Consider a Hilbert space H of finite dimension d=|H|. Let rho and sigma be states on H. Define varepsilon coloneqq T(rho,sigma),qquad nu coloneqq lVert rho-sigma rVert_∞. The function f_d(varepsilon,nu) is defined through the paper's equation~. Uniform continuity-bound conjecture. The entropy difference satisfies |S(rho)-S(sigma)|≤ f_d(varepsilon,nu), and this bound is tight. The conjecture proposes a tight uniform continuity bound for the von Neumann entropy in terms of the trace distance and operator-norm distance, particularly in the regime where 1/beta is not an integer, where the bound in Theorem~ is not tight.

Uniform tight continuity bound for von Neumann entropy

Consider a Hilbert space ${\mathcal{H}}$ of finite dimension $d=|{\mathcal{H}}|$ . Let $\rho$ and $\sigma$ be states on $\mathcal{H}$ . Define

\varepsilon \coloneqq \mathrm{T}(\rho,\sigma),\qquad \nu \coloneqq \lVert\rho-\sigma\rVert_{\infty}.

The function $f_d(\varepsilon,\nu)$ is defined through the paper's equation~. Uniform continuity-bound conjecture. The entropy difference satisfies

\left|S(\rho)-S(\sigma)\right|\leq f_d(\varepsilon,\nu),

and this bound is tight. The conjecture proposes a tight uniform continuity bound for the von Neumann entropy in terms of the trace distance and operator-norm distance, particularly in the regime where $1/\beta$ is not an integer, where the bound in Theorem~ is not tight.

A 2023 paper proposes a tight uniform bound for the entropy difference of two finite-dimensional quantum states using both their trace distance and operator-norm distance. It is settled in the integer-ratio regime, while the noninteger regime motivating the conjecture remains unresolved.

Known results

Theorem 4 proves $|S(\rho)-S(\sigma)|\leq \varepsilon\log((\nu/\varepsilon)d-1)+h(\varepsilon)$ under $\varepsilon\leq \nu d/(\nu d+3)$ ; it is tight when $\varepsilon/\nu$ is an integer.
The Audenaert–Petz inequality gives the established dimension-dependent trace-distance bound, including $\varepsilon\log(d-1)+h(\varepsilon)$ in its principal regime.

2023 conjecture

The paper explicitly conjectures that $|S(\rho)-S(\sigma)|\leq f_d(\varepsilon,\nu)$ and that this bound is tight. It discusses possible proof strategies but notes that a key monotonicity approach fails; no subsequent proof, disproof, or verification was found in the supplied sources.

Current status (as of August 2026): The conjectured bound is known to be tight when $\varepsilon/\nu$ is an integer, but the noninteger case remains open, with no recorded proof or counterexample.

Uniform tight continuity bound for von Neumann entropy

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