.
  • Non-equilibrium corrections: explicit, quantitative bounds on deviations from the standard second law when the internal degrees of freedom have not equilibrated relative to the chosen algebra (how large can β€œcorrection terms” be and how to estimate them operationally).
  • Interplay with ETH and thermalization: rigorous connections between the internal-equilibrium condition and eigenstate thermalization for local/coarse observables, including which classes of Hamiltonians and observables ensure emergence of the proposed thermodynamics.
  • Non-Markovian dynamics: validity conditions and modifications of the second law and fluctuation theorems when the system dynamics is non-Markovian (e.g., initially correlated environments, memory kernels) and how these manifest at the coarse-grained level.
  • Strong system–environment coupling: explicit treatment of interaction energy and partition ambiguities when defining work and heat under coarse-graining, and how the algebra choice affects heat/work splits in the strong-coupling regime.
  • Energy observable outside the algebra: how to define and interpret energy, work, and heat when the Hamiltonian (or parts of it) is not contained in the accessible algebra; whether β€œrepresentative” projections of the Hamiltonian are physically meaningful and how they impact the first and second laws.
  • Additional work from algebra manipulation: operational protocols and measurement schemes to implement, quantify, and experimentally verify the β€œwork-like” contribution arising from changing the accessible algebra (e.g., constraint changes or measurement backaction), including its energetic and entropic costs.
  • Protocol dependence with incompatible algebras: trajectory-level formulations (two-point or multi-point measurements) when initial and final algebras are incompatible, including systematic accounting of measurement backaction, memory-register costs, and their effects on fluctuation theorems.
  • Dependence on the Petz prior: exploration of non-uniform priors (e.g., KMS states) in the Petz recovery/Jaynes maximization, including conditions under which coarse-grained entropy still bounds or increases relative to von Neumann entropy and how the second-law statements change with alternative priors.
  • Formal properties of the algebraic entropy: proofs (or counterexamples) of subadditivity, strong subadditivity, and other thermodynamic concavity/convexity properties for Sπ’œ; characterization of data-processing inequalities and monotonicity under physically relevant CPTP maps that respect or coarsen the algebra.
  • Choice/optimization of the algebra: principled criteria or algorithms to select the β€œright” level of coarse-graining (i.e., an accessible algebra) that balances predictive power with experimental feasibility and minimizes non-equilibrium correction terms.
  • Gauge/representation dependence: analysis of how the algebra’s chosen representation (and unitary embedding U) affects the Boltzmann terms and thermodynamic quantities; conditions under which results are invariant or how to enforce representation-independent statements.
  • Composition and multipartite settings: general composition rules for interacting subsystems with different accessible algebras (e.g., π’œAβŠ—π’œB vs. joint algebras), and how to define and bound correlations, mutual information, and entropy production consistently across composite systems.
  • Time coarse-graining: formal development of the suggested macroscopic time-averaging (coarse time resolution) and its interplay with the algebra-based coarse-graining, including how time-averaging alters Petz reconstruction and entropy production.
  • Heat/work definitions along non-commutative, quasi-static paths: rigorous conditions for path-independence, integrability, and potential geometric (gauge/Berry-like) contributions when the accessible algebra varies along a quasi-static protocol; experimental strategies to detect such contributions.
  • Fluctuation theorem assumptions: explicit microreversibility/detailed-balance conditions required for the coarse-grained fluctuation theorems (open vs. autonomous cases), including how violations (e.g., control noise, feedback, algebra changes) modify the theorems.
  • Measurement memory and Landauer costs: incorporation of the energetic/entropic costs of recording and erasing measurement outcomes in the memory that stores macrostates, and how these costs interact with the second law at the coarse-grained level.
  • Scalability and computational tractability: algorithms for decomposing many-body Hilbert spaces into sectors, computing ρcgA, and evaluating Sπ’œ in large systems; approximate methods with guarantees (e.g., tensor networks, randomized measurements) and their error–complexity tradeoffs.
  • Benchmarks on many-body dynamics: systematic numerical and experimental case studies (quenches, transport, thermalization, chaos vs. integrability) quantifying when and how the algebra-dependent thermodynamics captures observed irreversibility and macroscopic laws.
  • Relation to other coarse-grained entropies: comparative analysis with diagonal, diagonal-in-energy, or coarse-grained phase-space entropies; conditions under which Sπ’œ coincides with or differs from these alternatives, and operational consequences for thermodynamic predictions.
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Practical Applications

Immediate Applications

The following use cases can be deployed with today’s experimental and computational capabilities, leveraging coarse-grained measurements and the algebraic framework (accessible subalgebras, Petz/Jaynes coarse-grained state, observational/algebraic entropy, two-point measurement protocols, and fluctuation theorems).

