--- title: "Sharp entropy continuity and stability under trace and operator constraints" date: 2026-11-06 version: "0.1.0-candidate" doi: 10.5281/zenodo.23997749 pdf: https://github.com/ipitchford/entropy-continuity-stability/releases/download/v0.1.0-candidate/entropy-continuity-stability-v0.1.0-candidate.pdf repository: https://github.com/ipitchford/entropy-continuity-stability archive: https://zenodo.org/records/23997749 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Sharp entropy continuity and stability under trace and operator constraints ## Summary in plain English How much can a quantum state's entropy change when the state changes only a little? The answer depends on what “a little” means—and on whether the distances are known exactly or merely bounded above. This paper separates those questions and asks a further one: if two states almost attain the largest possible entropy difference, must they resemble a pair that attains it exactly? At a fixed positive trace distance, the answer is yes. Both states are close to **one common extremal pair**, with an explicit error proportional to the square root of their entropy shortfall. There is also a useful warning. An error budget is not an exact measurement. Substituting two upper bounds into the formula for two exact distances can underestimate the possible entropy change. The paper gives a finite calculation for the correct upper-bound problem. This is an AI-assisted, unrefereed mathematical research candidate. Its proofs build on published entropy inequalities; the accompanying computations check examples and implementations, not the universal theorems. ## Summary for specialists For density matrices on a finite-dimensional space, write $\varepsilon=\tfrac12\|\rho-\sigma\|_1$, $\nu=\|\rho-\sigma\|_\infty$, and use natural-log entropy. The paper derives the prescribed-distance optimum $F_d(\varepsilon,\nu)$ from the fundamental entropic inequality of Audenaert and coauthors and Sason's capped-spectrum optimization. The formula and attaining family occur in earlier work by Jabbour and Datta. The additional analysis identifies all equality cases, evaluates the two-cap modulus $G_d(E,N)$ through finitely many endpoints and quadratic-root candidates, and proves common-pair stability. If the states are entropy-oriented and $\delta=F_d(\varepsilon,\nu)-[S(\rho)-S(\sigma)]$, there is an extremal pair at the same distances such that each state's **full trace-norm** error is at most $\min\{2,C_\varepsilon\sqrt\delta\}$, where $$ C_\varepsilon^2=\frac4{-\log(1-\varepsilon)}+\frac4\varepsilon-2 \quad(0<\varepsilon<1),\qquad C_1=\sqrt2. $$ No minimal-rank or rank-gap assumption is required. The coefficient is independent of dimension and operator distance, but not of trace distance. A three-dimensional example proves that the square-root exponent cannot generally be improved; the coefficient is not claimed optimal. ## Technical account ### Exact information and error budgets are different problems The exact problem fixes both distances. The cap problem allows every pair whose trace distance is at most $E$ and whose operator distance is at most $N$. It must therefore optimize over the distances as well as over the states. A four-dimensional example makes the distinction concrete. With caps $E=3/10$ and $N=1/5$, the cap optimum is attained at the smaller trace distance $1/5$: $$ \sigma=\operatorname{diag}(1,0,0,0),\qquad \rho=\operatorname{diag}(4/5,1/15,1/15,1/15). $$ Its entropy gap is $h(1/5)+(1/5)\log3$, strictly larger than the exact-distance optimum at $(3/10,1/5)$. Here $h$ is binary entropy with natural logarithms. Thus using the caps as exact inputs would give the wrong bound. The mechanism is a change in support dimension. The smallest allowed rank of one Jordan part of $\rho-\sigma$ is $\lceil\varepsilon/\nu\rceil$. When that rank increases, less space remains for the other part to spread uniformly. The exact-distance optimum can consequently jump down, even though entropy itself is continuous. Optimizing over upper bounds instead produces a continuous, nondecreasing modulus. ### A finite calculation, not a search over all matrices The cap modulus reduces to a one-variable maximization. On each rank interval, the only possible interior maximum is a specified quadratic root. Together with the interval endpoints, these give a candidate list whose size grows linearly with dimension. This is an exact mathematical reduction. It is not a guarantee about floating-point rounding or computational bit complexity. A certified implementation would need exact handling of rank boundaries and rigorous enclosures for the candidate values; nearly tied candidates must not be discarded on the basis of rounded decimals. ### What near equality reveals The stability proof keeps four nonnegative losses in the entropy argument: mass outside the relevant support, mixing within that support, excess entropy in the negative spectrum, and missing entropy in the positive spectrum. A small total deficit controls all four. Spectral rounding then selects a single support shared by both approximating states. The dependence on $\varepsilon$ matters: $C_\varepsilon$ behaves like $\sqrt{8/\varepsilon}$ near zero. A small raw entropy deficit alone does not identify an extremal pair uniformly