Relation Entropy in Quantum & Classical Theories

• Relation Entropy is a measure of distinguishability comparing actual states or distributions against maximum-entropy references in both classical and quantum contexts.
• It underpins unified formulations in quantum uncertainty, holography, and conformal field theory by providing UV-finite and continuum-compatible diagnostics.
• Researchers use this framework to derive analytic bounds and consistency checks across systems, enabling practical state discrimination and detailed error estimates.

Searching arXiv for recent and foundational papers on "relation entropy"/relative entropy to ground the article. Relation entropy, in the usage adopted in recent quantum-information work, denotes relative entropy, i.e. the Kullback-Leibler divergence, understood as a measure of distinguishability between probability distributions or quantum states. In this formulation it is not merely an auxiliary information-theoretic quantity: it provides a unified language for uncertainty relations for observables with discrete and continuous spectra, a UV-finite diagnostic in conformal field theory, a state-discrimination functional in holography, a control parameter for quantum thermodynamic uncertainty relations, and a tractable observable in Gaussian field theory and random-state theory (Floerchinger et al., 2020, Lashkari, 2014, Blanco et al., 2013, Salazar, 2023, Schröfl et al., 2023, Wei, 26 Sep 2025).

1. Terminology and formal definitions

In the quantum-uncertainty literature, relative entropy is explicitly identified with relation entropy and with the Kullback-Leibler divergence. The central idea is always the same: one compares an actual distribution or state to a reference distribution or state and quantifies how distinguishable they are (Floerchinger et al., 2020).

Setting	Objects	Definition
Discrete classical	$p(x)$, $\tilde p(x)$	$$S(p \| \tilde p) = \sum_x p(x)\ln \frac{p(x)}{\tilde p(x)}$$
Continuous classical	$f(x)$, $\tilde f(x)$	$$S(f \| \tilde f) = \int dx\, f(x)\ln \frac{f(x)}{\tilde f(x)}$$
Quantum	$\rho$, $\sigma$	$$S(\rho \| \sigma)= \operatorname{tr}(\rho \log \rho)-\operatorname{tr}(\rho \log \sigma)$$

A key technical point is that the continuous version is well-defined in contrast to the differential entropy, which is one reason the framework is useful for observables with continuous spectra. In quantum field-theoretic settings, the same quantity is described as a measure of distinguishability for quantum states and is emphasized to be free of ultraviolet divergences when both states arise from finite-energy reductions (Floerchinger et al., 2020, Lashkari, 2014).

The family of Rényi entropies extends to Rényi relative entropies. A form used in conformal field theory is the sandwiched Rényi relative entropy

$$S_\alpha (\rho \| \sigma) = \frac{1}{\alpha-1} \log \operatorname{tr} \left[ \left( \sigma^{\frac{1-\alpha}{2\alpha}} \rho \,\sigma^{\frac{1-\alpha}{2\alpha}} \right)^\alpha \right],$$

which reduces to ordinary relative entropy as $\tilde p(x)$0, and yields the fidelity at $\tilde p(x)$1 (Lashkari, 2014).

2. Relative-entropic uncertainty relations

The paper "Relative entropic uncertainty relation" reformulates quantum uncertainty directly in terms of relative entropy between measurement-outcome distributions and reference distributions with maximal entropy. The resulting relative entropic uncertainty relation (REUR) is

$\tilde p(x)$2

where $\tilde p(x)$3 and $\tilde p(x)$4 are the outcome distributions for two observables $\tilde p(x)$5 and $\tilde p(x)$6, $\tilde p(x)$7 and $\tilde p(x)$8 are maximum-entropy reference distributions, $\tilde p(x)$9 is the incompatibility parameter given for projective measurements by $$S(p \| \tilde p) = \sum_x p(x)\ln \frac{p(x)}{\tilde p(x)}$$0, and $$S(p \| \tilde p) = \sum_x p(x)\ln \frac{p(x)}{\tilde p(x)}$$1 is the von Neumann entropy of the state (Floerchinger et al., 2020).

