Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation

AI summary

Overview

Research area: Mathematical quantum information theory — quantitative continuity of entropy and entanglement measures, using functional inequalities (log-Sobolev) on finite-dimensional probability simplices.

Technical level: Advanced. The note works with trace-class operators, Hermitian bounded operators, and separable Hilbert spaces of arbitrary (including infinite) dimension, and relies on supporting-hyperplane arguments for convex roof functions.

Scope (one sentence): The paper derives sharp, state-dependent lower semicontinuity bounds for the von Neumann entropy and for the Entanglement of Formation (EoF) at states with a uniform positive spectrum, with optimal constants equal to the sharp log-Sobolev constant $K_d$ of the complete graph on $d$ vertices.

What This Paper Is About

The authors ask how fast the von Neumann entropy and the Entanglement of Formation can drop when a special state $\rho$ (one proportional to a projector, i.e., with spectrum $(1/d,\dots,1/d,0,0,\dots)$) is replaced by an arbitrary other state $\sigma$. Earlier work by the same authors showed that a bound of the form $S(\rho)-S(\sigma)\le C_\rho,\beta^2(\rho,\sigma)$ in the squared Bures distance cannot hold for every state, since the corresponding supremum can equal $+\infty$ for some $\rho$; the goal here is to identify a class of states for which such bounds not only hold but hold with the best possible constant. The answer is that the sharp constant is exactly the log-Sobolev constant $K_d$ for the complete graph, with $d$ equal to the rank of $\rho$ in the entropy case and to the Schmidt rank in the entanglement case.

Key Contributions

1. Sharp entropy bound. A proof that for any state $\rho$ of rank $d\ge 2$ with uniform positive spectrum, $S(\rho)-S(\sigma)\le K_d,(1-F(\rho,\sigma))\le K_d,\beta^2(\rho,\sigma)$ for every state $\sigma$, with $K_d$ the optimal log-Sobolev constant, and a proof that the first inequality is optimal (the supremum of the ratio equals $K_d$), including an explicit state attaining it when $d\ge 3$.
2. Sharp EoF bound. A proof that for any pure bipartite state $\rho$ whose Schmidt coefficients are all equal to $1/\sqrt{d}$ ($d\ge 2$, systems of any dimension), $E_F(\rho)-E_F(\sigma)\le K_d,(1-\operatorname{Tr}\rho\sigma)\le K_d|\rho-\sigma|_1/2$ for every state $\sigma$, again with $K_d$ optimal and an explicit optimiser for $d\ge 3$.
3. Explicit supporting affine functional for the EoF. The construction $\ell_\rho(\sigma)=\operatorname{Tr}\Lambda_\rho\sigma$ with $\Lambda_\rho=K_d\rho-aI_{AB}$ and $a=K_d-\ln d>0$, whose spectral diameter is exactly $K_d$ — directly connecting the log-Sobolev constant to the supporting-functional criterion for EoF continuity established in the authors' earlier paper.
4. An improved, off-support-aware bound. A strengthening in which the right-hand side is reduced by $a\operatorname{Tr}P_\rho^{\perp}\sigma$, where $P_\rho$ projects onto $\operatorname{supp}\rho_A\otimes\operatorname{supp}\rho_B$, using the alternative supporting functional $\widetilde{\Lambda}\rho=K_d\rho-aP\rho$.

