Finite-resource entanglement in continuum QFT
For specified local algebras, spacetime separation and energy/work budgets, bound entanglement distillation and catalytic embezzlement yields and errors under physically implementable local operations, including mixed states.
Problem statement
For specified local algebras, spacetime separation and energy/work budgets, bound entanglement distillation and catalytic embezzlement yields and errors under physically implementable local operations, including mixed states.
Lean 4
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Background
Local algebras in standard relativistic continuum QFT are typically type III. They have no intrinsic regional trace and density-matrix entropy of the finite-dimensional tensor-product kind. Cutoff entropies can diverge from short-distance correlations across the boundary; this does not prevent algebraic states or operational entanglement from being defined.
What is known: Modular theory and Araki relative entropy give regulator-independent tools. Relative entropy is an extended-valued quantity, not always finite; mutual information is finite only under appropriate state/region hypotheses. Longo–Xu rigorously calculate examples and their singular limits, including divergence as interval boundaries meet. Neither quantity should be described as unconditionally UV finite.
Published progress (2025): van Luijk, Stottmeister, Werner and Wilming formulate LOCC for commuting von Neumann algebras and extend Nielsen's pure-state conversion theorem to arbitrary factors. Under the stated Haag-duality assumptions, type III pure states can be converted into one another to arbitrary accuracy by LOCC; for type III_1 local operations without classical communication suffice. Universal entanglement embezzlement further shows how unconstrained algebraic operations can trivialize conversion costs.
The missing step is therefore not an unrestricted pure-state formalism. Fix an admissible finite-energy operation model and ask for matching bounds on work, localization, separation, output entanglement and approximation error. Determine which mixed-state and gauge-constrained resources remain accessible, and whether predictions have regulator-independent limits. Related: QF9, QF11, F7.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
published: LOCC and a generalized Nielsen theorem exist for arbitrary factors; unrestricted type-III pure-state conversion is characterized under Haag duality, without imposing finite-energy operation costs. Evidence
- [1]Witten, 'Notes on some entanglement properties of quantum field theory', Rev. Mod. Phys. 90, 045003 (2018) link
- [2]Faulkner, Leigh, Parrikar & Wang, 'Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition' (2016) link
- [3]Ceyhan & Faulkner, 'Recovering the QNEC from the ANEC', Comm. Math. Phys. (2020) link
- [4]van Luijk, Stottmeister, Werner & Wilming, 'Pure State Entanglement and von Neumann Algebras', CMP (2025) link
- [5]van Luijk et al., 'Embezzlement of entanglement, quantum fields, and the classification of von Neumann algebras' link
- [6]Longo & Xu, 'Relative Entropy in CFT', Advances in Mathematics 337 (2018) link
Recorded progress
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