This construction changes the usual emphasis of entropic uncertainty theory. Traditional entropic uncertainty relations lower-bound the sum of entropies; the REUR instead upper-bounds the combined distinguishability from maximally uninformative references. In the language of the paper, one cannot discern both measured distributions arbitrarily well from “maximal ignorance” models, and the attainable distinguishability is limited by incompatibility and mixedness. This upper-bound perspective is presented as dual and complementary to conventional lower-bound formulations (Floerchinger et al., 2020).

The framework is designed to treat discrete and continuous spectra uniformly. For discrete variables, uniform or Boltzmann-type models serve as maximum-entropy references; for continuous variables, Gaussian references arise when mean or variance constraints are imposed. With these choices, the REUR reduces to known results: the Maassen-Uffink bound in the discrete case, and the Białynicki-Birula/Mycielski and Frank-Lieb uncertainty relations in the continuous case. The paper also emphasizes that additional prior information can be encoded as constraints in the reference distributions, thereby tightening the bound in a natural way (Floerchinger et al., 2020).

An illustrative example is the angle–angular-momentum pair. Angular momentum may use a uniform model or, with moment constraints, a discrete Gaussian or Boltzmann-like model; the angle variable may use a uniform or von Mises distribution. The example is used to show that the construction remains meaningful and finite in continuum limits where ordinary entropies may diverge (Floerchinger et al., 2020).

3. Relative entropy in conformal field theory and holography

In conformal field theory, relative entropy is studied as a quantity that is both conceptually sharper and technically better behaved than entanglement entropy. A Euclidean path-integral and replica-trick construction expresses Rényi relative entropies through correlation functions on multi-sheeted geometries; in this setting, relative entropy is obtained as a limit of correlation functions and is UV finite, unlike entanglement entropy (Lashkari, 2014).

For reduced density matrices in QFT, Blanco, Casini, Hung, and Myers write the density matrix in modular form,

$$S(p \| \tilde p) = \sum_x p(x)\ln \frac{p(x)}{\tilde p(x)}$$2

so that the relative entropy between two states on the same region becomes

$$S(p \| \tilde p) = \sum_x p(x)\ln \frac{p(x)}{\tilde p(x)}$$3

For spherical regions in a CFT vacuum, the modular Hamiltonian is local: $$S(p \| \tilde p) = \sum_x p(x)\ln \frac{p(x)}{\tilde p(x)}$$4 Because relative entropy is positive and monotonic under inclusion, this relation yields strong constraints on entanglement and modular energy (Blanco et al., 2013).

A central consequence is the first-order equality

$$S(p \| \tilde p) = \sum_x p(x)\ln \frac{p(x)}{\tilde p(x)}$$5

for infinitesimally different states. In holography this furnishes a stringent consistency check on the Ryu-Takayanagi prescription, since both $$S(p \| \tilde p) = \sum_x p(x)\ln \frac{p(x)}{\tilde p(x)}$$6 and $$S(p \| \tilde p) = \sum_x p(x)\ln \frac{p(x)}{\tilde p(x)}$$7 can be computed for spherical regions in AdS/CFT. The same framework also leads to modified versions of the Bekenstein bound, recast as $$S(p \| \tilde p) = \sum_x p(x)\ln \frac{p(x)}{\tilde p(x)}$$8, with modular energy replacing ordinary energy in general regions (Blanco et al., 2013).

The CFT literature also stresses operational significance. Relative entropy provides an upper bound on trace distance through Pinsker’s inequality,

$$S(p \| \tilde p) = \sum_x p(x)\ln \frac{p(x)}{\tilde p(x)}$$9

so small relative entropy implies that states are close in trace distance. This makes the quantity simultaneously geometric, information-theoretic, and field-theoretic (Lashkari, 2014).

4. Quantum thermodynamics and random-state theory

Domingos S. P. Salazar generalized information-theoretic uncertainty bounds to the quantum domain by deriving a lower bound on observable uncertainty in terms of symmetrized quantum relative entropy. For arbitrary states $f(x)$0 and $f(x)$1 and any Hermitian observable $f(x)$2,

$f(x)$3

with $f(x)$4 and $f(x)$5 the inverse of $f(x)$6. The relation is tight for a minimal two-level system with commuting $f(x)$7, $f(x)$8, and $f(x)$9, and it leads to a quantum thermodynamic uncertainty relation in terms of entropy production for arbitrary dynamics and non-thermal environments (Salazar, 2023).