Main Findings

• The constant is the log-Sobolev constant. In both results the optimal constant $C_\rho$ equals $K_d$, defined by $K_d=2$ for $d=2$ and $K_d=\frac{d}{d-2}\ln(d-1)$ for $d>2$. In the entropy result $d=\operatorname{rank}\rho$; in the EoF result $d=\operatorname{rank}\rho_A=\operatorname{rank}\rho_B$ (the Schmidt rank).
• Exact form of the entropy bound (Proposition 1). For $\rho=\frac{1}{d}\sum_{k=1}^{d}|\phi_k\rangle\langle\phi_k|$ on a Hilbert space of any dimension, $S(\rho)-S(\sigma)\le K_d(1-F(\rho,\sigma))\le K_d\beta^2(\rho,\sigma)$ for all states $\sigma$, and $\sup_{\sigma\neq\rho}\frac{S(\rho)-S(\sigma)}{1-F(\rho,\sigma)}=K_d$.
• Where the entropy supremum is attained. For $d\ge 3$ it is attained at $\sigma=\sum_k\lambda_k|\phi_k\rangle\langle\phi_k|$ with $\lambda_1=\frac{d-1}{d}$ and $\lambda_2=\dots=\lambda_d=\frac{1}{d(d-1)}$. For $d=2$ no optimal state exists; the value $2$ is approached along $\sigma_\varepsilon=\mathrm{diag}(\frac12+\varepsilon,\frac12-\varepsilon)$, $\varepsilon\in(0,\frac12)$, where $\ln 2-S(\sigma_\varepsilon)=2\varepsilon^2+O(\varepsilon^4)$ and $1-F(\rho,\sigma_\varepsilon)=\varepsilon^2+O(\varepsilon^4)$.
• Exact form of the EoF bound (Proposition 2 and Corollary 4). For a pure $\rho$ with Schmidt representation $\rho=\frac{1}{d}\sum_{i,j=1}^{d}|\varphi_i\rangle\langle\varphi_j|\otimes|\psi_i\rangle\langle\psi_j|$, $E_F(\rho)-E_F(\sigma)\le K_d(1-\operatorname{Tr}\rho\sigma)\le K_d|\rho-\sigma|1/2$; since $\operatorname{Tr}\rho\sigma=F(\rho,\sigma)$ for pure $\rho$, this is equivalently $E_F(\rho)-E_F(\sigma)\le K_d(1-F(\rho,\sigma))\le K_d\beta^2(\rho,\sigma)$. The supremum of the ratio equals $K_d$, attained for $d\ge3$ at the pure state $\sigma=\sum{i,j}\sqrt{\lambda_i\lambda_j}|\varphi_i\rangle\langle\varphi_j|\otimes|\psi_i\rangle\langle\psi_j|$ with the same $\lambda$ pattern.
• The supporting functional and its diameter. With $a=K_d-\ln d$, Lemma 7 guarantees $a>0$ for every natural $d\ge2$. The functional $\ell_\rho(\sigma)=\operatorname{Tr}\Lambda_\rho\sigma$, $\Lambda_\rho=K_d\rho-aI_{AB}=(\ln d)\rho\ominus a(I_{AB}-\rho)$, satisfies $\ell_\rho(\rho)=E_F(\rho)=\ln d$ and $\ell_\rho(\sigma)\le E_F(\sigma)$ for all $\sigma$; its spectrum is discrete with minimum $-a$ and maximum $\ln d$, so its spectral diameter equals $K_d$.
• Extension lemma (Lemma 5). If a supporting operator exists on the face $\mathfrak{S}(\mathcal{H}A^\rho\otimes\mathcal{H}B^\rho)$, it can be extended to the whole space via $\Lambda^{\mathrm{ext}}{\rho,c}=P\rho^A\otimes P_\rho^B\Lambda_\rho P_\rho^A\otimes P_\rho^B-c(I_{AB}-P_\rho^A\otimes P_\rho^B)$, $c\ge0$, with diameter $D(\Lambda^{\mathrm{ext}}{\rho,c})=D(\Lambda\rho)+\max{0,c+\inf\mathrm{Sp}(\Lambda_\rho)}$.
• Improved bound (Proposition 8). With $P_\rho$ the projector onto $\operatorname{supp}\rho_A\otimes\operatorname{supp}\rho_B$ and $P_\rho^{\perp}=I_{AB}-P_\rho$, $E_F(\rho)-E_F(\sigma)\le K_d(1-\operatorname{Tr}\rho\sigma)-a\operatorname{Tr}P_\rho^{\perp}\sigma\le K_d|\rho-\sigma|1/2-a\operatorname{Tr}P\rho^{\perp}\sigma$, and the first inequality is sharp (supremum of the ratio equals $1$, attained at the state of equation (21) when $d\ge3$). This can also be derived from the first bound via selective LOCC-monotonicity of the EoF, using the measurement induced by $P_\rho$.
• No datasets or numerical benchmarks. The paper reports no experiments, datasets, or benchmark comparisons; all results are theorems and explicit closed-form examples.

Methodology in Plain English

The strategy is to convert a classical inequality into a quantum one at almost no cost.

First, the authors take the sharp log-Sobolev inequality for the uniform measure on the complete graph with $d$ vertices and rewrite it in the elementary form $\ln d+\sum_{i=1}^{d}\lambda_i\ln\lambda_i\le K_d\left[1-\frac{1}{d}\left(\sum_{i=1}^{d}\sqrt{\lambda_i}\right)^2\right]$, valid for any probability distribution with $d$ outcomes. Notably, they point out this same inequality appeared independently as a special case of optimality conditions for accessible information of a quantum ensemble.

