Project: Holographic embezzlement capacity & the entanglement cost of f-routing
Target: IQOQI Open Quantum Problem #48 (Alex May, The entanglement cost of f-routing). Parallel track to the Bell projects; feeds the wormholes ↔ QI thread.
Unlike the other two projects this one is already partly in flight: a manuscript is drafted for holographic-dual-embezzlement-protocol, and the open question embezzlement-capacity-lower-bound is the capacity statement. The group’s advantage over May’s discrete (rank / span-program) cutting edge is the type III₁ continuum and the Araki–Uhlmann relative entropy toolkit.
1. Thesis statement
Derive an operationally meaningful lower bound on the entanglement cost of f-routing / non-local quantum computation in the continuum (type III) limit, using Araki–Uhlmann relative entropy and the crossed-product (type II) construction, and determine whether the holographic generalized entropy is the operational capacity — closing embezzlement-capacity-lower-bound. This complements the discrete rank/span-program bounds (ITCS 2025) with a continuum algebraic bound.
2. Why it fits the group (already mid-flight)
- A manuscript is drafted for holographic-dual-embezzlement-protocol (catalyst geometry, bulk operation, cost theorem).
- The open question embezzlement-capacity-lower-bound is the capacity statement.
- Relative-entropy results for coherent / squeezed states (Eur. Phys. J. C 85; Nucl. Phys. B 1018) give the cost toolkit; the cocycle-perturbation machinery (the algebraic-QFT course Master’s project, Phase C1) computes embezzlement cost on the crossed product directly.
- The two PhD students (embezzlement; vN algebras and many-body) are already on exactly this material.
3. Phased plan (~12–15 months)
Phase 0 — Dictionary (3–4 wks)
Map f-routing ↔ conditional disclosure of secrets ↔ non-local computation, and the ITCS-2025 rank lower bound. Identify the continuum limit in which “entanglement cost” becomes a type III relative-entropy quantity. Deliverable: positioning note linking May’s problem to the embezzlement program; tie into the drafted manuscript.
Phase 1 — Relative-entropy lower bound (6–8 wks)
Via the crossed product (III II, trace restored), define a finite cost for the routing/embezzlement operation and bound it below by Araki–Uhlmann relative entropy using monotonicity / data-processing. Builds on the coherent-state relative-entropy results. Deliverable: a continuum lower bound — standalone result, independent of the capacity claim. People: the two PhD students.
Phase 2 — Capacity theorem (6–8 wks)
Test whether the Phase-1 bound matches the holographic generalized-entropy rate from the drafted manuscript. Match generalized entropy is the operational capacity (the headline of embezzlement-capacity-lower-bound); gap a sharp, publishable gap statement. Deliverable: capacity theorem or quantified gap.
Phase 3 — Bridge to May’s discrete bounds (3–4 wks)
Show the continuum bound is the scaling limit of the rank bound — or that it captures structure the rank bound misses. Deliverable: paper bridging QI-cryptography and AQFT.
4. Risks
| Risk | Likelihood | Mitigation |
|---|---|---|
| Continuum limit of f-routing cost is non-standard | High | Phase 1 relative-entropy bound stands alone; doesn’t depend on the holographic claim |
| Holographic capacity match fails | Medium | Drafted manuscript de-risks Phase 2; a gap result is still publishable |
5. Targets & people
JHEP / Quantum / PRD. Lead: Marcelo + the two PhD students. This is the project that feeds the wormholes ↔ QI research thread directly.
6. Connection to existing wiki questions
- embezzlement-capacity-lower-bound — the capacity question this project closes
- holographic-dual-embezzlement-protocol — the drafted manuscript this project builds on
- embezzlement-cost-relative-entropy — the relative-entropy cost toolkit
- relative-entropy-interacting-theories — the cocycle route beyond free fields
7. Sequencing within the OQP program
Parallel track, driven by the two PhD students plus the drafted manuscript — does not compete with the long-range-bell-decay-project spine for the senior-collaborator bottleneck.
Project plan committed 2026-06-19. Parallel to long-range-bell-decay-project and bell-in-type-iii-project. Part of the relative-entropy-qft and bell-inequalities-qft lines.