Formalize Open Quantum Problem #20: Reversible entanglement manipulation

### What is the conjecture

This is problem [#20](https://oqp.iqoqi.oeaw.ac.at/reversible-entanglement-manipulation) in [Reinhard F. Werner’s collection](https://arxiv.org/abs/quant-ph/0504166), later [collected by a community of quantum researchers](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems).

> **Problem (Open Quantum Problem #20: “Reversible entanglement manipulation”).**
> Are PPT operations sufficient to ensure asymptotically reversible interconversion of all, i.e. pure and mixed, bipartite entangled states?
> What is the smallest non-trivial class of operations that permits asymptotically reversible interconversion of all, i.e. pure and mixed, bipartite entangled states?

A standard formalization is in terms of **asymptotic conversion rates** under a chosen class of allowed operations:

* Fix finite-dimensional bipartite Hilbert spaces `H_A ⊗ H_B` and `H_{A'} ⊗ H_{B'}`, and let `ρ_AB` and `σ_A'B'` be bipartite density operators on them.
* For a class `C` of allowed quantum channels (e.g. `LOCC`, `PPT`, ...), define the asymptotic conversion rate
`r_C(ρ -> σ) := sup { r ≥ 0 : lim_{n→∞} inf_{Λ_n ∈ C} || Λ_n(ρ^{⊗ n}) - σ^{⊗ floor(rn)} ||_1 = 0 }`.
* Taking a two-qubit maximally entangled state `Φ^+` as the unit resource, define the **distillable entanglement** and **entanglement cost**
`E_{d,C}(ρ) := r_C(ρ -> Φ^+)`,
`E_{c,C}(ρ) := r_C(Φ^+ -> ρ)^{-1}`.

Then `C` yields an **asymptotically reversible** bipartite entanglement theory iff for all bipartite states `ρ` and `σ`,
`r_C(ρ -> σ) = r_C(σ -> ρ)^{-1}`,
equivalently iff
`E_{d,C}(ρ) = E_{c,C}(ρ)`
for every bipartite state `ρ`.

The benchmark case is `LOCC`:

* For bipartite **pure** states, asymptotic interconversion under `LOCC` is reversible, governed by the entropy of entanglement (equivalently the relative entropy of entanglement on pure states).
* For general **mixed** states, reversibility under `LOCC` fails in general: one can have
`E_{d,LOCC}(ρ) < E_{c,LOCC}(ρ)`.

The OQP problem asks whether reversibility can be restored by enlarging the free operation class. A central candidate is the class of **PPT operations** (often formalized as completely PPT-preserving channels):

* `Λ` is **PPT-preserving** if `(id ⊗ Λ)(τ)` is PPT whenever `τ` is PPT.
* **Original first question:** does
`E_{d,PPT}(ρ) = E_{c,PPT}(ρ)`
hold for every bipartite state `ρ`?
* **More general question:** what is the **smallest non-trivial** class `C` for which all bipartite entangled states become asymptotically reversibly interconvertible?

The OQP page also highlights important partial results / refinements:

* `PPT` does **not** suffice in full generality: irreversibility persists even under PPT operations in the bipartite setting.
* More strongly, irreversibility also persists under **non-entangling** (`NE`) operations, i.e. channels `Λ` such that `Λ(σ)` is separable whenever `σ` is separable.
* A natural candidate resolution is the class of **asymptotically non-entangling** operations. In the Brandão–Plenio framework, one measures the entanglement generated on separable inputs using the **global robustness**
`R_g(ρ) := inf { λ ≥ 0 : (ρ + λ ω)/(1+λ) is separable for some state ω }`,
and allows sequences `Λ_n` with
`R_g(Λ_n(σ_n)) ≤ δ_n`
for all separable `σ_n`, where `δ_n -> 0`.

A precise candidate theorem in that framework is:

* **Brandão–Plenio candidate second-law statement:** under asymptotically non-entangling operations,
`E_{d,ANE}(ρ) = E_{c,ANE}(ρ) = E_R^∞(ρ)`
for every bipartite state `ρ`, where `E_R^∞` is the regularized relative entropy of entanglement.

Historically, this candidate solution was claimed in the 2008/2010 Brandão–Plenio program, but later a gap was identified in the generalized quantum Stein’s lemma argument. So, in the original OQP framing, the remaining open problem is to determine whether the asymptotically non-entangling resolution is correct, or else to identify the true minimal reversible class.

