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Entanglement

What is settled, what is only partly settled, and where the questions still stand open.

Deep map: Classifying bipartite entanglement

43 claims and questions · 8 on the frontier · checked 2026-10-30

EntanglementEntanglementDetectionDetectionMeasuresMeasuresConversionConversionMultipartite entanglementMultipartiteEvery separable state has a positive partial transpose (Established)Separable states are PPTIn 2×2 and 2×3 systems, PPT is equivalent to separability (Established)PPT decides 2×2 and 2×3PPT entangled states exist in 3×3 and 2×4 (Established)PPT entangled states existEntanglement can be certified by witness operators (Established)Witnesses detect entanglementThe cross norm and realignment criterion (Established)Realignment criterionA hierarchy of semidefinite tests detects every entangled state (Established)A complete test hierarchyDeciding separability is NP-hard (Established)Separability is NP-hardIs the composition of two PPT maps always entanglement breaking? (Conjectured)PPT² conjectureHow powerful are two unentangled quantum proofs? (Open)How powerful is QMA(2)?A measure must not increase under local operations (Established)Measures must be monotonesWhen a measure must equal the reduced entropy on pure states (Established)Pure states: entropy is uniqueEntanglement of formation has a closed form for two qubits (Established)Two-qubit formation formulaEntanglement of formation is additive (Refuted)Formation is additiveEntanglement cost equals regularised entanglement of formation (Established)Cost is regularised formationNegativity and log-negativity can be computed for any state (Established)Negativity is computableRelative entropy of entanglement is additive (Refuted)Relative entropy is additiveSquashed entanglement is additive and faithful (Established)Squashed entanglement is faithfulCan distillable entanglement be computed in general? (Open)Formula for distillable entanglement?Partly entangled pure states concentrate into singlets (Established)Pure states concentrateMajorization decides single-copy pure state conversion (Established)Majorization decides pure LOCCEntangled catalysts enable otherwise impossible conversions (Established)Catalysts enable conversionsWith catalysts, entanglement entropy decides pure state conversion (Established)Entropy rules catalysed conversionNoisy entanglement can be purified by LOCC (Established)Noisy pairs can be distilledEvery entangled two-qubit state is distillable (Established)All two-qubit entanglement distillsBound entanglement exists (Established)PPT entanglement can't be distilledBound entanglement can improve teleportation (Established)Bound entanglement can be activatedDo bound entangled states with a negative partial transpose exist? (Open)Do NPT bound states exist?NPT Werner states are two-copy undistillable in every dimension (Claimed)Two copies of NPT Werner failBell nonlocality implies distillability (the Peres conjecture) (Refuted)Bell nonlocality implies distillableEntanglement manipulation under LOCC is irreversible (Established)LOCC is irreversibleEntanglement is reversible under all non-entangling operations (Refuted)Non-entangling ops are reversibleThe generalised quantum Stein's lemma, and reversibility under asymptotically non-entangling operations (Established)Generalised Stein's lemmaCatalysis cannot distill PPT bound entanglement (Established)Catalysts can't free PPT boundThree qubits can be entangled in two inequivalent ways (Established)Three qubits: GHZ and WFour qubits fall into nine families (Established)Four qubits: nine familiesEntanglement between qubits is monogamous (Established)Qubit entanglement is monogamousNo single maximally entangled multipartite state exists (Established)Maximally entangled setsAlmost all multipartite pure states are isolated under LOCC (Established)Almost all states are isolatedTwo bound entangled states can together be distillable (Established)Bound states can superactivateStandard genuine multipartite entanglement can be built from pairs (Established)Network entanglementFor which parties and dimensions do AME states exist? (Partial)Which AME states exist?Seven-party AME states exist exactly when d ≥ 3 (Claimed)AME(7,d) exists for all d ≥ 3When can many copies of one tensor produce another? (Partial)Asymptotic tensor restriction

Statuses were checked against each source’s abstract or introduction, not by re-deriving the proofs. The map is a growing selection, so a missing topic means it has not been mapped yet, not that nothing is known about it.