We consider families of tight upper bounds on the difference
S(ρ)−S(σ) with a rank/energy constraint imposed on the state
ρ, which are valid provided that the state
ρ partially majorizes the state
σ and is close to the state
σ with respect to the trace norm. The upper bounds within these families depend on the parameter
m of partial majorization. The upper bounds corresponding to
m=0 coincide with the (unconditional) optimal semicontinuity bounds for the von Neumann entropy with a rank/energy constraint obtained in [1] and [2]. State-dependent improvements of these semicontinuity bounds are proposed and analysed numerically. The notion of
ε-sufficient majorization dimension of the set of states with bounded energy is introduced and analysed. Classical versions of the above results formulated in terms of probability distributions and Shannon entropy are also considered. Bibliography: 20 titles.
Using techniques proposed in [Sason, IEEE Trans. Inf. Th. 59, 7118 (2013)] and [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)], and based on the results from the latter, we construct a globally optimal continuity bound for the von Neumann entropy. This bound applies to any state under energy constraints imposed by arbitrary Hamiltonians that satisfy the Gibbs hypothesis. This completely solves the problem of finding an optimal continuity bound for the von Neumann entropy in this setting, previously known only for pairs of states that are sufficiently close to each other. Our main technical result, a globally optimal semicontinuity bound for the von Neumann entropy under general energy constraints, leads to this continuity bound. To prove it, we also derive an optimal Fano-type inequality for random variables with a countably infinite alphabet and a general constraint, as well as optimal semicontinuity and continuity bounds for the Shannon entropy in the same setting. In doing so, we improve the results derived in [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)].
We present two results derived from the sharp log-Sobolev inequality for the uniform measure on a complete graph which concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems.
The first result is a sharp Lipschitz lower semicontinuity bound for the von Neumann entropy at any mixed state
ρ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality
S(ρ)−S(σ)≤Cρ(1−F(ρ,σ)) valid for any state
σ, where
Cρ is a constant depending on the rank of
ρ.
The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state
ρ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality
EF(ρ)−EF(σ)≤Cρ(1−Trρσ) valid for any state
σ, where
Cρ is a constant depending on the Schmidt rank of
ρ.
In both cases the optimal constant
Cρ is equal to the optimal constant
Kd in the log-Sobolev inequality for the complete graph with
d vertices: in the first case
d=rankρ, in the second one
d=rankρA=rankρB.
The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.
Abstract We describe an advanced version of the AFW-technique proposed in [M. E. Shirokov, ‘‘Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki–Fannes–Winter technique’’, Lett. Math. Phys. 113, 121 (2023); M. E. Shirokov, ‘‘Local lower bounds on characteristics of quantum and classical systems’’, Lobachevskii J. Math. 44, 2169–2191 (2023)] which allows us to obtain lower semicontinuity bounds, continuity bounds and local lower bounds for characteristics of quantum systems and discrete random variables. We consider applications of the new version of the AFW-technique to several basic characteristics of quantum systems (the von Neumann entropy, the energy-type functionals, the quantum relative entropy, the conditional entropy and the entanglement of formation).