Hidden Rules of Physics Revealed by Limiting Information Access

Scientists are increasingly recognising the fundamental role of information in shaping physical laws, and a new study by Tiago Pernambuco (Federal University of Rio Grande do Norte) and Lucas Chibebe Céleri (Federal University of Goiás) proposes a geometric framework demonstrating how restricted access to microscopic information underpins quantum thermodynamics. Their work models measurement constraints as a gauge symmetry, effectively creating a reduced space of physically distinguishable states and yielding a gauge-invariant formulation with significant implications for entropy and fluctuation theorems. This research unifies the first and second laws through invariant work and coherent heat, identifying entropy production as a geometric consequence of limited observability and offering a novel perspective on irreversibility, making it a substantial contribution to our understanding of the foundations of thermodynamics and its quantum extensions.

Measurement constraints are modelled as a gauge symmetry acting on density operators, inducing a gauge-reduced space of physically distinguishable states. The specific contribution lies in demonstrating how limitations on information access directly lead to the emergence of physical laws, particularly within the realm of quantum thermodynamics.

Stochastic thermodynamics from gauge invariance and trajectory geometry

Scientists construct a gauge-invariant formulation wherein the invariant entropy possesses a stochastic description and satisfies a general detailed fluctuation theorem. From this result, an integrated fluctuation theorem and a Clausius-like inequality are derived, unifying the first and second laws in terms of invariant work and coherent heat.
Entropy production is identified with the relative entropy between forward and backward probability measures on the gauge-reduced space of thermodynamic trajectories, revealing irreversibility as a geometric consequence of limited observability. The third law emerges as a singular zero-temperature limit in which thermodynamic orbits collapse and entropy production vanishes.

Since the framework applies to arbitrary information constraints, it encompasses energy-based thermodynamics as a particular case of more general measurement scenarios. Physical laws are ultimately formulated relative to the information accessible to an observer, and thermodynamics provides a paradigmatic example of how such restrictions give rise to irreversibility.

Thermodynamics is fundamentally based on a coarse-grained description of physical systems. Although microscopic dynamics are governed by reversible laws, macroscopic observations inevitably discard detailed information, leading to the emergence of thermodynamic variables and irreversibility. In classical physics, this transition is often justified by statistical arguments, such as the central limit theorem, which suppresses relative fluctuations in large systems.

In contrast, many standard formulations of quantum thermodynamics assume near-complete control over the state of the system and define thermodynamic quantities using full microscopic information [2, 3] (see also Ref. and References therein). This raises a fundamental question: how can irreversibility emerge in quantum systems when only limited information is operationally accessible.

Recent works have addressed this question by modeling restricted access to quantum degrees of freedom as a gauge symmetry acting on the space of density operators [5, 7]. In this framework, the density operator is viewed as a carrier of information, part of which is operationally redundant given the measurement capabilities of the agent.

For instance, if the agent can measure only energy, the information associated with coherences or basis choices within degenerate eigenspaces becomes physically irrelevant. This redundancy is removed by the action of a thermodynamic gauge group, producing gauge-invariant quantities that faithfully represent the accessible thermodynamic information.

Gauge reduction thus provides a fundamental and geometric implementation of coarse-graining in quantum thermodynamics. In the present article, we show that the invariant entropy introduced in admits a stochastic formulation and satisfies a general detailed fluctuation theorem. From this result, a Clausius-like inequality is derived that unifies the first and second laws of thermodynamics within the gauge-invariant framework.

These results reveal the geometric origin of irreversibility. Moreover, because the construction applies to arbitrary sets of accessible observables, the resulting fluctuation theorem and thermodynamic laws are not restricted to energy measurements but hold for general measurement constraints, thus going far beyond existing relations [8, 9].

In this way, the present work establishes a unified and geometrically grounded formulation of quantum thermodynamics that is consistent with both stochastic thermodynamics and fundamental symmetry principles. Here we focus on the case where the agent has access only to energy measurements, leading to a geometric formulation of quantum thermodynamics.