  • Coarse-grained thermodynamic monitoring of quantum hardware
    • Sectors: software, hardware (quantum computing), energy
    • Use: Estimate entropy production, heat, and work in superconducting qubits, trapped ions, or spin ensembles using only readily available measurement data (e.g., populations, stabilizer outcomes, parity checks), avoiding full state tomography.
    • Tools/workflows: A software library integrating with Qiskit/Cirq/PennyLane to compute algebra-dependent entropy S_A and observational entropy from measurement counts; dashboards for per-experiment entropy production and irreversibility diagnostics.
    • Assumptions/dependencies: Choice of accessible subalgebra aligned with actual measurement operators; sufficient statistics from repeated runs; approximate projective readout or calibrated POVMs; internal equilibrium approximation if one seeks β€œstandard” second-law forms.
  • Thermodynamic calibration of quantum heat engines and refrigerators with partial access
    • Sectors: quantum thermodynamics, energy, experimental physics
    • Use: Quantify work/heat in circuit QED or trapped-ion engines via two-point coarse-grained measurements; include the β€œalgebra manipulation” work due to constraint/measurement changes.
    • Tools/workflows: Two-point measurement protocols across coarse observables; data processing pipelines that compute heat/work splits from quasi-static paths of measurement schemes; uncertainty estimates via fluctuation theorems.
    • Assumptions/dependencies: Repeatable preparation and measurement; knowledge of sector volumes V_J; control over measurement settings to define quasi-static paths; partial tomographic access within sectors for noncommutative algebras.
  • Many-body equilibration and thermalization tests via observational/algebraic entropy
    • Sectors: academia (quantum many-body physics), materials/condensed matter, cold atoms
    • Use: Evaluate equilibration in cold-atom lattices or spin chains with only coarse observables (e.g., binned number counts, parity, structure factors); diagnose internal vs environment-induced irreversibility.
    • Tools/workflows: Protocols computing Sob and S_A from coarse measurement histograms over time; comparison across different subalgebras to separate internal equilibration from open-system effects.
    • Assumptions/dependencies: Appropriate coarse-graining reflecting experimental resolution; ergodic/typical dynamics at the microlevel; stability against detection noise.
  • Energy-aware error-correction and syndrome processing
    • Sectors: quantum computing, energy
    • Use: Use algebras generated by stabilizer measurements to quantify entropy/heat contributions per QEC cycle, including measurement backaction as β€œalgebra work,” informing cryogenic budget and cycle optimization.
    • Tools/workflows: Integration into QEC pipelines to log S_A increments per round; automatic alerts when entropy production per cycle exceeds thresholds; cross-correlation with physical qubit energy dissipation.
    • Assumptions/dependencies: Reliable stabilizer measurement statistics; mapping of syndrome outcomes to sectors J; internal equilibrium for ancillary degrees of freedom, or inclusion of the correction terms when not satisfied.
  • Partial-observability diagnostics in open systems
    • Sectors: experimental quantum physics, hardware diagnostics
    • Use: Disentangle irreversibility from environment-induced noise vs internal equilibration by varying the accessible subalgebra and comparing entropy production; validate noise models (e.g., dephasing vs relaxation).
    • Tools/workflows: Systematic campaigns with alternative measurement bases; entropy production estimators that attribute contributions to environment vs internal degrees of freedom.
    • Assumptions/dependencies: Ability to switch measurement setups across noncommuting bases; sufficiently slow drift to compare runs; identified commutant structure for interpretation.
  • Materials and spectroscopy: nonequilibrium thermodynamics from limited probes
    • Sectors: materials science, NMR/ESR, neutron and X-ray scattering
    • Use: Quantify entropy production and equilibration during quenches or pump–probe sequences using accessible coarse observables (e.g., a few moments or correlators), avoiding full density-matrix reconstruction.
    • Tools/workflows: Data analysis routines mapping measured correlators to representatives in the accessible algebra and estimating S_A changes and fluctuation-theorem bounds.
    • Assumptions/dependencies: Valid representatives for measured observables; reliable estimation of sector dimensions; stationarity or controlled protocols to apply two-point schemes.
  • Resource accounting in cryogenic testbeds
    • Sectors: energy, hardware operations
    • Use: Track cumulative entropy production and inferred heat loads during device operation using coarse-grained metrics, complementing direct thermal sensors.
    • Tools/workflows: Continuous streaming of measurement counts to compute rolling S_A trends; integration with cryostat telemetry to correlate thermodynamic metrics with cooling performance.
    • Assumptions/dependencies: Calibration linking entropy production to heat under the chosen coarse-graining; repeated-run statistics; stable measurement channels.
  • Representative-operator estimation under coarse access
    • Sectors: software, academia, hardware
    • Use: Compute β€œrepresentatives” of desired observables within the accessible algebra to estimate expectation values from coarse data (e.g., block-diagonal projections), enabling performance metrics without finer measurements.
    • Tools/workflows: Software implementing o_A = (Petz ∘ C)(o) and expectation estimation on ρcg_A; uncertainty quantification from finite sampling.
    • Assumptions/dependencies: Known accessible algebra and Petz prior (typically uniform); Type-I algebra decomposition for finite systems; invertibility of C(u) on support.
  • Teaching/verification of fluctuation theorems with limited measurements
    • Sectors: education, academia
    • Use: Lab modules using two-point macroscopic measurements to verify coarse-grained fluctuation theorems and entropy production inequalities without full tomography.
    • Tools/workflows: Open-source datasets and scripts; step-by-step labs on observational entropy and algebraic entropy construction.
    • Assumptions/dependencies: Stable preparation and readout; sufficient sample sizes.

