Is the holographic embezzlement rate optimal? (the capacity lower bound)

Statement

The manuscript Generalized Entropy as the Capacity for Holographic Entanglement Embezzlement (see holographic-dual-embezzlement-protocol) proves a conditional achievability bound: in a finite sector of the crossed product of logarithmic trace volume $\mathcal{C}$ (with a left–right mirror realization), a $d$-dimensional maximally entangled target can be extracted with infidelity

$1-F_d\le C_d\frac{\log d}{\mathcal{C}}.$

The open question is the matching lower bound:

$1-F_d\ge c_d\frac{\log d}{\mathcal{C}}$

for every protocol in the finite-sector class, for fixed $d$ and large $\mathcal{C}$.

This is the difference between a bound and a capacity. The upper bound says $\mathcal{C}$ (hence, under $\mathcal{C}\simeq S_{\text{gen}}$, the generalized entropy) suffices to embezzle at this rate. Only the lower bound says it is necessary — i.e. that $S_{\text{gen}}$ is the operational capacity. The manuscript title claims the latter; the body currently proves only the former. Closing this gap is what earns the title.

Why It Matters

• It converts the paper’s contribution from “a generalized-entropy bound on holographic embezzlement” into a genuine operational characterization of $S_{\text{gen}}$ — a new meaning distinct from RT/FLM (entanglement entropy), Wall (the GSL), and CLPW/CPW (entropy of the dressed algebra). None of those tie $S_{\text{gen}}$ to the rate of a task; this would.
• It is the part least exposed to being scooped by the algebraic-holography insiders (Witten/Liu/CPW), because it requires one-shot / smooth-entropy quantum-information machinery closer to the group’s idiom than to theirs.
• It is a natural thesis result for one of the group’s PhD students.

What We Know

The problem reduces to a gravity-free spectral statement. Embezzling a $d$-dimensional maximally entangled target forces the catalyst Schmidt spectrum $\{p_n\}$ to be approximately invariant under the $d$-fold block stretch $n\mapsto (m,r)$ with $n=d(m-1)+r+1$. In the logarithmic coordinate $s=\log n$ this is approximate invariance under the translation $s\mapsto s-\log d$. So the lower bound is exactly:

A probability distribution supported on a log-window of width $\mathcal{C}$ cannot be invariant under a $\log d$ translation to better than $O(\log d/\mathcal{C})$ in fidelity.

No gravity enters this statement; the holographic content is entirely in the separate identification $\mathcal{C}\simeq S_{\text{gen}}$.

Finite-dimensional anchors. The optimality/uniqueness of slowly varying embezzling families in finite dimensions is established by Leung–Wang (Phys. Rev. A 90, 042331, 2014) and, more completely, by Zanoni–Theurer–Gour (Complete Characterization of Entanglement Embezzlement, Quantum 8, 1368, 2024). These characterize which Schmidt tails are admissible and single out the van Dam–Hayden family — exactly the input needed to control the smooth-tail case below.

The continuum picture already in the manuscript ($\S$4): for a state flat in $s$ over $[0,L]$, a translation by $\Delta$ has overlap $(L-\Delta )/L$, giving $1-F\le 2\Delta /L$ with ${\Delta }_d=\log d$. The lower bound must run this as a variational lower bound over all admissible spectra, not just for the flat profile.

Possible Approaches

Primary route — sharp-cutoff / bounded-tail sectors (expected to be provable):

1. Reduce to Hellinger distance. Standard embezzlement bookkeeping (Leung–Wang; Zanoni–Theurer–Gour) gives $\sqrt{F_d}\le {\sum }_n\sqrt{p_n{p}_n^{\prime }}$, where $p^{\prime }$ is the $d$-fold reindexed (log-shifted) spectrum. Hence $1-F_d\ge \frac{1}{2}\Vert \sqrt{p}-\sqrt{p^{\prime }}{\Vert }^2$ — the squared Hellinger distance between a distribution and its $\log d$ log-translate.
2. Forced boundary leakage. If $p$ is supported on $s\in [0,\mathcal{C}]$, the shift $p^{\prime }$ lives on $[\log d,\mathcal{C}+\log d]$; the support mismatch has log-measure $\sim 2\log d$ out of $\mathcal{C}$, and any normalized $p$ must place fidelity weight $\gtrsim \log d/\mathcal{C}$ in the mismatch region.
3. Variational optimization. Minimize the Hellinger mismatch over admissible $p$ at fixed support width $\mathcal{C}$ and shift $\log d$. The minimizer is the near-flat-in-$s$ (van Dam–Hayden) profile, with mismatch $\Theta (\log d/\mathcal{C})$ — matching the upper bound and fixing $c_d$.

Where it breaks — and the honest scope. Step 2 uses a hard support cutoff. For a smooth wavepacket (e.g. Gaussian tails in $s$) there is no sharp boundary and the translation overlap can decay faster, so the naive bound is evadable. This is exactly why Zanoni–Theurer–Gour matters: it constrains which tails are admissible. The provable theorem is therefore conditional on a regularity / cutoff class for the sector — which is physically the right class, because the holographic sector has finite trace volume set by $S_{\text{gen}}$ (a genuine cutoff). Honest statement of the target: within the finite-trace-window class, the rate $\log d/\mathcal{C}$ is optimal.

Recommended order of work: first pin down the precise regularity class on the sector that makes the bound tight (what “finite-trace-window” must mean for the variational argument to close), then push steps 1–3 to a rigorous inequality with an explicit $c_d$. Getting the hypothesis right is what makes the proof go through cleanly.

Tooling. Smooth max-entropy / smooth support-trace methods from one-shot quantum information (Tomamichel; Hayden–Penington alpha-bit) are the natural formal language for the finite-trace-window cutoff and for lifting from finite type-I regulators to the type II${}_{\infty }$ algebra.

Related Questions

• holographic-dual-embezzlement-protocol — the umbrella project page; this is its single most important open sub-question.
• embezzlement-cost-relative-entropy — the original group-internal cost question; the lower bound is the sharp-rate refinement of it.
• bell-chsh-in-holographic-setting — shares the Type III₁ / modular-flow toolkit; relevant to the Q4 “Bell × embezzlement” follow-up once the capacity result is in hand.

Notes

• Reduction and strategy worked out in discussion 2026-06-05 (with the assistant).
• Key external anchors: Zanoni–Theurer–Gour (Quantum 8, 1368, 2024, arXiv:2303.17749); Leung–Wang (PRA 90, 042331, 2014); Tomamichel (arXiv:1203.2142); Hayden–Penington (CMP 374, 369, 2020).