Different choices of accessible observables, including non-commuting sets, define distinct thermodynamic theories through changes in the underlying gauge group, while the geometric construction and methods remain unchanged. Only the physical interpretation of measurable quantities is modified, reflecting the specific information constraints.

The theory of quantum thermodynamics proposed in Refs. [5, 6] received a rigorous geometric formulation in terms of fiber bundles in Ref. We employ this formulation to unravel the geometric structure behind stochastic quantum thermodynamics. Consider a quantum system described by a time-dependent density operator ρt and a Hamiltonian Ht.

When accessible measurements are restricted to a given set of observables (e.g. energy measurements), different density operators may become thermodynamically indistinguishable. This redundancy defines an emerging thermodynamic gauge group GT, whose action leaves all physically accessible quantities invariant.

In the case of energy measurements, GT is given by GT(t) ≃U(n1−t)× U(n2 t)×· · ·×U(nk t), where ni t are the degeneracies of the instantaneous Hamiltonian eigenvalues and P i ni t = d, with d the Hilbert-space dimension. Physical thermodynamic functionals are required to be invariant under the conjugation action ρt →VtρtV † t, with Vt ∈GT(t).

To formalize this structure, two distinct but related geometric objects are needed. First, a trivial principal U(d)-bundle over time ξ = (R × U(d), π, U(d), R) is introduced, where the base space is time and the fiber is the full unitary group acting on the system Hilbert space. A connection in this bundle is defined by the (right-invariant) Maurer, Cartan form At = utu† t, where ut diagonalizes Ht.

This connection induces the covariant derivative ∇t(·) = ∂t(·)+[At, ·], which plays the role of a gauge-covariant time derivative. Within this framework, time-dependent Hermitian operators are interpreted as sections of an associated vector bundle with fiber given by the space of Hermitian matrices, transforming under the adjoint representation of U(d).

In particular, the density operator ρt is treated as a matter field. Gauge-invariant definitions of work and heat naturally follow as Winv[ρt] = Z τ 0 dt Tr(ρt∇tHt) , (1) Qinv[ρt] = Z τ 0 dt Tr(Ht∇tρt) . (2) One detail of particular importance to the present work is that, by defining the parallel transport of a density matrix in the associated bundle through the usual equation ∇tρt = 0, we see that it corresponds to paths without invariant heat exchange.

For a closed system, where no heat is exchanged between the system and some environment, Qinv = Qc, where Qc, called coherent heat, is associated with the generation of coherences and is defined as [5, 6] Qc[ρ] = Z τ 0 dt Tr ρ uthtu† t + ρutht u† t, (3) where ht = utHtu† t. Thus, following the arguments presented in [5, 6], parallel transport in closed systems corresponds to paths where no information about the system is lost in the form of effective heat, thus defining the adiabatic transformations.

A second geometric structure is associated with the emergent group GT(t) itself. Since GT depends explicitly on time through the spectral degeneracies of Ht, it does not define a single principal bundle over R. Instead, it gives rise to a family of trivial principal bundles over individual time instants (or time intervals when degeneracies are constant), each equipped with its own Maurer-Cartan connection.

This structure encodes the coarse-graining induced by the measurement constraints and formalizes the emergent quantum thermodynamics. Equation (4), we define the stochastic gauge-invariant entropy associated with a measurement outcome k at time t as (see also Eq. (A10) in Appendix A) s(k) = −ln pk nk t . (5) This definition captures two distinct contributions to the uncertainty: the classical probability distribution (−ln pk) and the intrinsic quantum uncertainty due to degeneracy (ln nk t), which leads to the Holevo asymmetry term SΓ discussed in Ref. (see also Appendix A).

The stochastic entropy production σinv for a single trajectory is the difference between the final and initial stochastic entropies σinv = sR(l) −sF (k), which directly leads to the detailed fluctuation relation for agents who are restricted to energy measurements σinv = ln pk F pl R + ln nl τ nk 0 . (6) The theorem admits a natural geometric interpretation within the gauge-invariant quantum thermodynamics. An energy measurement does not select a quantum state, but rather an entire orbit of states within a degenerate eigenspace under the action of GT.