Long-Term Applications

These opportunities require further research and/or scaling, including improved measurement control, larger devices, standards, and integration with industrial workflows.

  • Energy-efficient quantum computing and scheduling
    • Sectors: software, hardware, energy
    • Use: Incorporate coarse-grained thermodynamic metrics into compilers/schedulers to minimize entropy production and heat per algorithm, with constraints on measurement backaction and β€œalgebra work.”
    • Tools/products: Thermo-aware transpilers; runtime policies that select measurement/control algebras to reduce dissipation.
    • Assumptions/dependencies: Mature models linking S_A trajectories to physical heat; robust estimation under time-varying noise; verified quasi-static manipulation sequences.
  • Measurement-induced work engines and thermodynamic devices
    • Sectors: quantum thermodynamics, hardware innovation
    • Use: Design engines that exploit the identified β€œalgebra manipulation” work (from changing constraints/measurement backaction) to extract useful work or enhance refrigeration.
    • Tools/products: Prototype devices using controlled changes in accessible algebras (e.g., measurement-induced constraints) as a resource.
    • Assumptions/dependencies: Fast, accurate control of measurement bases; low-loss ancillary systems; validated accounting of measurement backaction as work at scale.
  • Standards for thermodynamic reporting under partial observability
    • Sectors: policy, metrology, industry consortia
    • Use: Define standardized protocols for reporting entropy production, work, and heat based on accessible subalgebras and coarse measurements (analogous to benchmarking standards).
    • Tools/products: Reference implementations, conformance test suites, and certification processes.
    • Assumptions/dependencies: Community consensus on accessible-algebra choices for given platforms; traceability to physical units; error budgets for coarse-grained metrics.
  • Scalable telemetry for quantum data centers
    • Sectors: energy, operations, cloud services
    • Use: Deploy continuous coarse-grained thermodynamic monitoring of large quantum fleets to optimize cooling loads and scheduling, and to detect drifts/failures via irreversibility signatures.
    • Tools/products: Fleet-level dashboards; predictive maintenance models using S_A trends and fluctuation-theorem deviations.
    • Assumptions/dependencies: Aggregation of statistically meaningful measurement data; cyber-physical integration with cooling and power systems.
  • Thermodynamic control of macroscopic quantum sensors and oscillators
    • Sectors: sensing, metrology
    • Use: Maintain internal equilibrium and minimal dissipation in macroscopic quantum devices (e.g., optomechanical resonators, superconducting sensors) using control policies derived from algebraic thermodynamics.
    • Tools/products: Controllers that adapt accessible algebras and measurement strengths in real time to keep entropy production low while meeting sensing performance.
    • Assumptions/dependencies: Real-time state-of-statistics estimation; accurate models for commutant dynamics and equilibration rates.
  • Energy-aware design of quantum network nodes
    • Sectors: communications, distributed systems
    • Use: Apply bipartite autonomous viewpoint to audit and optimize energy/entropy exchanges between interacting nodes under partial measurement constraints.
    • Tools/products: Node-level protocols that log S_A-based metrics alongside link-level QoS; thermodynamic budgets for entanglement distribution.
    • Assumptions/dependencies: Distributed synchronization of two-point measurements; cross-node calibration of accessible algebras.
  • Quantum materials engineering guided by equilibration diagnostics
    • Sectors: materials science, manufacturing
    • Use: Use algebraic/observational entropy as design metrics to favor fast internal equilibration with weak macroscopic dissipation, informing material selection and processing.
    • Tools/products: Screening workflows for candidate materials using coarse-grained entropy production in quenches/anneals as figures of merit.
    • Assumptions/dependencies: Correlation between coarse-grained metrics and device-level durability/efficiency; scalable measurement pipelines.
  • Robust control under partial observability
    • Sectors: control engineering, robotics (quantum-enabled), industrial automation
    • Use: Develop controllers that explicitly reason with accessible subalgebras and entropy bounds to stabilize large quantum systems without full observability.
    • Tools/products: Control algorithms that exploit the entropy hierarchy (Sob β‰₯ S_A β‰₯ S_vN) to trade off measurement burden vs performance.
    • Assumptions/dependencies: Verified models of the algebra-dependent second law and fluctuation bounds for the target system; reliable inference of ρcg_A in feedback loops.
  • Privacy-respecting thermodynamic auditing
    • Sectors: compliance, cloud services
    • Use: Provide high-level thermodynamic compliance guarantees (e.g., energy use, waste heat) using only coarse aggregate measurements that avoid exposing intellectual property or sensitive states.
    • Tools/products: Auditing APIs exposing standardized S_A-based summaries; third-party verification services using fluctuation-theorem tests.
    • Assumptions/dependencies: Acceptance of coarse-grained metrics in regulatory frameworks; agreed-upon priors and sector decompositions.

