Quantum Noncommutativity Uniquely Determines Relative Entropy

Abstract: Quantum relative entropy is a core concept in physics, governing the limits of communication, thermodynamic irreversibility and quantum resource conversion. However, the requirement that physical processes cannot increase state distinguishability, the data-processing inequality, permits an infinite family of alternative divergence measures. Here we show that quantum relative entropy is uniquely selected by a sharper operational principle. We evaluate distinguishability through binary guessing games, in which an observer discriminates between pairs of quantum states using the optimal measurement. We prove that any additive measure that respects the odds revealed by these optimal measurements must coincide with the Umegaki relative entropy. This rigidity is a purely quantum phenomenon. Whereas classical theory permits a continuous family of valid divergence measures, including Rényi divergences, quantum noncommutativity. collapses this mathematical freedom. The result is exact, requiring neither a thermodynamic limit of infinitely many copies nor super-additivity assumptions for correlated states. It establishes quantum relative entropy not merely as an asymptotic quantity, but as the unique additive distinguishability measure compatible with single-shot quantum discrimination.

• The paper uniquely establishes the Umegaki relative entropy as the solitary measure for quantum state distinguishability by leveraging quantum noncommutativity and Lorenz preorder.
• It employs binary hypothesis testing and a layer-cake construction to derive additivity and eliminate Renyi mixtures without asymptotic or super-additive assumptions.
• The work impacts resource theories by showing that noncommutative structures strictly constrain divergence measures, leading to a unique operational foundation for quantum information.

Quantum Noncommutativity and the Unique Determination of Relative Entropy

Overview

The paper "Quantum Noncommutativity Uniquely Determines Relative Entropy" (2607.01712) addresses the long-standing foundational question in quantum information theory: what precisely singles out the Umegaki quantum relative entropy as the operationally and mathematically canonical measure of quantum state distinguishability? Despite the sufficiency of data-processing inequalities (DPIs) and a multitude of DPI-monotone quantum divergences—including Renyi and $f$-divergences—the paper demonstrates that quantum noncommutativity, in tandem with an operationally sharpened definition of distinguishability, eliminates this degeneracy entirely, uniquely specifying the Umegaki relative entropy.

This result transcends previous uniqueness theorems by removing asymptotic, super-additivity, and correlation structure assumptions. Instead, it leverages the finer Lorenz preorder induced by binary statistical testing and decision theory, together with additivity, to derive this uniqueness at the single-shot, finite-copy level.

Motivation and Background

The Umegaki relative entropy,

$$D(\rho\|\sigma) = \operatorname{Tr}[\rho (\log \rho - \log \sigma)]$$

is fundamental in a variety of quantum protocols, from hypothesis testing and channel coding to thermodynamic irreversibility and resource theories. While its operational significance is largely due to properties such as DPI and additivity, these requirements alone are not sufficient for uniqueness: many inequivalent quantum divergences satisfy them.

Prior characterizations relied on additional regularity conditions such as lower semicontinuity, full additivity, and super-additivity for correlated systems [Matsumoto 2010, Wilming et al. 2017]. However, these often depend on infinite-copy (thermodynamic) or structure-imposed constraints. The present work departs fundamentally by introducing a directly operational and geometric axiom: dominance in all binary discrimination games, formalized through Lorenz majorization.

Lorenz Preorder and Binary Statistical Experiments

The core technical apparatus is the Lorenz preorder, which operationalizes quantum comparison through a family of binary hypothesis tests parametrized by prior biases. For quantum states $\rho$ and $\sigma$, and prior $t \in [0,1]$, the optimal guessing probability is

$$P_{\rm guess}^{(t)}(\rho, \sigma) = \max_{0 \leq \Lambda \leq I} \big\{ t \operatorname{Tr}[\Lambda \rho] + (1-t)\operatorname{Tr}[(I-\Lambda) \sigma] \big\}.$$

The Lorenz preorder $(\rho, \sigma) \succ_L (\rho',\sigma')$ holds iff $$P_{\rm guess}^{(t)}(\rho, \sigma) \geq P_{\rm guess}^{(t)}(\rho', \sigma')$$ for all $t$. Geometrically, this corresponds to inclusions of testing regions and allows the operational contrast to be phrased in terms of convex order among associated layer-cake measures.

Notably, while this preorder collapses to stochastic convertibility classically (by Blackwell's theorem), quantum noncommutativity ensures the Lorenz preorder is strictly finer than channel convertibility for noncommuting pairs.

