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Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation
A. S. Holevo, M. E. Shirokov
TL;DR
The paper asks for sharp lower semicontinuity bounds for entropy and entanglement of formation, where such bounds do not hold at every state. It applies the sharp complete-graph log-Sobolev inequality and identifies uniform-spectrum state classes with sharp bounds whose optimal constants match the graph constant.
Problem
A finite Bures-distance entropy bound is not available at every state, and the paper investigates state-dependent lower semicontinuity bounds for entropy and entanglement of formation.
Source: §1 · p. 2, §3.1 · p. 5Method
The paper applies the sharp log-Sobolev inequality for the uniform measure on a complete graph to derive lower semicontinuity bounds for entropy and entanglement of formation.
Source: §1 · p. 2Results
For finite-rank uniform-spectrum states, the entropy bound in fidelity deficit is sharp, and for pure states with uniform-spectrum marginals the optimal EoF constant equals K_d.
Source: §1 · pp. 2–4Takeaways & Limitations
The complete-graph log-Sobolev constant supplies the optimal constant for the entropy bound and for the specified pure-state EoF class.
Source: §1 · pp. 2, 4The text does not claim a finite-constant bound of the stated kind for all pure bipartite states.
Source: §3.1 · p. 5
Abstract
from arXiv · showhide
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 $\rho$ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality $\,S(\rho)-S(\sigma)\leq C_\rho(1-F(\rho,\sigma))\,$ valid for any state $\sigma$, where $C_{\rho}$ is a constant depending on the rank of $\rho$. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state $\rho$ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality $\,E_F(\rho)-E_F(\sigma)\leq C_\rho(1-\mathrm{Tr}\rho\sigma)\,$ valid for any state $\sigma$, where $C_{\rho}$ is a constant depending on the Schmidt rank of $\rho$. In both cases the optimal constant $C_\rho$ is equal to the optimal constant $K_{d}$ in the log-Sobolev inequality for the complete graph with $d$ vertices: in the first case $d=\mathrm{rank}\rho$, in the second one $d=\mathrm{rank}\rho_A=\mathrm{rank}\rho_B$. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.
1 Introduction
The introduction uses the sharp log-Sobolev inequality for a complete graph to obtain sharp lower semicontinuity bounds for entropy and entanglement of formation. It identifies uniform-spectrum states as classes where these bounds have finite, optimal constants.
1 Introduction
- A finite Bures-distance entropy bound does not hold at every state, but the relevant supremum is finite for finite-rank states with uniform positive spectrum.The introduction says the supremum is infinite for some states and finite for the specified class, including in infinite-dimensional systems.Source: §1 · p. 2
- For finite-rank states with uniform positive spectrum, the entropy bound in fidelity deficit is sharp, with optimal constant K_d for d equal to the state's rank.The result applies in finite- and infinite-dimensional quantum systems.Source: §1 · pp. 2–3
- In infinite-dimensional bipartite systems, EoF is defined as the closed convex-roof extension of marginal entropy from pure states to all states.For pure states, the marginal entropies agree; the finite-dimensional definition reduces to an infimum over pure-state ensembles.Source: §1 · p. 3
- The EoF bound problem asks for a state-dependent constant independent of the comparison state, and its validity is equivalent to a supporting affine functional with bounded spectral diameter.The introduction notes that a general proof or counterexample for supporting functionals at pure bipartite states was unresolved.Source: §1 · p. 3
- Some bipartite states make the EoF lower semicontinuity bound impossible for every finite constant.The cited result includes bipartite systems generally and, in particular, two-qubit systems.Source: §1 · p. 3
- For pure bipartite states whose marginals have uniform positive spectrum, the optimal entanglement-of-formation bound constant equals K_d, with d the Schmidt rank.The introduction identifies this class through its Schmidt numbers and marginal spectra.Source: §1 · p. 4
2 Preliminaries
The preliminaries set the quantum-state and entropy conventions, then state the complete-graph log-Sobolev inequality used in the results.
2 Preliminaries
- A quantum system is represented on a separable Hilbert space, with states defined as positive trace-class operators of unit trace.The text also introduces the real Banach space of bounded Hermitian operators and the trace-class operator space.Source: §2 · p. 4
- The von Neumann entropy is concave and lower semicontinuous on the state space, taking values in [0, +∞].The preliminaries also introduce its homogeneous extension to the positive cone and invoke entropy concavity.Source: §2 · p. 4
- The sharp log-Sobolev inequality is stated for a real random variable with d equally likely values, then specialized to non-negative variables with E[f^2] = 1.The derivation uses the complete-graph Dirichlet form and a distribution with probabilities i/d.Source: §2 · pp. 4–5
- The text notes that the inequality also arose as a special case of optimality conditions for accessible information of a quantum ensemble [5].The cited connection is to Eq. (26) in [5].Source: §2 · p. 5
3 Main results
The section gives sharp lower semicontinuity bounds for von Neumann entropy and Entanglement of Formation, with constants tied to the complete-graph log-Sobolev inequality. It also sharpens the EoF estimate using the support of the marginal states.
3 Main results
- The paper does not claim a finite EoF bound of this kind for every pure state; the stated condition may fail for some states.The motivation includes an unresolved question about supporting affine functionals at arbitrary pure states in finite-dimensional composite systems.Source: §3.1 · p. 5
- For a rank-d state, the entropy bound in terms of the fidelity deficit is sharp, with optimal constant K_d from the complete-graph log-Sobolev inequality.The proposition applies to Hilbert spaces of any dimension.Source: §3.1 · p. 6
3.2 Lipschitz semicontinuity bound for the EoF
- For a pure bipartite state with Schmidt representation, the EoF bound is sharp and uses the complete-graph log-Sobolev constant K_d.The proof obtains the bound using an entropy estimate for the marginal state.Source: §3.2 · pp. 8–9
- The EoF result also yields a global supporting functional at ρ whose operator has spectral diameter K_d.The functional takes the form ℓ_ρ(σ) = Tr Λ_ρσ.Source: §3.2 · p. 8
3.3 Improved version of Proposition 2
- The improved EoF estimate uses P_ρ, the projector onto supp ρ_A ⊗ supp ρ_B, and its first inequality is sharp.The paper notes that the same improvement applies to Corollary 4.Source: §3.3 · pp. 11–12
- The improved EoF result leaves open whether K_d is optimal in the second inequalities of the original and improved estimates.The paper frames this as the existence of a global supporting functional with spectral diameter below K_d.Source: §3.3 · p. 13