Cross-cutting assumptions and dependencies

  • Accessible subalgebra identification: Practical deployment depends on aligning the chosen subalgebra with what the measurement/control hardware truly accesses; for noncommutative cases, sufficient coverage across incompatible bases is needed.
  • Internal equilibrium: β€œTextbook” second-law forms emerge when unresolved degrees of freedom (commutant algebra) equilibrate; otherwise, correction terms must be accounted for.
  • Finite-dimensional/type-I setting: The decomposition into sectors relies on type-I von Neumann algebras (typical for finite, engineered systems); extensions to infinite-dimensional cases need care.
  • Petz/Jaynes prior: Using the uniform prior ensures monotonicity of entropy under coarse-graining; nonuniform priors (e.g., KMS/thermal) are possible but alter interpretations.
  • Experimental considerations: Approximate projective measurements or well-characterized POVMs, repeated-run statistics, and stable protocols are required to estimate p_J and ρ_J reliably.
  • Model validation: Translating S_A variations into heat/work requires validated models (e.g., Hamiltonians, coupling to environments) and careful accounting for measurement backaction.
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Glossary

  • Accessible subalgebra: A specified subset of observables (closed as a von Neumann subalgebra) that the observer can measure and use to describe the system. "we introduce the notion of coarse-graining of a system based on the specification of an accessible subalgebra."
  • Algebra-dependent entropy: An entropy defined by maximizing von Neumann entropy subject to constraints from a chosen subalgebra of accessible observables; it interpolates between von Neumann and observational entropies. "and use Jaynes principle to define an algebra-dependent entropy which interpolates between the von Neumann and observational entropies."
  • Automorphism: A structure-preserving map from an algebra to itself; here, a one-parameter group acting on the commutant to formalize equilibrium (KMS) conditions. "satisfy a KMS condition with respect to a prescribed one-parameter group of automorphisms Ξ±_t"
  • Boltzmann entropy: The entropy associated with the logarithm of the number of microstates compatible with a macrostate (state counting). "as quantified by Boltzmann entropy and the second law"
  • Commutant: The set of all operators that commute with every operator in a given algebra; it represents the inaccessible degrees of freedom relative to the accessible algebra. "It is useful to introduce the commutant π’œβ€² of the algebra π’œ"
  • Commutative algebra: An algebra in which all elements commute; here generated by a single measurement basis. "we first review this framework in the case of a commutative algebra generated by the measurement of a single (coarse) observable"
  • Complete Positive Trace Preserving (CPTP) map: A quantum channel that maps density operators to density operators while preserving positivity on extended systems and trace. "via the Complete Positive Trace Preserving (CPTP) micro-macro channel"
  • Coarse-grained state: The maximum-entropy state consistent with the statistics of accessible observables, representing a representative microstate compatible with macroscopic data. "the coarse-grained state is defined as the state reaching this maximal von Neumann entropy"
  • Coarse-graining: A reduction of description that retains only statistics of selected observables, grouping many microstates into macrostates. "We achieve this program by proposing a notion of coarse-graining formally based upon defining a subalgebra of accessible observables of the system."
  • Density operator: The operator describing a quantum state (possibly mixed), from which all measurement statistics are derived. "which is a functional of the full system density operator."
  • Ergodicity: The assumption that microscopic degrees of freedom explore all compatible microstates, justifying maximum-entropy assignments at fixed macroscopic constraints. "Under such assumption of ergodicity, the unresolved degrees of freedom encoded in π’œβ€² can therefore be described"