Quantum Lorenz Divergences: Rigidity and Uniqueness

A quantum Lorenz divergence (QLD) is a scalar function on $\mathcal{D}(A) \times \mathcal{D}(A)$ that is monotone under Lorenz majorization. The principal advancement is the demonstration of the following rigidity phenomena:

• Extension Rigidity: Fix any Lorenz-continuous classical divergence. There is at most one QLD extending it to arbitrary finite-dimensional noncommuting quantum states—namely, the "layer-cake" construction based on the pair's associated Stieltjes measure. Thus, noncommutativity sharply restricts the quantum degrees of freedom relative to the classical case.
• Additive Uniqueness Collapse: Imposing additivity (for tensor products) and normalization, the only QLD is the Umegaki relative entropy. In the quantum regime, all measures outside relative entropy (i.e., mixtures of Renyi divergences with order $$D(\rho\|\sigma) = \operatorname{Tr}[\rho (\log \rho - \log \sigma)]$$0) are eliminated by the constraints of noncommutative additivity.

These conclusions are derived without appealing to asymptotics, regularization, or super-additive constraints. It is shown that only the Umegaki entropy, considered through its integral representation over binary-testing and hockey-stick divergences, respects both the full Lorenz monotonicity and additivity axioms.

Main Results and Technical Framework

Lorenz Extension Theorem

Given a classical divergence monotone under relative majorization and Lorenz-continuous on likelihood ratios supported in $$D(\rho\|\sigma) = \operatorname{Tr}[\rho (\log \rho - \log \sigma)]$$1, its quantum Lorenz divergence extension to noncommuting quantum states is unique:

$$D(\rho\|\sigma) = \operatorname{Tr}[\rho (\log \rho - \log \sigma)]$$2

for some convex $$D(\rho\|\sigma) = \operatorname{Tr}[\rho (\log \rho - \log \sigma)]$$3 determined by the classical restriction, where $$D(\rho\|\sigma) = \operatorname{Tr}[\rho (\log \rho - \log \sigma)]$$4 is the layer-cake measure. The proof uses a geometric bracketing argument: quantum Lorenz curves can be uniformly sandwiched by converging sequences of classical Lorenz curves.

Additivity and MPST Reduction

In the classical setting, the set of normalized, additive, monotone, Lorenz-continuous divergences forms a simplex of Renyi integrals as proven by Mu, Pomatto, Strack, and Tamuz (MPST). In the quantum setting, however, once the unique Lorenz extension is imposed, the only additive QLD is the Umegaki entropy; any attempt at a nontrivial mixture is obstructed by quantum tensor-product noncommutativity, which uniquely fixes the functional form.

Implications and Theoretical Significance

• Operational Foundations: The paper demonstrates that the operational principle of distinguishability via binary games, when strengthened to ask for order-dominance in every such game, is sufficient—together with additivity—to single out the Umegaki entropy among all conceivable quantum divergences.
• Classical–Quantum Disparity: Whereas classical theory accommodates a one-parameter family of additive divergences (Renyi mixtures), quantum mechanics collapses this structure to a single point for noncommuting states, owing to incompatibility of non-Renyi-1 divergences with the demands of noncommutative additivity.
• Resource Theoretic Consequences: The Lorenz preorder is more refined than standard resource-theoretic monotones under channels, suggesting a pathway toward a resource theory of binary quantum experiments in which transformations can act asymmetrically on the two hypotheses.
• Extension to Infinite Dimensions and Multiparty Settings: The tools developed suggest plausible generalizations to infinite-dimensional settings and possibly to multiterminal hypothesis testing or conditional/mutual information, although technical difficulties regarding unbounded ratios and topological continuity must be addressed.

Notable Claims

• The Umegaki quantum relative entropy is the unique normalized additive QLD compatible with classical Lorenz continuity.
• This uniqueness holds at the single-shot (finite-copy) level, requiring neither thermodynamic limits nor super-additivity assumptions.
• Classically, additivity and data-processing allow a simplex of Renyi mixtures, but quantum noncommutativity eliminates this freedom for noncommuting pairs.

Numerical and Structural Results

• Explicit separation arguments for qubit states demonstrate that all regularization gaps outside order $$D(\rho\|\sigma) = \operatorname{Tr}[\rho (\log \rho - \log \sigma)]$$5 vanish only in the Umegaki case; no mixture of other orders survives the constraints of tensor-product additivity, as shown by detailed asymptotic behavior of Petz, sandwiched, and layer-cake divergences.
• Lorenz continuous classical Renyi divergences are shown to exclude endpoint contributions, reinforcing the uniqueness of quantum extensions.

Future Directions

• Lorenz-geometric classifications for multi-hypothesis testing and resource conversion metrics in quantum information may benefit from the present operational/majorization framework.
• Potential for resource theories beyond symmetric channel actions, where transformations may independently process the two branches of hypotheses, informed by the Lorenz geometry of binary experiments.
• Generalization to infinite-dimensional or continuous-variable settings requires handling unbounded likelihood ratios and carefully extending the Lorenz continuity concept.

Conclusion

The work establishes the privileged status of Umegaki relative entropy as a consequence of quantum noncommutativity and an operationally sharpened notion of statistical experiment comparison. The resulting theory closes fundamental gaps in the axiomatic foundation of quantum information, rigorously anchoring the exceptional role of quantum relative entropy and offering a unified, operationally meaningful perspective on quantum divergence measures.