The states belonging to the same orbit are thermodynamically indistinguishable since they are related by gauge transformations. Consequently, pk represents the probability assigned to an orbit (fiber), rather than to a point in the Hilbert space, with the factor 1/nk t corresponding to the normalized measure induced by group averaging.

Equation (6) shows that irreversibility has two distinct geometric origins. The term involving the ratio of degeneracies quantifies the entropy production arising from the loss of information as one moves from a fine-grained to a coarse-grained description.

Thermodynamic laws emerge from informational geometry and limited observability

Researchers have established a geometric framework wherein physical laws arise from restricted access to microscopic information. Measurement constraints are modelled as a gauge symmetry acting on density operators, inducing a gauge-reduced space of physically distinguishable states. In the specific case examined, the invariant entropy admits a stochastic formulation and satisfies a general detailed fluctuation theorem.

From this, an integrated fluctuation theorem and a Clausius-like inequality were derived, unifying the first and second laws of thermodynamics in terms of invariant work and coherent heat. Entropy production is identified with the relative entropy between forward and backward probability measures on the gauge-reduced space of trajectories, revealing irreversibility as a geometric consequence of limited observability.

The third law emerges as a singular limit at zero temperature, where orbits collapse and entropy production vanishes. This framework applies to arbitrary information constraints, encompassing energy-based measurements as a specific case within broader measurement scenarios. Gauge reduction fundamentally implements coarse-graining in quantum thermodynamics.

The invariant entropy introduced in the work possesses a stochastic formulation and satisfies a detailed fluctuation theorem. A Clausius-like inequality was subsequently derived, unifying the first and second laws within the gauge-invariant framework. These results reveal the geometric origin of irreversibility, extending beyond existing thermodynamic relations.

The research establishes a unified and geometrically grounded formulation of quantum thermodynamics consistent with stochastic thermodynamics and fundamental symmetry principles. Different choices of accessible observables define distinct thermodynamic theories through changes in the underlying gauge group, while the geometric construction and methods remain unchanged.

Only the physical interpretation of measurable quantities is modified, reflecting the specific information constraints. Time-dependent Hermitian operators are interpreted as sections of an associated vector bundle, transforming under the adjoint representation of U(d). Gauge-invariant definitions of work are given by the integral of the trace of density operators and the covariant derivative of the Hamiltonian, calculated from zero to a time τ. Similarly, coherent heat is defined as an integral involving the density operator, unitary transformations, and the Hamiltonian, revealing that parallel transport corresponds to paths without invariant heat exchange.

Geometric Constraints Define Thermodynamic Laws and Irreversibility

Thermodynamic laws originate from limitations in accessing microscopic information, according to a newly formulated geometric framework. Measurement constraints are modelled as a gauge symmetry which reduces the space of physically distinguishable states, leading to a gauge-invariant formulation of thermodynamics.

This construction yields a stochastic description of invariant entropy and satisfies a detailed fluctuation theorem, subsequently deriving an integrated fluctuation theorem and a Clausius-like inequality that unifies the first and second laws of thermodynamics in terms of invariant work and coherent heat. Irreversibility arises as a geometric consequence of limited observability, identified with the relative entropy between forward and backward probability measures on the gauge-reduced space of trajectories.

The third law emerges as a singular limit at zero temperature where orbits collapse and entropy production vanishes, reflecting the impossibility of reaching this state through physically allowed transport. This framework extends beyond energy-based thermodynamics, encompassing more general measurement scenarios and establishing thermodynamics as a realization of a geometric theory of restricted information.

The findings demonstrate that thermodynamics is intrinsically linked to the set of accessible observables, with different measurement restrictions defining distinct thermodynamic frameworks. Entropy production is directly related to the distinguishability of thermodynamic trajectories, providing a unified geometric formulation of all four laws. The framework has direct relevance to experimental quantum systems where complete control over degrees of freedom is limited, such as trapped-ion systems and quantum optical setups, and may find further applications in these areas.