In the thermodynamic application, the entropy production is defined by

$\tilde f(x)$0

with a corresponding dual quantity

$\tilde f(x)$1

The resulting bound is stated to apply to arbitrary initial states, arbitrary global dynamics, strong coupling, and regimes far from thermal equilibrium. A notable interpretive point is that coherence generally prevents the bound from being saturated even though it remains valid (Salazar, 2023).

A different line of work studies relative entropy statistically over ensembles of random density matrices. "Average relative entropy of random states" derives exact, explicit formulas for the average relative entropy of independent random states from Hilbert-Schmidt and Bures-Hall ensembles, both for states drawn from the same ensemble and from different ensembles. A crucial technical ingredient is a factorization of ensemble averages after evaluation of the relevant unitary integral. In the Hilbert-Schmidt case, the finite-dimensional formula is presented as complementing Kudler-Flam’s asymptotic replica-method result for equal dimensions (Wei, 26 Sep 2025).

5. Gaussian statistical field theory and continuum structure

In Gaussian scalar field theory, relative entropy is used to compare probability measures on field configurations. For finite-dimensional Gaussians $\tilde f(x)$2 and $\tilde f(x)$3,

$\tilde f(x)$4

while in infinite-dimensional settings finiteness depends on spectral properties of the covariance operators (Schröfl et al., 2023).

For Gaussian scalar field theories in a bounded region with the same classical boundary conditions but different masses, the relative entropy is finite if and only if $\tilde f(x)$5. If the theories differ only by Robin boundary conditions, finiteness holds if and only if $\tilde f(x)$6. By contrast, Dirichlet versus Robin or Dirichlet versus Neumann gives relative entropy that is always infinite. The paper therefore shows that the informational comparison of continuum theories depends sharply on spatial dimension and on boundary conditions (Schröfl et al., 2023).

The same work defines mutual information between disjoint regions by relative entropy: $\tilde f(x)$7 If the regions are separated by a finite distance, the mutual information is finite and satisfies an area law. If the regions are touching, the mutual information is infinite. The explanation advanced in the paper is the Markov property of Gaussian scalar fields: conditioned on boundary data, the field inside a region is independent of the exterior, so the boundary structure controls both finiteness and divergence (Schröfl et al., 2023).

6. Interpretation, scope, and terminological ambiguity

Across these literatures, relation entropy functions as a measure of distance from a reference. In uncertainty theory the reference is a maximum-entropy model; in CFT and holography it is a comparison state on the same region; in thermodynamics it may be a factorized or post-evolution reference state; in Gaussian field theory it is another continuum measure with different parameters. This suggests a unifying interpretation: relation entropy measures how much structure remains after the most neutral or most relevant reference has been specified (Floerchinger et al., 2020, Blanco et al., 2013, Salazar, 2023, Schröfl et al., 2023).

A common misconception is to identify all entropy relations with lower bounds on uncertainty or disorder. The REUR shows the opposite orientation can be natural: the sum of relative entropies is bounded from above, because the issue is not minimal uncertainty but maximal distinguishability from ignorance models. A second source of confusion is terminological. The phrase “relation entropy” may suggest any relation involving entropy, and arXiv indeed contains distinct literatures on relations between entropy and extremality, thermal entropy and entanglement entropy, entropy and kinetics, entropy and pair correlations, and entropy and dynamical invariants. Those works study equations involving entropy, but they are conceptually different from the use of relation entropy as an alias of relative entropy (Floerchinger et al., 2020, Goon et al., 2019, Chen et al., 2014, Sorkin et al., 2022, Watson et al., 2024, Kawan, 2018, Keller et al., 2014, Acharya et al., 2024).

Taken in its precise information-theoretic sense, relation entropy is therefore best understood as a reference-dependent, UV-finite in many field-theoretic contexts, and continuum-compatible measure of distinguishability. Its current significance lies not only in state discrimination, but in the fact that it organizes uncertainty relations, thermodynamic bounds, holographic consistency conditions, and continuum field-theoretic comparisons within a single formal vocabulary (Lashkari, 2014, Floerchinger et al., 2020).