as the states approach one another. For a desired full-trace-norm error $\eta$, the stated estimate suffices when $\delta\le\eta^2/C_\varepsilon^2$. The paper also treats near saturation under upper caps, provided the actual trace distance has a positive lower bound. The nearby pair is an extremizer at the **actual distances**, not necessarily a maximizer for the cap problem. ## Evidence, assurance and limitations The general statements rest on written proofs, not simulations. The package separates finite arithmetic examples, numerical diagnostics and checker guard tests from those proofs. Producer-side checking and internal AI-assisted review are not unaffiliated reproduction, external peer review or proof-assistant verification. The results are finite-dimensional. The stability coefficient is not claimed optimal and does not remain bounded as trace distance tends to zero. The conditional-entropy appendix gives attainment with a common maximally mixed conditioning marginal when the first subsystem's dimension is at least the second's; it does not establish sharpness for arbitrary prescribed marginals or unequal marginals. The novelty assessment is bounded by the sources inspected. It does not establish that no equivalent statement exists elsewhere. ## Relationship to earlier work The foundational noncommutative inequalities are due to [Audenaert, Bergh, Datta, Jabbour, Capel and Gondolf](https://doi.org/10.1109/TIT.2025.3586478). The exact-distance formula, feasibility conditions and attaining construction are identified in [Jabbour and Datta](https://doi.org/10.1109/JSAIT.2024.3469929), while the capped-spectrum entropy optimization is in [Sason](https://doi.org/10.1109/TIT.2013.2274515). The trace-distance-only case recovers [Audenaert's sharp continuity estimate](https://doi.org/10.1088/1751-8113/40/28/S18). The proposed contribution is the equality-necessity analysis, explicit cap evaluation and quantitative closeness to a common extremal pair. The fixed-difference and restricted conditional-entropy statements are consequences, not separate foundational inequalities. ## Who should care and why | Audience | Potential use | Required caution | | --- | --- | --- | | Quantum information theorists | Analyze equality and near equality in two-distance entropy bounds | Preserve the trace-distance conditioning and finite-dimensional assumptions | | Researchers using state-approximation error budgets | Calculate the worst entropy change allowed by two upper bounds | Use the cap modulus, not the exact formula with caps substituted | | Validated-numerics developers | Implement a finite candidate calculation with explicit rank boundaries | Numerical evaluations are not certified enclosures without additional rounding control | ## Why the problem matters A continuity bound turns information about state error into information about entropy error. A stability theorem goes in a different direction: it extracts structural information from near equality. Keeping these two uses distinct prevents a sharp mathematical formula from becoming an unreliable error estimate. ## How to inspect the result Start with the introduction's distinction between exact distances and upper caps, then the four-dimensional example in Section 6. Theorem 4 states the stability guarantee and its conditioning. Sections 2 and 5 contain the equality and stability arguments; Section 4 gives the candidate calculation. For computational checks, use the package's verification instructions and recorded environment. Compare the scope of each receipt with the claim being inspected: a checked example does not prove the theorem, and a successful script run does not establish priority. ## Most valuable next projects The most direct implementation project is an interval-certified cap evaluator, including rank-boundary and tied-candidate tests. A sharper stability coefficient would strengthen the theorem without changing its interpretation. Extending sharp conditional-entropy attainment to other prescribed marginals is a separate mathematical problem; the present construction does not settle it. ## What is in the evidence package The package contains the self-contained manuscript and source, claim and assurance registers, primary-source locators, example and stability checkers, guard tests, and verification instructions. The public argument does not depend on access to private reviews. ## Verification status Unrefereed AI-assisted mathematical candidate. Internal checking is not external peer review, unaffiliated reproduction or formal verification; bounded source comparison does not guarantee priority. ## References 1. Sason (2013). Entropy Bounds for Discrete Random Variables via Maximal Coupling. IEEE Transactions on Information Theory 59, 7118–7131. 2. Jabbour and Datta (2024). Tightening continuity bounds for entropies and bounds on quantum capacities. IEEE Journal on Selected Areas in Information Theory 5, 645–658. 3. Audenaert, Bergh, Datta, Jabbour, Capel and Gondolf (2025). Continuity bounds for quantum entropies arising from a fundamental entropic inequality. IEEE Transactions on Information Theory 71, 7029–7038. 4. Audenaert (2007). A sharp continuity estimate for the von Neumann entropy. Journal of Physics A: Mathematical and Theoretical 40, 8127–8136.