Statement
Problem 5.1 (the sole problem of Theme 4) of Goto et al.: what are the appropriate mathematical toolsets and arguments for canonically extending quantum information theory to infinite-dimensional settings, such as those encountered in QFT? Finite-dimensional QI leans on density matrices, von Neumann entropy, and Schmidt decomposition — none of which survive intact on the type III₁ (and, after regularization, type II) von Neumann algebras that describe local regions in QFT and gravity.
Why It Matters
This is, almost verbatim, the algebraic frontier the group’s whole current program sits on — the single most on-target problem in the paper. Every one of the group’s active tools is a partial answer to it: Araki–Uhlmann relative entropy is finite precisely where von Neumann entropy diverges; embezzlement is native to type III₁; the crossed product restores a trace by promoting III₁ to type II. That the holography/QI community independently names this as the open problem of infinite-dimensionality is strong outside validation of the group’s bet.
What We Know
- Local algebras in QFT are hyperfinite type III₁; individual entropies are UV-divergent but differences (relative entropy, mutual information) are finite and well-defined (Araki, Uhlmann; the group’s 2025 papers compute them explicitly for coherent/squeezed/cat states).
- The crossed product regularizes III₁ to type II, restoring a (renormalized) trace and a finite generalized entropy (Witten 2022, CPW, CLPW).
- Embezzlement of entanglement is possible on type III₁ with vanishing standard-monotone cost (van Luijk–Stottmeister–Werner–Wilming 2024; the group’s embezzlement line).
- The Bell-CHSH / Tsirelson story in infinite dimensions carries genuine subtleties (commuting-operator vs. tensor models) — see the Tsirelson thread below.
Possible Approaches
- Frame the group’s existing finite-quantity results (relative entropy across dimensions, squeezed/cat states, embezzlement rate) as a coherent contribution to this program — see relative-entropy-embezzlement-as-infinite-dim-qi.
- Push one-shot / smooth-entropy and resource-theoretic tools (Tomamichel-style) onto type II crossed-product windows — the tooling already flagged in embezzlement-capacity-lower-bound.
- Isolate where the tensor-factorization failure genuinely changes an operational number — see tsirelson-problem-vacuum-bell-chsh.