Effective membership in LOCC and its closure
For finitely specified bipartite channels or instruments, develop certified -membership tests for the closure of LOCC in a stated operational norm, and distinguish this task from exact finite-round or unbounded-round LOCC implementability.
Problem statement
For finitely specified bipartite channels or instruments, develop certified -membership tests for the closure of LOCC in a stated operational norm, and distinguish this task from exact finite-round or unbounded-round LOCC implementability.
Lean 4
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Background
Local operations and classical communication (LOCC) are the free operations of entanglement theory, but their geometry is subtle. LOCC is not closed, and separable operations can fail to be LOCC — the operational phenomenon of nonlocality without entanglement.
What is known: Chitambar–Leung–Mančinska–Ozols–Winter carefully distinguish bounded-round protocols, unbounded-round LOCC, its closure and separable operations. These distinctions must be fixed before membership is a well-defined task. For channels one can use diamond-norm approximation; for instruments retain the classical outcome register. For pure bipartite state conversion, Nielsen's majorization criterion already gives an exact single-copy solution.
Preprint progress (August 2026): Zhao and Chen encode complete multiqubit orthogonal product bases as edge-coloured multigraphs and claim that a complete colour-splitting tree exactly characterizes perfect finite-round LOCC discrimination. Their algorithm constructs a protocol in this structured family; it is not a membership algorithm for arbitrary LOCC maps.
Useful residual questions ask for certified approximation with a specified promise gap, structured instrument classes, and communication-round complexity. Exact membership and membership in the closure must not be interchanged.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
Recorded progress
Add yours →Nothing recorded yet. A pointer to a paper the entry is missing counts.
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Discussion
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