OP-3027Partially solvedQuantum information

Irreversibility of mixed-state entanglement

Give a structural characterization of finite-dimensional bipartite states satisfying $E_C(\rho)=E_D(\rho)$ under ordinary asymptotic LOCC with vanishing error and no catalysts.

Source-reported statusAs of 2026-09-04Source & evidence ↓

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Problem statement

Give a structural characterization of finite-dimensional bipartite states satisfying $E_C(\rho)=E_D(\rho)$ under ordinary asymptotic LOCC with vanishing error and no catalysts.

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Background

Pure-state entanglement is reversible: entropy of entanglement governs both formation and distillation. Mixed-state irreversibility is established by explicit examples, including bound-entangled states with $E_D=0<E_C$ . It is not correct to treat every mixed state as irreversible: locally distinguishable flagged mixtures of suitable pure resources provide reversible mixed subclasses.

Vidal and Cirac exhibited irreversible states and clarified the gap $E_C-E_D$ . Changing the free operations changes the question: strict non-entangling, asymptotically non-entangling, PPT and LOCC conventions are not interchangeable. Results on entanglement “second laws”, including Lami–Regula's irreversibility theorem, must retain their operation and error hypotheses.

Preprint progress (August 2026): Ao, Philip and Streltsov claim full additivity and strong superadditivity of regularized PPT relative entropy, and an explicit state with an irreversible formation–distillation gap even with correlated catalysts under PPT operations. This does not supply the missing equality criterion for ordinary LOCC.

The target is the boundary between reversible and irreversible states under the fixed convention, not merely another example of a gap.

Source disclosures

Ao, Philip and Streltsov report ChatGPT 5.6 assistance with editing, organization, references and exploration/verification of mathematical arguments, while asserting independent author review and responsibility. Source

Progress & source assessment

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

preprint: Irreversibility persists for an explicit state under PPT operations with correlated catalysts; this is not an LOCC equality classification. Evidence

References

[1]Vidal & Cirac, 'Irreversibility in asymptotic manipulations of entanglement', PRL 86, 5803 (2001) link
[2]Lami & Regula, 'No second law of entanglement manipulation after all', Nat. Phys. 19 (2023) link
[3]Ao, Philip & Streltsov, 'PPT Entanglement with Correlated Catalysis: Monotones and Irreversibility' (2026 preprint) link

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