### Where to find the details / references

Primary sources:

* [Open Quantum Problems site (Problem #20)](https://oqp.iqoqi.oeaw.ac.at/reversible-entanglement-manipulation)
* [Open Quantum Problems master list (for numbering/metadata)](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems)
* O. Krüger & R. F. Werner, “Some Open Problems in Quantum Information Theory” (Problem 20):
[https://arxiv.org/abs/quant-ph/0504166](https://arxiv.org/abs/quant-ph/0504166)

Key related references (as listed on or directly relevant to the OQP page):

* C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations” (Phys. Rev. A 53, 2046 (1996); arXiv: quant-ph/9511030)
(pure-state asymptotic reversibility under `LOCC`)
* G. Vidal and J. I. Cirac, “Irreversibility in asymptotic manipulations of entanglement” (Phys. Rev. Lett. 86, 5803 (2001); arXiv: quant-ph/0102036)
and M. Horodecki, A. Sen De, and U. Sen, “Rates of asymptotic entanglement transformations for bipartite mixed states: maximally entangled states are not special” (Phys. Rev. A 67, 062314 (2003); arXiv: quant-ph/0207031)
(mixed-state `LOCC` irreversibility)
* E. M. Rains, “A semidefinite program for distillable entanglement” (IEEE Trans. Inf. Theory 47, 2921 (2001); arXiv: quant-ph/0008047)
(standard reference for PPT-preserving operations / Rains-type bounds)
* K. Audenaert, M. B. Plenio, and J. Eisert, “Entanglement Cost under Positive-Partial-Transpose-Preserving Operations” (Phys. Rev. Lett. 90, 027901 (2003))
(shows PPT operations can restore reversibility for certain states)
* S. Ishizaka and M. B. Plenio, “Multiparticle entanglement under asymptotic positive-partial-transpose-preserving operations” (Phys. Rev. A 72, 042325 (2005); arXiv: quant-ph/0503025)
(multipartite pure states are not all reversibly interconvertible under PPT)
* X. Wang and R. Duan, “Irreversibility of Asymptotic Entanglement Manipulation Under Quantum Operations Completely Preserving Positivity of Partial Transpose” (Phys. Rev. Lett. 119, 180506 (2017); arXiv: 1606.09421)
(answers the PPT part in the negative in the bipartite setting)
* L. Lami and B. Regula, “No second law of entanglement manipulation after all” (Nat. Phys. 19, 184–189 (2023); arXiv: 2111.02438)
(shows irreversibility under all non-entangling operations)
* F. G. S. L. Brandão and M. B. Plenio, “A Reversible Theory of Entanglement and its Relation to the Second Law” (Commun. Math. Phys. 295, 829–851 (2010); arXiv: 0710.5827)
(the asymptotically non-entangling candidate solution)
* M. Berta, F. G. S. L. Brandão, G. Gour, L. Lami, M. B. Plenio, B. Regula, and M. Tomamichel, “On a gap in the proof of the generalised quantum Stein’s lemma and its consequences for the reversibility of quantum resources” (Quantum 7, 1103 (2023); arXiv: 2205.02813)
(explains the gap that re-opened the Brandão–Plenio approach)

(If you want to include a brief “recent status” pointer section, there are also newer post-2023 papers on the generalized quantum Stein’s lemma / second-law program; but the OQP page above, plus the PPT and non-entangling no-go papers, are the core references for writing the issue.)

### Prerequisites needed

* Finite-dimensional quantum mechanics: density matrices, tensor products, bipartite states
* `LOCC` and basic entanglement manipulation tasks (distillation, dilution, conversion)
* Asymptotic state-conversion rates; tensor powers and trace-norm convergence
* Partial transpose, PPT criterion, PPT-preserving maps
* Entanglement measures such as distillable entanglement, entanglement cost, relative entropy of entanglement, robustness measures
* (Optional) Resource-theoretic / thermodynamic viewpoint on reversible state conversion

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-81 (Quantum theory)
* ams-94 (Information and communication theory)
* ams-15 (Linear and multilinear algebra; matrix theory)
* ams-47 (Operator theory)

### Choose either option

* [ ] I plan on adding this conjecture to the repository
* [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

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No target file or test is named. Start by reviewing existing formalized conjectures in the repository, then read the Open Quantum Problems page and the listed references to determine the precise statement to encode. Done means the reversible-entanglement-manipulation conjecture is added in the repository’s established Lean style, with any required metadata and no unformalized scope left ambiguous.

Written by the indexing model from the issue text.