Second, they observe that for a state $\rho$ proportional to a rank-$d$ projector, the left-hand side of that inequality is exactly the entropy difference $S(\rho)-S(\sigma)$ when $\sigma$ is supported inside the support of $\rho$, while the bracket on the right is exactly the fidelity deficit $1-F(\rho,\sigma)$.

Third, to handle an arbitrary $\sigma$ (possibly with larger or infinite support), they apply a carefully built pinching channel made of partial isometries that leaves $\rho$ unchanged, cannot decrease the fidelity by monotonicity, and produces a state that is majorized by $\sigma$ — so that Schur concavity of the entropy gives $S(\tilde\sigma)\le S(\sigma)$. This reduces the general case to the supported case.

Fourth, for the Entanglement of Formation, they exploit its definition as a convex roof: they verify the pointwise bound on pure states in a fixed subspace, integrate it against an optimal ensemble (a probability measure with barycenter $\sigma$), and thereby obtain the bound for arbitrary mixed $\sigma$ on the face. An extension lemma then lifts the resulting supporting affine functional from that face to the entire state space, which is what turns the estimate into the clean trace and trace-norm forms.

Finally, sharpness is settled by explicit calculations: a specific diagonal or Schmidt-diagonal state saturates the bound for $d\ge3$, and for $d=2$ a one-parameter family with a Taylor expansion shows the constant $2$ is approached but not attained.

Why This Matters

These results pin down the exact "exchange rate" between a distance-like quantity (fidelity deficit or trace distance) and a drop in entropy or entanglement, for a distinguished family of maximally mixed on their support states. Continuity bounds with sharp constants are the standard currency of quantum Shannon theory, where they convert approximate statements about states into approximate statements about capacities and rates.

• Quantum Shannon theory: continuity of entropies underlies converse and achievability arguments for channel capacities and for entanglement-assisted communication.
• Entanglement quantification and certification: knowing the modulus of continuity of the EoF lets one estimate entanglement from a finite-precision tomographic reconstruction, with an error budget that is provably tight.
• Security proofs in quantum cryptography: device-imperfections are usually modelled as small deviations of states; sharp continuity of entropy-type quantities translates those deviations into key-rate penalties.
• Mathematical statistics of quantum states: the bound clarifies which states admit a supporting hyperplane for the EoF, a structural property of the convex set of states.

Industry relevance: the work is foundational rather than applied; it is relevant to groups building quantum communication hardware and to standards efforts that require certified,误差-controlled statements about entropy and entanglement, not to near-term product development. (The paper itself does not discuss applications.)

Future Directions

• The pure-state supporting-functional question. The authors state they found neither a proof nor a counterexample for the existence of a supporting affine functional for the EoF at an arbitrary pure bipartite state (including the 2-qubit case), and they explicitly flag an open question at the end of the note (whose full statement lies beyond the truncated text provided here).
• Beyond the uniform spectrum. The whole construction hinges on $\rho$ having a flat positive spectrum; the natural question is what replaces $K_d$ for general finite-rank $\rho$, and whether the ratio can remain finite more broadly than for the flat case.
• Generalising the inequality beyond the complete graph. Only the complete-graph log-Sobolev inequality is used; other graphs and measures (and hence other constants) may yield bounds for states with different symmetry or spectrum structure.
• Other entanglement measures and other divergences. The supporting-functional machinery was developed for the EoF; whether analogous sharp Lipschitz bounds hold for related convex-roof measures, and with respect to other distances, is left open.

Target Audience

Researchers in mathematical quantum information and quantum Shannon theory, particularly those working on continuity bounds for entropies and entanglement measures, on convex-roof entanglement quantifiers, and on functional inequalities applied to quantum states. The note is short and self-contained enough for graduate students with a background in operator theory and quantum information, but it assumes fluency with trace-class operators, fidelity and Bures distance, majorization, and the convex-roof definition of the EoF. Readers looking for experimental results, numerics, or algorithmic proposals will not find them here.

Authors’ abstract

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 $\,S(ρ)-S(σ)\leq C_ρ(1-F(ρ,σ))\,$ valid for any state $σ$, where $C_ρ$ 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 $\,E_F(ρ)-E_F(σ)\leq C_ρ(1-\mathrm{Tr}ρσ)\,$ valid for any state $σ$, where $C_ρ$ is a constant depending on the Schmidt rank of $ρ$. In both cases the optimal constant $C_ρ$ is equal to the optimal constant $K_{d}$ in the log-Sobolev inequality for the complete graph with $d$ vertices: in the first case $d=\mathrm{rank}ρ$, in the second one $d=\mathrm{rank}ρ_A=\mathrm{rank}ρ_B$. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.