  • Fluctuation theorems: Exact relations that constrain the statistics of thermodynamic fluctuations, extending the second law beyond averages. "We also derive fluctuation theorems constraining the fluctuations of thermodynamic functions at the coarse-grained level."
  • Jaynes principle: The maximum-entropy principle for assigning states given partial information (constraints) about observables. "and use Jaynes principle to define an algebra-dependent entropy"
  • KMS condition: A mathematical criterion characterizing thermal (equilibrium) states in algebraic quantum theory relative to a time-automorphism group. "satisfy a KMS condition with respect to a prescribed one-parameter group of automorphisms Ξ±_t"
  • Macrostate: A coarse description of the system specified by accessible measurement outcomes (or reduced sectors), typically of much lower dimension than the full Hilbert space. "This information can be encoded in a macrostate belonging to a smaller Hilbert space"
  • Micro-macro map: The channel mapping a microscopic quantum state to its macroscopic (coarse) description defined by the accessible measurement outcomes. "The micro-macro map C maps a microstate from 𝓗 onto the corresponding macrostate in 𝖧̄."
  • Micro-canonical distribution: The uniform distribution over all microstates compatible with fixed macroscopic constraints (e.g., energy and particle number). "when the variables one chose to ignore in the coarse-graining reach a micro-canonical distribution"
  • Noncommutative algebra: An algebra whose elements do not necessarily commute; accessing such an algebra combines information from incompatible measurements within sectors. "general coarse-graining associated to noncommutative algebras."
  • Observational entropy: The maximal von Neumann entropy consistent with coarse measurement outcomes; decomposes into Shannon and Boltzmann-like contributions. "the so-called observational entropy"
  • Partial trace: An operation that discards (traces out) some degrees of freedom to obtain a reduced state for the remaining subsystem. "or from a partial trace"
  • Petz recovery map: A canonical, CPTP β€œinverse” of a quantum channel relative to a chosen reference state, recovering the maximum-entropy state consistent with coarse data. "The Petz recovery map P_{C,u}, or equivalently, Jaynes maximization principle, can be employed to recover the most probable microstate compatible with the observation"
  • POVM: A positive-operator valued measure; a general (non-projective) quantum measurement described by positive operators summing to identity. "one typically employs a POVM"
  • Projective measurement: An ideal measurement described by orthogonal projectors, used here to define coarse observables and macrostates. "via a projective measurement associated to a set of orthogonal projectors"
  • Pure-dephasing channel: A unital, non-unitary quantum channel that suppresses coherences in a given basis without changing populations. "non-unitary and unital (i.e. a pure-dephasing channel)"
  • Quasi-static path: A sequence of infinitesimal changes (here, of measurement schemes or constraints) slow enough to define thermodynamic work/heat consistently. "By considering a quasi-static path of measurement schemes"
  • Tomographically-complete set: A set of measurements sufficient to reconstruct a quantum state uniquely. "combine results from a tomographically-complete set of measurements to re-build a density operator."
  • Type I von Neumann algebra: A class of von Neumann algebras that decompose into direct sums of full matrix algebras, enabling sector-wise descriptions. "the type I von Neumann subalgebra π’œ βŠ‚ ℬ(β„‹)"
  • Unital channel: A quantum channel that preserves the identity operator; it does not decrease the maximally mixed state. "non-unitary and unital (i.e. a pure-dephasing channel)"
  • von Neumann algebra: A weakly closed, self-adjoint operator algebra on a Hilbert space, central in the algebraic formulation of quantum theory. "type-I von Neumann algebras."
  • von Neumann entropy: The quantum analogue of Shannon entropy defined as S(ρ) = βˆ’Tr(ρ ln ρ). "the von Neumann entropy"
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